3.15.91 \(\int \frac {(1-x^3+x^5) (-3+2 x^5)}{x^3 (1+x^3+x^5) \sqrt [4]{x+x^6}} \, dx\)

Optimal. Leaf size=104 \[ \frac {4 \left (x^6+x\right )^{3/4}}{3 x^3}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^6+x}}{\sqrt {x^6+x}-x^2}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\frac {\sqrt {x^6+x}}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}}{x \sqrt [4]{x^6+x}}\right ) \]

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Rubi [F]  time = 2.43, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^3*(1 + x^3 + x^5)*(x + x^6)^(1/4)),x]

[Out]

(4*(1 + x^5)^(1/4)*Hypergeometric2F1[-9/20, 1/4, 11/20, -x^5])/(3*x^2*(x + x^6)^(1/4)) - (16*x*(1 + x^5)^(1/4)
*Hypergeometric2F1[3/20, 1/4, 23/20, -x^5])/(3*(x + x^6)^(1/4)) + (8*x^3*(1 + x^5)^(1/4)*Hypergeometric2F1[1/4
, 11/20, 31/20, -x^5])/(11*(x + x^6)^(1/4)) + (40*x^(1/4)*(1 + x^5)^(1/4)*Defer[Subst][Defer[Int][x^2/((1 + x^
20)^(1/4)*(1 + x^12 + x^20)), x], x, x^(1/4)])/(x + x^6)^(1/4) + (16*x^(1/4)*(1 + x^5)^(1/4)*Defer[Subst][Defe
r[Int][x^14/((1 + x^20)^(1/4)*(1 + x^12 + x^20)), x], x, x^(1/4)])/(x + x^6)^(1/4)

Rubi steps

\begin {align*} \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^{13/4} \sqrt [4]{1+x^5} \left (1+x^3+x^5\right )} \, dx}{\sqrt [4]{x+x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{12}+x^{20}\right ) \left (-3+2 x^{20}\right )}{x^{10} \sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \left (-\frac {3}{x^{10} \sqrt [4]{1+x^{20}}}-\frac {4 x^2}{\sqrt [4]{1+x^{20}}}+\frac {2 x^{10}}{\sqrt [4]{1+x^{20}}}+\frac {2 x^2 \left (5+2 x^{12}\right )}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (5+2 x^{12}\right )}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}-\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{20}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \sqrt [4]{1+x^5} \, _2F_1\left (-\frac {9}{20},\frac {1}{4};\frac {11}{20};-x^5\right )}{3 x^2 \sqrt [4]{x+x^6}}-\frac {16 x \sqrt [4]{1+x^5} \, _2F_1\left (\frac {3}{20},\frac {1}{4};\frac {23}{20};-x^5\right )}{3 \sqrt [4]{x+x^6}}+\frac {8 x^3 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {11}{20};\frac {31}{20};-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \left (\frac {5 x^2}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )}+\frac {2 x^{14}}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \sqrt [4]{1+x^5} \, _2F_1\left (-\frac {9}{20},\frac {1}{4};\frac {11}{20};-x^5\right )}{3 x^2 \sqrt [4]{x+x^6}}-\frac {16 x \sqrt [4]{1+x^5} \, _2F_1\left (\frac {3}{20},\frac {1}{4};\frac {23}{20};-x^5\right )}{3 \sqrt [4]{x+x^6}}+\frac {8 x^3 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {11}{20};\frac {31}{20};-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}+\frac {\left (40 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{1+x^{20}} \left (1+x^{12}+x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.62, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^3 \left (1+x^3+x^5\right ) \sqrt [4]{x+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^3*(1 + x^3 + x^5)*(x + x^6)^(1/4)),x]

[Out]

Integrate[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^3*(1 + x^3 + x^5)*(x + x^6)^(1/4)), x]

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IntegrateAlgebraic [A]  time = 2.81, size = 104, normalized size = 1.00 \begin {gather*} \frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x+x^6}}{-x^2+\sqrt {x+x^6}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x+x^6}}{\sqrt {2}}}{x \sqrt [4]{x+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 - x^3 + x^5)*(-3 + 2*x^5))/(x^3*(1 + x^3 + x^5)*(x + x^6)^(1/4)),x]

[Out]

(4*(x + x^6)^(3/4))/(3*x^3) + 2*Sqrt[2]*ArcTan[(Sqrt[2]*x*(x + x^6)^(1/4))/(-x^2 + Sqrt[x + x^6])] + 2*Sqrt[2]
*ArcTanh[(x^2/Sqrt[2] + Sqrt[x + x^6]/Sqrt[2])/(x*(x + x^6)^(1/4))]

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fricas [B]  time = 156.51, size = 682, normalized size = 6.56 \begin {gather*} \frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{10} + 2 \, x^{8} + x^{6} + 2 \, x^{5} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (x^{5} - 3 \, x^{3} + 1\right )} + 2 \, \sqrt {2} {\left (3 \, x^{7} - x^{5} + 3 \, x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{4} + x\right )} \sqrt {x^{6} + x} + {\left (16 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} x^{3} + 2 \, \sqrt {2} {\left (x^{6} - 3 \, x^{4} + x\right )} \sqrt {x^{6} + x} + \sqrt {2} {\left (x^{10} - 8 \, x^{8} - x^{6} + 2 \, x^{5} - 8 \, x^{3} + 1\right )} + 4 \, {\left (x^{7} + x^{5} + x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} + x^{3} + 1}} + 1}{x^{10} - 14 \, x^{8} + x^{6} + 2 \, x^{5} - 14 \, x^{3} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{10} + 2 \, x^{8} + x^{6} + 2 \, x^{5} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (x^{5} - 3 \, x^{3} + 1\right )} - 2 \, \sqrt {2} {\left (3 \, x^{7} - x^{5} + 3 \, x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{4} + x\right )} \sqrt {x^{6} + x} + {\left (16 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} x^{3} - 2 \, \sqrt {2} {\left (x^{6} - 3 \, x^{4} + x\right )} \sqrt {x^{6} + x} - \sqrt {2} {\left (x^{10} - 8 \, x^{8} - x^{6} + 2 \, x^{5} - 8 \, x^{3} + 1\right )} + 4 \, {\left (x^{7} + x^{5} + x^{2}\right )} {\left (x^{6} + x\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1}{x^{5} + x^{3} + 1}} + 1}{x^{10} - 14 \, x^{8} + x^{6} + 2 \, x^{5} - 14 \, x^{3} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{5} + x^{3} + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x + 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1\right )}}{x^{5} + x^{3} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{5} + x^{3} - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {x^{6} + x} x - 2 \, \sqrt {2} {\left (x^{6} + x\right )}^{\frac {3}{4}} + 1\right )}}{x^{5} + x^{3} + 1}\right ) + 8 \, {\left (x^{6} + x\right )}^{\frac {3}{4}}}{6 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-x^3+1)*(2*x^5-3)/x^3/(x^5+x^3+1)/(x^6+x)^(1/4),x, algorithm="fricas")

[Out]

1/6*(12*sqrt(2)*x^3*arctan((x^10 + 2*x^8 + x^6 + 2*x^5 + 2*x^3 + 2*sqrt(2)*(x^6 + x)^(3/4)*(x^5 - 3*x^3 + 1) +
2*sqrt(2)*(3*x^7 - x^5 + 3*x^2)*(x^6 + x)^(1/4) + 4*(x^6 + x^4 + x)*sqrt(x^6 + x) + (16*(x^6 + x)^(3/4)*x^3 +
2*sqrt(2)*(x^6 - 3*x^4 + x)*sqrt(x^6 + x) + sqrt(2)*(x^10 - 8*x^8 - x^6 + 2*x^5 - 8*x^3 + 1) + 4*(x^7 + x^5 +
x^2)*(x^6 + x)^(1/4))*sqrt((x^5 + x^3 + 2*sqrt(2)*(x^6 + x)^(1/4)*x^2 + 4*sqrt(x^6 + x)*x + 2*sqrt(2)*(x^6 +
x)^(3/4) + 1)/(x^5 + x^3 + 1)) + 1)/(x^10 - 14*x^8 + x^6 + 2*x^5 - 14*x^3 + 1)) - 12*sqrt(2)*x^3*arctan((x^10
+ 2*x^8 + x^6 + 2*x^5 + 2*x^3 - 2*sqrt(2)*(x^6 + x)^(3/4)*(x^5 - 3*x^3 + 1) - 2*sqrt(2)*(3*x^7 - x^5 + 3*x^2)*
(x^6 + x)^(1/4) + 4*(x^6 + x^4 + x)*sqrt(x^6 + x) + (16*(x^6 + x)^(3/4)*x^3 - 2*sqrt(2)*(x^6 - 3*x^4 + x)*sqrt
(x^6 + x) - sqrt(2)*(x^10 - 8*x^8 - x^6 + 2*x^5 - 8*x^3 + 1) + 4*(x^7 + x^5 + x^2)*(x^6 + x)^(1/4))*sqrt((x^5
+ x^3 - 2*sqrt(2)*(x^6 + x)^(1/4)*x^2 + 4*sqrt(x^6 + x)*x - 2*sqrt(2)*(x^6 + x)^(3/4) + 1)/(x^5 + x^3 + 1)) +
1)/(x^10 - 14*x^8 + x^6 + 2*x^5 - 14*x^3 + 1)) + 3*sqrt(2)*x^3*log(4*(x^5 + x^3 + 2*sqrt(2)*(x^6 + x)^(1/4)*x^
2 + 4*sqrt(x^6 + x)*x + 2*sqrt(2)*(x^6 + x)^(3/4) + 1)/(x^5 + x^3 + 1)) - 3*sqrt(2)*x^3*log(4*(x^5 + x^3 - 2*s
qrt(2)*(x^6 + x)^(1/4)*x^2 + 4*sqrt(x^6 + x)*x - 2*sqrt(2)*(x^6 + x)^(3/4) + 1)/(x^5 + x^3 + 1)) + 8*(x^6 + x)
^(3/4))/x^3

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} + x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-x^3+1)*(2*x^5-3)/x^3/(x^5+x^3+1)/(x^6+x)^(1/4),x, algorithm="giac")

[Out]

integrate((2*x^5 - 3)*(x^5 - x^3 + 1)/((x^6 + x)^(1/4)*(x^5 + x^3 + 1)*x^3), x)

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maple [C]  time = 18.15, size = 213, normalized size = 2.05

method	result	size
trager	\(\frac {4 \left (x^{6}+x \right )^{\frac {3}{4}}}{3 x^{3}}-2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \sqrt {x^{6}+x}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{5}+x^{3}+1}\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-2 \sqrt {x^{6}+x}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (x^{6}+x \right )^{\frac {3}{4}}}{x^{5}+x^{3}+1}\right )\)	\(213\)
risch	\(\frac {\frac {4 x^{5}}{3}+\frac {4}{3}}{x^{2} \left (x \left (x^{5}+1\right )\right )^{\frac {1}{4}}}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}-2 \sqrt {x^{6}+x}\, \RootOf \left (\textit {\_Z}^{4}+1\right ) x +\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}-2 \left (x^{6}+x \right )^{\frac {3}{4}}}{x^{5}+x^{3}+1}\right )+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \sqrt {x^{6}+x}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x -\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{6}+x \right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{6}+x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{5}+x^{3}+1}\right )\)	\(221\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5-x^3+1)*(2*x^5-3)/x^3/(x^5+x^3+1)/(x^6+x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

4/3*(x^6+x)^(3/4)/x^3-2*RootOf(_Z^4+1)*ln((2*(x^6+x)^(1/2)*RootOf(_Z^4+1)^3*x-RootOf(_Z^4+1)*x^5-2*RootOf(_Z^4
+1)^2*(x^6+x)^(1/4)*x^2+RootOf(_Z^4+1)*x^3+2*(x^6+x)^(3/4)-RootOf(_Z^4+1))/(x^5+x^3+1))+2*RootOf(_Z^4+1)^3*ln(
-(RootOf(_Z^4+1)^3*x^5-RootOf(_Z^4+1)^3*x^3+2*RootOf(_Z^4+1)^2*(x^6+x)^(1/4)*x^2-2*(x^6+x)^(1/2)*RootOf(_Z^4+1
)*x+RootOf(_Z^4+1)^3+2*(x^6+x)^(3/4))/(x^5+x^3+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + x\right )}^{\frac {1}{4}} {\left (x^{5} + x^{3} + 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5-x^3+1)*(2*x^5-3)/x^3/(x^5+x^3+1)/(x^6+x)^(1/4),x, algorithm="maxima")

[Out]

integrate((2*x^5 - 3)*(x^5 - x^3 + 1)/((x^6 + x)^(1/4)*(x^5 + x^3 + 1)*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^5-3\right )\,\left (x^5-x^3+1\right )}{x^3\,{\left (x^6+x\right )}^{1/4}\,\left (x^5+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^5 - 3)*(x^5 - x^3 + 1))/(x^3*(x + x^6)^(1/4)*(x^3 + x^5 + 1)),x)

[Out]

int(((2*x^5 - 3)*(x^5 - x^3 + 1))/(x^3*(x + x^6)^(1/4)*(x^3 + x^5 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (2 x^{5} - 3\right ) \left (x^{5} - x^{3} + 1\right )}{x^{3} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (x^{5} + x^{3} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5-x**3+1)*(2*x**5-3)/x**3/(x**5+x**3+1)/(x**6+x)**(1/4),x)

[Out]

Integral((2*x**5 - 3)*(x**5 - x**3 + 1)/(x**3*(x*(x + 1)*(x**4 - x**3 + x**2 - x + 1))**(1/4)*(x**5 + x**3 + 1
)), x)

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