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Solve improvements.
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@corywalker
corywalker committed Mar 17, 2018
1 parent 54618b5 commit 481b959
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Showing 6 changed files with 67 additions and 11 deletions.
42 changes: 41 additions & 1 deletion expreduce/builtin_power.go
View file
Expand Up @@ -57,6 +57,27 @@ func NthRoot(x *big.Int, n *big.Int) *big.Int {
}
}

func RadSimp(radicand *big.Int, index *big.Int) (*big.Int, *big.Int) {
//xPositivity := x.Cmp(big.NewInt(0))
//nPositivity := n.Cmp(big.NewInt(0))
i := big.NewInt(2)
pow := big.NewInt(0)
mod := big.NewInt(0)
div := big.NewInt(0)
for true {
pow.Exp(i, index, nil)
mod.Mod(radicand, pow)
cmpRes := mod.Cmp(big.NewInt(0))
if cmpRes == 0 {
div = div.Div(radicand, pow)
return i, div
} else if cmpRes > 0 {
break
}
}
return nil, nil
}

func GetPowerDefinitions() (defs []Definition) {
defs = append(defs, Definition{
Name: "Power",
Expand Down Expand Up @@ -171,8 +192,27 @@ func GetPowerDefinitions() (defs []Definition) {
}
return E(S("Power"), NewInteger(root), NewInteger(m))
}
return this
}
if nPositivity == 1 {
absX := big.NewInt(0)
absX.Abs(x)
extracted, radicand := RadSimp(absX, n)
if extracted != nil {
if xPositivity == -1 {
radicand.Neg(radicand)
}
return E(
S("Times"),
NewInteger(extracted),
E(
S("Power"),
NewInteger(radicand),
powerRat,
),
)
}
}
return this
}
return this
},
Expand Down
16 changes: 8 additions & 8 deletions expreduce/resources.go
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Large diffs are not rendered by default.

8 changes: 7 additions & 1 deletion expreduce/resources/arithmetic.m
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Expand Up @@ -112,10 +112,14 @@
};

Times::usage = "`(e1 * e2 * ...)` computes the product of all expressions in the function.";
Verbatim[Times][beg___, a_^Optional[m_], a_^Optional[n_], end___] := beg*a^(m+n)*end;
(* This is likely the most compute-intensive rule in the system. Modify with
care. *)
Verbatim[Times][beg___, a_^Optional[m_], a_^Optional[n_], end___] := beg*a^(m+n)*end /; (!NumberQ[a] || !NumberQ[m] || !NumberQ[n]);
(*Verbatim[Times][beg___, a_^Optional[m_], a_^Optional[n_], end___] := beg*a^(m+n)*end;*)
Verbatim[Times][Rational[1, a_Integer], inner___, a_Integer^n_, end___] := inner*a^(n-1)*end;
Times[den_Integer^-1, num_Integer, rest___] := Rational[num,den] * rest;
Times[ComplexInfinity, rest___] := ComplexInfinity;
a_Integer?Negative^b_Rational*c_Integer^d_Rational*rest___ := (-1)^b*rest /; (a == -c && b == -d);
Sin[x_]*Cos[x_]^(-1)*rest___ := Tan[x]*rest;
Cos[x_]*Sin[x_]^(-1)*rest___ := Cot[x]*rest;
Attributes[Times] = {Flat, Listable, NumericFunction, OneIdentity, Orderless, Protected};
Expand Down Expand Up @@ -182,6 +186,8 @@

(*Conversion of exact numeric functions to reals*)
ESameTest[True, MatchQ[Sqrt[2*Pi]*.1, _Real]],

ESameTest[(-1)^(1/3) a b c, (-2)^(1/3)*2^(-1/3)*a*b*c],
]
};

Expand Down
2 changes: 1 addition & 1 deletion expreduce/resources/equationdata.m
View file
@@ -1,3 +1,3 @@
solveQuadratic[a_, b_, c_, x_] := {{x->(-b-Sqrt[b^2-4 a c])/(2 a)},{x->(-b+Sqrt[b^2-4 a c])/(2 a)}};
solveCubic[d_,c_,b_,a_,x_] := {{x->-(c/(3 d))-(2^(1/3) (-c^2+3 b d))/(3 d (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3))+(-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3)/(3 2^(1/3) d)},{x->-(c/(3 d))+((1+I Sqrt[3]) (-c^2+3 b d))/(3 2^(2/3) d (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3))-(1/(6 2^(1/3) d))(1-I Sqrt[3]) (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3)},{x->-(c/(3 d))+((1-I Sqrt[3]) (-c^2+3 b d))/(3 2^(2/3) d (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3))-(1/(6 2^(1/3) d))(1+I Sqrt[3]) (-2 c^3+9 b c d-27 a d^2+Sqrt[4 (-c^2+3 b d)^3+(-2 c^3+9 b c d-27 a d^2)^2])^(1/3)}};
solveQuartic[e_.*x_^4 + d_.*x_^3 + c_.*x_^2 + b_.*x_ + a_., x_] := {{x->-(d/(4 e))-1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))-1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))-1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))+1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))+1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))-1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))+1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))+1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))}}/;FreeQ[{a,b,c,d,e},x];
solveQuartic[e_,d_,c_,b_,a_,x_] := {{x->-(d/(4 e))-1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))-1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))-1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))+1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))+1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))-1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))},{x->-(d/(4 e))+1/2 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3)))+1/2 \[Sqrt](d^2/(2 e^2)-(4 c)/(3 e)-(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))-(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+Sqrt[-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2])^(1/3))+(-(d^3/e^3)+(4 c d)/e^2-(8 b)/e)/(4 \[Sqrt](d^2/(4 e^2)-(2 c)/(3 e)+(2^(1/3) (c^2-3 b d+12 a e))/(3 e (2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3))+(1/(3 2^(1/3) e))((2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e+\[Sqrt](-4 (c^2-3 b d+12 a e)^3+(2 c^3-9 b c d+27 a d^2+27 b^2 e-72 a c e)^2))^(1/3)))))}}/;FreeQ[{a,b,c,d,e},x];
4 changes: 4 additions & 0 deletions expreduce/resources/power.m
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Expand Up @@ -187,6 +187,10 @@
ESameTest[Power, 9^(-1/3)//Head],
ESameTest[27, 9^(3/2)],
ESameTest[1/27, 9^(-3/2)],

(* Test simplifying radicals *)
ESameTest[4 2^(1/3), (128)^(1/3)],
ESameTest[4 (-2)^(1/3), (-128)^(1/3)],
], EKnownFailures[
ESameTest[(3+I Sqrt[29]) E^(-((2 I \[Pi])/3)), ((3 + I*Sqrt[29])^3)^(1/3)],
]
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6 changes: 6 additions & 0 deletions expreduce/resources/solve.m
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Expand Up @@ -175,6 +175,12 @@
solveCubic[d,c,b,a,x]/;FreeQ[{a,b,c,d},x];
solveCubic[d_.*x_^3 + c_.*x_^2 + a_., x_] :=
solveCubic[d,c,0,a,x]/;FreeQ[{a,c,d},x];
solveCubic[d_.*x_^3 + b_.*x_ + a_., x_] :=
solveCubic[d,0,b,a,x]/;FreeQ[{a,b,d},x];
solveQuartic[e_.*x_^4 + d_.*x_^3 + c_.*x_^2 + b_.*x_ + a_., x_] :=
solveQuartic[e,d,c,b,a,x]/;FreeQ[{a,b,c,d,e},x];
solveQuartic[e_.*x_^4 + d_.*x_^3 + a_., x_] :=
solveQuartic[e,d,0,0,a,x]/;FreeQ[{a,d,e},x];

(* Solve using u-substitution for polynomial-like forms.*)
uSubstitute::usage =
Expand Down

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