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

Optimal. Leaf size=95 \[ -2\ 2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \left (x^5-x\right )^{3/4}}{x^4-1}\right )-2\ 2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \left (x^5-x\right )^{3/4}}{x^4-1}\right )+\frac {4 \left (x^5-x\right )^{3/4} \left (3 x^4+7 x^3-3\right )}{21 x^6} \]

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Rubi [F]  time = 2.45, 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^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(4*(1 - x^4)^(1/4)*Hypergeometric2F1[-21/16, -3/4, -5/16, x^4])/(7*x^5*(-x + x^5)^(1/4)) - (4*(1 - x^4)^(1/4)*
Hypergeometric2F1[-3/4, -9/16, 7/16, x^4])/(3*x^2*(-x + x^5)^(1/4)) + (4*(1 - x^4)^(1/4)*Hypergeometric2F1[-3/
4, -5/16, 11/16, x^4])/(5*x*(-x + x^5)^(1/4)) - (24*x^(1/4)*(-1 + x^4)^(1/4)*Defer[Subst][Defer[Int][(x^2*(-1
+ x^16)^(3/4))/(-1 - 2*x^12 + x^16), x], x, x^(1/4)])/(-x + x^5)^(1/4) + (16*x^(1/4)*(-1 + x^4)^(1/4)*Defer[Su
bst][Defer[Int][(x^6*(-1 + x^16)^(3/4))/(-1 - 2*x^12 + x^16), x], x, x^(1/4)])/(-x + x^5)^(1/4)

Rubi steps

\begin {align*} \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \int \frac {\left (-1+x^4\right )^{3/4} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^{25/4} \left (-1-2 x^3+x^4\right )} \, dx}{\sqrt [4]{-x+x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4} \left (3+x^{16}\right ) \left (-1-x^{12}+x^{16}\right )}{x^{22} \left (-1-2 x^{12}+x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {3 \left (-1+x^{16}\right )^{3/4}}{x^{22}}-\frac {3 \left (-1+x^{16}\right )^{3/4}}{x^{10}}+\frac {\left (-1+x^{16}\right )^{3/4}}{x^6}+\frac {2 x^2 \left (3-2 x^4\right ) \left (-1+x^{16}\right )^{3/4}}{1+2 x^{12}-x^{16}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}\\ &=\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4}}{x^6} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}+\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (3-2 x^4\right ) \left (-1+x^{16}\right )^{3/4}}{1+2 x^{12}-x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}+\frac {\left (12 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4}}{x^{22}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}-\frac {\left (12 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {\left (-1+x^{16}\right )^{3/4}}{x^{10}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}\\ &=\frac {\left (8 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {3 x^2 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}}+\frac {2 x^6 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}+\frac {\left (4 \sqrt [4]{x} \left (-1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{16}\right )^{3/4}}{x^6} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x+x^5}}+\frac {\left (12 \sqrt [4]{x} \left (-1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{16}\right )^{3/4}}{x^{22}} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x+x^5}}-\frac {\left (12 \sqrt [4]{x} \left (-1+x^4\right )\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^{16}\right )^{3/4}}{x^{10}} \, dx,x,\sqrt [4]{x}\right )}{\left (1-x^4\right )^{3/4} \sqrt [4]{-x+x^5}}\\ &=\frac {4 \sqrt [4]{1-x^4} \, _2F_1\left (-\frac {21}{16},-\frac {3}{4};-\frac {5}{16};x^4\right )}{7 x^5 \sqrt [4]{-x+x^5}}-\frac {4 \sqrt [4]{1-x^4} \, _2F_1\left (-\frac {3}{4},-\frac {9}{16};\frac {7}{16};x^4\right )}{3 x^2 \sqrt [4]{-x+x^5}}+\frac {4 \sqrt [4]{1-x^4} \, _2F_1\left (-\frac {3}{4},-\frac {5}{16};\frac {11}{16};x^4\right )}{5 x \sqrt [4]{-x+x^5}}+\frac {\left (16 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}-\frac {\left (24 \sqrt [4]{x} \sqrt [4]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^{16}\right )^{3/4}}{-1-2 x^{12}+x^{16}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right ) \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-1-2 x^3+x^4\right ) \sqrt [4]{-x+x^5}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.78, size = 95, normalized size = 1.00 \begin {gather*} \frac {4 \left (-3+7 x^3+3 x^4\right ) \left (-x+x^5\right )^{3/4}}{21 x^6}-2\ 2^{3/4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \left (-x+x^5\right )^{3/4}}{-1+x^4}\right )-2\ 2^{3/4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \left (-x+x^5\right )^{3/4}}{-1+x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-3 + 7*x^3 + 3*x^4)*(-x + x^5)^(3/4))/(21*x^6) - 2*2^(3/4)*ArcTan[(2^(1/4)*(-x + x^5)^(3/4))/(-1 + x^4)] -
2*2^(3/4)*ArcTanh[(2^(1/4)*(-x + x^5)^(3/4))/(-1 + x^4)]

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fricas [B]  time = 101.94, size = 280, normalized size = 2.95 \begin {gather*} -\frac {84 \cdot 8^{\frac {1}{4}} x^{6} \arctan \left (\frac {16 \cdot 8^{\frac {1}{4}} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (8^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 8 \cdot 8^{\frac {1}{4}} \sqrt {x^{5} - x} x\right )} + 4 \cdot 8^{\frac {3}{4}} {\left (x^{5} - x\right )}^{\frac {3}{4}}}{8 \, {\left (x^{4} - 2 \, x^{3} - 1\right )}}\right ) + 21 \cdot 8^{\frac {1}{4}} x^{6} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} + 8^{\frac {3}{4}} \sqrt {x^{5} - x} x + 8^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 21 \cdot 8^{\frac {1}{4}} x^{6} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} - x\right )}^{\frac {1}{4}} x^{2} - 8^{\frac {3}{4}} \sqrt {x^{5} - x} x - 8^{\frac {1}{4}} {\left (x^{4} + 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{5} - x\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} - 1}\right ) - 8 \, {\left (x^{5} - x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} + 7 \, x^{3} - 3\right )}}{42 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/42*(84*8^(1/4)*x^6*arctan(1/8*(16*8^(1/4)*(x^5 - x)^(1/4)*x^2 + 2^(3/4)*(8^(3/4)*(x^4 + 2*x^3 - 1) + 8*8^(1
/4)*sqrt(x^5 - x)*x) + 4*8^(3/4)*(x^5 - x)^(3/4))/(x^4 - 2*x^3 - 1)) + 21*8^(1/4)*x^6*log(-(4*sqrt(2)*(x^5 - x
)^(1/4)*x^2 + 8^(3/4)*sqrt(x^5 - x)*x + 8^(1/4)*(x^4 + 2*x^3 - 1) + 4*(x^5 - x)^(3/4))/(x^4 - 2*x^3 - 1)) - 21
*8^(1/4)*x^6*log(-(4*sqrt(2)*(x^5 - x)^(1/4)*x^2 - 8^(3/4)*sqrt(x^5 - x)*x - 8^(1/4)*(x^4 + 2*x^3 - 1) + 4*(x^
5 - x)^(3/4))/(x^4 - 2*x^3 - 1)) - 8*(x^5 - x)^(3/4)*(3*x^4 + 7*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 (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}}{{\left (x^{5} - x\right )}^{\frac {1}{4}} {\left (x^{4} - 2 \, x^{3} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 24.69, size = 277, normalized size = 2.92

method	result	size
trager	\(\frac {4 \left (3 x^{4}+7 x^{3}-3\right ) \left (x^{5}-x \right )^{\frac {3}{4}}}{21 x^{6}}-\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x +2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}+x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}-8\right )}{x^{4}-2 x^{3}-1}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{5}-x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x +2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}-4 \left (x^{5}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}-2 x^{3}-1}\right )\)	\(277\)
risch	\(\frac {\frac {4}{7} x^{8}-\frac {8}{7} x^{4}+\frac {4}{7}+\frac {4}{3} x^{7}-\frac {4}{3} x^{3}}{x^{5} \left (x \left (x^{4}-1\right )\right )^{\frac {1}{4}}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\sqrt {x^{5}-x}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x -2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}+x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}-4 \left (x^{5}-x \right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{4}-8\right )}{x^{4}-2 x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}-x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \left (x^{5}-x \right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{5}-x \right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}-2 x^{3}-1}\right )\)	\(285\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/21*(3*x^4+7*x^3-3)*(x^5-x)^(3/4)/x^6-RootOf(_Z^4-8)*ln(-((x^5-x)^(1/2)*RootOf(_Z^4-8)^3*x+2*RootOf(_Z^4-8)^2
*(x^5-x)^(1/4)*x^2+x^4*RootOf(_Z^4-8)+2*RootOf(_Z^4-8)*x^3+4*(x^5-x)^(3/4)-RootOf(_Z^4-8))/(x^4-2*x^3-1))-Root
Of(_Z^2+RootOf(_Z^4-8)^2)*ln(((x^5-x)^(1/2)*RootOf(_Z^2+RootOf(_Z^4-8)^2)*RootOf(_Z^4-8)^2*x+2*RootOf(_Z^4-8)^
2*(x^5-x)^(1/4)*x^2-RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^4-2*RootOf(_Z^2+RootOf(_Z^4-8)^2)*x^3-4*(x^5-x)^(3/4)+Root
Of(_Z^2+RootOf(_Z^4-8)^2))/(x^4-2*x^3-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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