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

Optimal. Leaf size=63 \[ -4 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^6-x}}\right )-4 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^6-x}}\right )+\frac {4 \left (x^6-x\right )^{3/4} \left (3 x^5+14 x^3-3\right )}{21 x^6} \]

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Rubi [F]  time = 2.32, 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^5\right ) \left (-1+x^3+x^5\right ) \left (3+2 x^5\right )}{x^6 \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^5)*(-1 + x^3 + x^5)*(3 + 2*x^5))/(x^6*(-1 - x^3 + x^5)*(-x + x^6)^(1/4)),x]

[Out]

(4*(1 - x^5)^(1/4)*Hypergeometric2F1[-21/20, -3/4, -1/20, x^5])/(7*x^5*(-x + x^6)^(1/4)) - (8*(1 - x^5)^(1/4)*
Hypergeometric2F1[-3/4, -9/20, 11/20, x^5])/(3*x^2*(-x + x^6)^(1/4)) + (8*(1 - x^5)^(1/4)*Hypergeometric2F1[-3
/4, -1/20, 19/20, x^5])/(-x + x^6)^(1/4) - (24*x^(1/4)*(-1 + x^5)^(1/4)*Defer[Subst][Defer[Int][(x^2*(-1 + x^2
0)^(3/4))/(-1 - x^12 + x^20), x], x, x^(1/4)])/(-x + x^6)^(1/4) + (40*x^(1/4)*(-1 + x^5)^(1/4)*Defer[Subst][De
fer[Int][(x^10*(-1 + x^20)^(3/4))/(-1 - x^12 + x^20), x], x, x^(1/4)])/(-x + x^6)^(1/4)

Rubi steps

\begin {align*} \int \frac {\left (-1+x^5\right ) \left (-1+x^3+x^5\right ) \left (3+2 x^5\right )}{x^6 \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^5\right )^{3/4} \left (-1+x^3+x^5\right ) \left (3+2 x^5\right )}{x^{25/4} \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^{20}\right )^{3/4} \left (-1+x^{12}+x^{20}\right ) \left (3+2 x^{20}\right )}{x^{22} \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 \left (-1+x^{20}\right )^{3/4}}{x^{22}}-\frac {6 \left (-1+x^{20}\right )^{3/4}}{x^{10}}+\frac {2 \left (-1+x^{20}\right )^{3/4}}{x^2}+\frac {2 x^2 \left (-3+5 x^8\right ) \left (-1+x^{20}\right )^{3/4}}{-1-x^{12}+x^{20}}\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 {\left (-1+x^{20}\right )^{3/4}}{x^2} \, 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 (-3+5 x^8\right ) \left (-1+x^{20}\right )^{3/4}}{-1-x^{12}+x^{20}} \, 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 {\left (-1+x^{20}\right )^{3/4}}{x^{22}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^6}}-\frac {\left (24 \sqrt [4]{x} \sqrt [4]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^{20}\right )^{3/4}}{x^{10}} \, 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 \left (-\frac {3 x^2 \left (-1+x^{20}\right )^{3/4}}{-1-x^{12}+x^{20}}+\frac {5 x^{10} \left (-1+x^{20}\right )^{3/4}}{-1-x^{12}+x^{20}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^6}}+\frac {\left (8 \sqrt [4]{x} \left (-1+x^5\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{20}\right )^{3/4}}{x^2} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^5\right )^{3/4} \sqrt [4]{-x+x^6}}+\frac {\left (12 \sqrt [4]{x} \left (-1+x^5\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{20}\right )^{3/4}}{x^{22}} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^5\right )^{3/4} \sqrt [4]{-x+x^6}}-\frac {\left (24 \sqrt [4]{x} \left (-1+x^5\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{20}\right )^{3/4}}{x^{10}} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^5\right )^{3/4} \sqrt [4]{-x+x^6}}\\ &=\frac {4 \sqrt [4]{1-x^5} \, _2F_1\left (-\frac {21}{20},-\frac {3}{4};-\frac {1}{20};x^5\right )}{7 x^5 \sqrt [4]{-x+x^6}}-\frac {8 \sqrt [4]{1-x^5} \, _2F_1\left (-\frac {3}{4},-\frac {9}{20};\frac {11}{20};x^5\right )}{3 x^2 \sqrt [4]{-x+x^6}}+\frac {8 \sqrt [4]{1-x^5} \, _2F_1\left (-\frac {3}{4},-\frac {1}{20};\frac {19}{20};x^5\right )}{\sqrt [4]{-x+x^6}}-\frac {\left (24 \sqrt [4]{x} \sqrt [4]{-1+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^{20}\right )^{3/4}}{-1-x^{12}+x^{20}} \, 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^{10} \left (-1+x^{20}\right )^{3/4}}{-1-x^{12}+x^{20}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^6}}\\ \end {align*}

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Mathematica [F]  time = 0.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^5\right ) \left (-1+x^3+x^5\right ) \left (3+2 x^5\right )}{x^6 \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^5)*(-1 + x^3 + x^5)*(3 + 2*x^5))/(x^6*(-1 - x^3 + x^5)*(-x + x^6)^(1/4)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 2.74, size = 63, normalized size = 1.00 \begin {gather*} \frac {4 \left (-3+14 x^3+3 x^5\right ) \left (-x+x^6\right )^{3/4}}{21 x^6}-4 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^6}}\right )-4 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-3 + 14*x^3 + 3*x^5)*(-x + x^6)^(3/4))/(21*x^6) - 4*ArcTan[x/(-x + x^6)^(1/4)] - 4*ArcTanh[x/(-x + x^6)^(1
/4)]

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fricas [B]  time = 79.30, size = 137, normalized size = 2.17 \begin {gather*} -\frac {2 \, {\left (21 \, x^{6} \arctan \left (\frac {2 \, {\left ({\left (x^{6} - x\right )}^{\frac {1}{4}} x^{2} + {\left (x^{6} - x\right )}^{\frac {3}{4}}\right )}}{x^{5} - x^{3} - 1}\right ) - 21 \, x^{6} \log \left (-\frac {x^{5} + x^{3} - 2 \, {\left (x^{6} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{6} - x} x - 2 \, {\left (x^{6} - x\right )}^{\frac {3}{4}} - 1}{x^{5} - x^{3} - 1}\right ) - 2 \, {\left (x^{6} - x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} + 14 \, x^{3} - 3\right )}\right )}}{21 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/21*(21*x^6*arctan(2*((x^6 - x)^(1/4)*x^2 + (x^6 - x)^(3/4))/(x^5 - x^3 - 1)) - 21*x^6*log(-(x^5 + x^3 - 2*(
x^6 - x)^(1/4)*x^2 + 2*sqrt(x^6 - x)*x - 2*(x^6 - x)^(3/4) - 1)/(x^5 - x^3 - 1)) - 2*(x^6 - x)^(3/4)*(3*x^5 +
14*x^3 - 3))/x^6

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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^{5} - 1\right )}}{{\left (x^{6} - x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [C]  time = 16.77, size = 185, normalized size = 2.94

method	result	size
trager	\(\frac {4 \left (3 x^{5}+14 x^{3}-3\right ) \left (x^{6}-x \right )^{\frac {3}{4}}}{21 x^{6}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}-x}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}-x \right )^{\frac {3}{4}}-2 \left (x^{6}-x \right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}-1}\right )-2 \ln \left (-\frac {x^{5}+2 \left (x^{6}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{6}-x}+2 \left (x^{6}-x \right )^{\frac {1}{4}} x^{2}+x^{3}-1}{x^{5}-x^{3}-1}\right )\)	\(185\)
risch	\(\frac {\frac {4}{7} x^{10}+\frac {8}{3} x^{8}-\frac {8}{7} x^{5}-\frac {8}{3} x^{3}+\frac {4}{7}}{x^{5} \left (x \left (x^{5}-1\right )\right )^{\frac {1}{4}}}+2 \ln \left (-\frac {-x^{5}+2 \left (x^{6}-x \right )^{\frac {3}{4}}-2 x \sqrt {x^{6}-x}+2 \left (x^{6}-x \right )^{\frac {1}{4}} x^{2}-x^{3}+1}{x^{5}-x^{3}-1}\right )+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{6}-x}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{6}-x \right )^{\frac {3}{4}}-2 \left (x^{6}-x \right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{5}-x^{3}-1}\right )\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/21*(3*x^5+14*x^3-3)*(x^6-x)^(3/4)/x^6+2*RootOf(_Z^2+1)*ln(-(-RootOf(_Z^2+1)*x^5+2*RootOf(_Z^2+1)*(x^6-x)^(1/
2)*x-RootOf(_Z^2+1)*x^3+2*(x^6-x)^(3/4)-2*(x^6-x)^(1/4)*x^2+RootOf(_Z^2+1))/(x^5-x^3-1))-2*ln(-(x^5+2*(x^6-x)^
(3/4)+2*x*(x^6-x)^(1/2)+2*(x^6-x)^(1/4)*x^2+x^3-1)/(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^{5} - 1\right )}}{{\left (x^{6} - x\right )}^{\frac {1}{4}} {\left (x^{5} - x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {\left (x^5-1\right )\,\left (2\,x^5+3\right )\,\left (x^5+x^3-1\right )}{x^6\,{\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(-((x^5 - 1)*(2*x^5 + 3)*(x^3 + x^5 - 1))/(x^6*(x^6 - x)^(1/4)*(x^3 - x^5 + 1)),x)

[Out]

int(-((x^5 - 1)*(2*x^5 + 3)*(x^3 + x^5 - 1))/(x^6*(x^6 - x)^(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 (x - 1\right ) \left (2 x^{5} + 3\right ) \left (x^{5} + x^{3} - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )}{x^{6} \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**5-1)*(x**5+x**3-1)*(2*x**5+3)/x**6/(x**5-x**3-1)/(x**6-x)**(1/4),x)

[Out]

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

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