3.12.51 \(\int \frac {(4+x^5) \sqrt [4]{-2+x^4+2 x^5} (2-4 x^5+x^8+2 x^{10})}{x^{10} (-1+x^5)} \, dx\)

Optimal. Leaf size=85 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{2 x^5+x^4-2}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2 x^5+x^4-2}}\right )+\frac {4 \sqrt [4]{2 x^5+x^4-2} \left (10 x^{10}+x^9+43 x^8-20 x^5-x^4+10\right )}{45 x^9} \]

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Rubi [F]  time = 1.96, 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 (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/(x^10*(-1 + x^5)),x]

[Out]

2*Defer[Int][(-2 + x^4 + 2*x^5)^(1/4), x] + Defer[Int][(-2 + x^4 + 2*x^5)^(1/4)/(-1 + x), x] - 8*Defer[Int][(-
2 + x^4 + 2*x^5)^(1/4)/x^10, x] + 6*Defer[Int][(-2 + x^4 + 2*x^5)^(1/4)/x^5, x] - 4*Defer[Int][(-2 + x^4 + 2*x
^5)^(1/4)/x^2, x] + Defer[Int][(-2 + x^4 + 2*x^5)^(1/4)/(1 + x + x^2 + x^3 + x^4), x] + 2*Defer[Int][(x*(-2 +
x^4 + 2*x^5)^(1/4))/(1 + x + x^2 + x^3 + x^4), x] + 3*Defer[Int][(x^2*(-2 + x^4 + 2*x^5)^(1/4))/(1 + x + x^2 +
x^3 + x^4), x] - Defer[Int][(x^3*(-2 + x^4 + 2*x^5)^(1/4))/(1 + x + x^2 + x^3 + x^4), x]

Rubi steps

\begin {align*} \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx &=\int \left (2 \sqrt [4]{-2+x^4+2 x^5}+\frac {\sqrt [4]{-2+x^4+2 x^5}}{-1+x}-\frac {8 \sqrt [4]{-2+x^4+2 x^5}}{x^{10}}+\frac {6 \sqrt [4]{-2+x^4+2 x^5}}{x^5}-\frac {4 \sqrt [4]{-2+x^4+2 x^5}}{x^2}+\frac {\left (1+2 x+3 x^2-x^3\right ) \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=2 \int \sqrt [4]{-2+x^4+2 x^5} \, dx-4 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^2} \, dx+6 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^5} \, dx-8 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^{10}} \, dx+\int \frac {\sqrt [4]{-2+x^4+2 x^5}}{-1+x} \, dx+\int \frac {\left (1+2 x+3 x^2-x^3\right ) \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4} \, dx\\ &=2 \int \sqrt [4]{-2+x^4+2 x^5} \, dx-4 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^2} \, dx+6 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^5} \, dx-8 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^{10}} \, dx+\int \frac {\sqrt [4]{-2+x^4+2 x^5}}{-1+x} \, dx+\int \left (\frac {\sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4}+\frac {2 x \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4}+\frac {3 x^2 \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4}-\frac {x^3 \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4}\right ) \, dx\\ &=2 \int \sqrt [4]{-2+x^4+2 x^5} \, dx+2 \int \frac {x \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4} \, dx+3 \int \frac {x^2 \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4} \, dx-4 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^2} \, dx+6 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^5} \, dx-8 \int \frac {\sqrt [4]{-2+x^4+2 x^5}}{x^{10}} \, dx+\int \frac {\sqrt [4]{-2+x^4+2 x^5}}{-1+x} \, dx+\int \frac {\sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4} \, dx-\int \frac {x^3 \sqrt [4]{-2+x^4+2 x^5}}{1+x+x^2+x^3+x^4} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^5\right ) \sqrt [4]{-2+x^4+2 x^5} \left (2-4 x^5+x^8+2 x^{10}\right )}{x^{10} \left (-1+x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/(x^10*(-1 + x^5)),x]

[Out]

Integrate[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/(x^10*(-1 + x^5)), x]

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IntegrateAlgebraic [A]  time = 2.66, size = 85, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-2+x^4+2 x^5} \left (10-x^4-20 x^5+43 x^8+x^9+10 x^{10}\right )}{45 x^9}+2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right )-2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-2+x^4+2 x^5}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((4 + x^5)*(-2 + x^4 + 2*x^5)^(1/4)*(2 - 4*x^5 + x^8 + 2*x^10))/(x^10*(-1 + x^5)),x]

[Out]

(4*(-2 + x^4 + 2*x^5)^(1/4)*(10 - x^4 - 20*x^5 + 43*x^8 + x^9 + 10*x^10))/(45*x^9) + 2*ArcTan[x/(-2 + x^4 + 2*
x^5)^(1/4)] - 2*ArcTanh[x/(-2 + x^4 + 2*x^5)^(1/4)]

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fricas [B]  time = 51.69, size = 161, normalized size = 1.89 \begin {gather*} \frac {45 \, x^{9} \arctan \left (\frac {{\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {3}{4}} x}{x^{5} - 1}\right ) + 45 \, x^{9} \log \left (-\frac {x^{5} + x^{4} - {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} x^{3} + \sqrt {2 \, x^{5} + x^{4} - 2} x^{2} - {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {3}{4}} x - 1}{x^{5} - 1}\right ) + 4 \, {\left (10 \, x^{10} + x^{9} + 43 \, x^{8} - 20 \, x^{5} - x^{4} + 10\right )} {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}}}{45 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(2*x^5+x^4-2)^(1/4)*(2*x^10+x^8-4*x^5+2)/x^10/(x^5-1),x, algorithm="fricas")

[Out]

1/45*(45*x^9*arctan(((2*x^5 + x^4 - 2)^(1/4)*x^3 + (2*x^5 + x^4 - 2)^(3/4)*x)/(x^5 - 1)) + 45*x^9*log(-(x^5 +
x^4 - (2*x^5 + x^4 - 2)^(1/4)*x^3 + sqrt(2*x^5 + x^4 - 2)*x^2 - (2*x^5 + x^4 - 2)^(3/4)*x - 1)/(x^5 - 1)) + 4*
(10*x^10 + x^9 + 43*x^8 - 20*x^5 - x^4 + 10)*(2*x^5 + x^4 - 2)^(1/4))/x^9

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{10} + x^{8} - 4 \, x^{5} + 2\right )} {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + 4\right )}}{{\left (x^{5} - 1\right )} x^{10}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(2*x^5+x^4-2)^(1/4)*(2*x^10+x^8-4*x^5+2)/x^10/(x^5-1),x, algorithm="giac")

[Out]

integrate((2*x^10 + x^8 - 4*x^5 + 2)*(2*x^5 + x^4 - 2)^(1/4)*(x^5 + 4)/((x^5 - 1)*x^10), x)

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maple [C]  time = 13.55, size = 235, normalized size = 2.76

method	result	size
trager	\(\frac {4 \left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} \left (10 x^{10}+x^{9}+43 x^{8}-20 x^{5}-x^{4}+10\right )}{45 x^{9}}+\ln \left (-\frac {-x^{5}+\left (2 x^{5}+x^{4}-2\right )^{\frac {3}{4}} x -\sqrt {2 x^{5}+x^{4}-2}\, x^{2}+\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} x^{3}-x^{4}+1}{\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{5}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {2 x^{5}+x^{4}-2}\, x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\left (2 x^{5}+x^{4}-2\right )^{\frac {3}{4}} x +\left (2 x^{5}+x^{4}-2\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right )}\right )\)	\(235\)
risch	\(\frac {\frac {116}{15} x^{13}+\frac {172}{45} x^{12}-\frac {32}{15} x^{9}-\frac {116}{15} x^{8}+\frac {16}{15} x^{4}-\frac {16}{3} x^{10}+\frac {16}{3} x^{5}-\frac {16}{9}+\frac {16}{9} x^{15}+\frac {16}{15} x^{14}}{x^{9} \left (2 x^{5}+x^{4}-2\right )^{\frac {3}{4}}}+\frac {\left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-4 x^{15}-8 x^{14}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{11}-5 x^{13}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{10}-x^{12}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{9}+12 x^{10}+2 \sqrt {8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8}\, x^{7}+16 x^{9}+\sqrt {8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8}\, x^{6}+8 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{6}+5 x^{8}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {3}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{5}-12 x^{5}-2 \sqrt {8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8}\, x^{2}-8 x^{4}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x +4}{\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (2 x^{5}+x^{4}-2\right )^{2}}\right )-\ln \left (-\frac {4 x^{15}+8 x^{14}+5 x^{13}+4 \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{11}+x^{12}+4 \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{10}+\left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{9}-12 x^{10}+2 \sqrt {8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8}\, x^{7}-16 x^{9}+\sqrt {8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8}\, x^{6}-5 x^{8}-8 \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{6}+\left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {3}{4}} x^{3}-4 \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x^{5}+12 x^{5}-2 \sqrt {8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8}\, x^{2}+8 x^{4}+4 \left (8 x^{15}+12 x^{14}+6 x^{13}+x^{12}-24 x^{10}-24 x^{9}-6 x^{8}+24 x^{5}+12 x^{4}-8\right )^{\frac {1}{4}} x -4}{\left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (2 x^{5}+x^{4}-2\right )^{2}}\right )\right ) \left (\left (2 x^{5}+x^{4}-2\right )^{3}\right )^{\frac {1}{4}}}{\left (2 x^{5}+x^{4}-2\right )^{\frac {3}{4}}}\)	\(1337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+4)*(2*x^5+x^4-2)^(1/4)*(2*x^10+x^8-4*x^5+2)/x^10/(x^5-1),x,method=_RETURNVERBOSE)

[Out]

4/45*(2*x^5+x^4-2)^(1/4)*(10*x^10+x^9+43*x^8-20*x^5-x^4+10)/x^9+ln(-(-x^5+(2*x^5+x^4-2)^(3/4)*x-(2*x^5+x^4-2)^
(1/2)*x^2+(2*x^5+x^4-2)^(1/4)*x^3-x^4+1)/(-1+x)/(x^4+x^3+x^2+x+1))+RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)*x^5+Root
Of(_Z^2+1)*(2*x^5+x^4-2)^(1/2)*x^2-RootOf(_Z^2+1)*x^4-(2*x^5+x^4-2)^(3/4)*x+(2*x^5+x^4-2)^(1/4)*x^3+RootOf(_Z^
2+1))/(-1+x)/(x^4+x^3+x^2+x+1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{10} + x^{8} - 4 \, x^{5} + 2\right )} {\left (2 \, x^{5} + x^{4} - 2\right )}^{\frac {1}{4}} {\left (x^{5} + 4\right )}}{{\left (x^{5} - 1\right )} x^{10}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(2*x^5+x^4-2)^(1/4)*(2*x^10+x^8-4*x^5+2)/x^10/(x^5-1),x, algorithm="maxima")

[Out]

integrate((2*x^10 + x^8 - 4*x^5 + 2)*(2*x^5 + x^4 - 2)^(1/4)*(x^5 + 4)/((x^5 - 1)*x^10), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^5+4\right )\,{\left (2\,x^5+x^4-2\right )}^{1/4}\,\left (2\,x^{10}+x^8-4\,x^5+2\right )}{x^{10}\,\left (x^5-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 4)*(x^4 + 2*x^5 - 2)^(1/4)*(x^8 - 4*x^5 + 2*x^10 + 2))/(x^10*(x^5 - 1)),x)

[Out]

int(((x^5 + 4)*(x^4 + 2*x^5 - 2)^(1/4)*(x^8 - 4*x^5 + 2*x^10 + 2))/(x^10*(x^5 - 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**5+4)*(2*x**5+x**4-2)**(1/4)*(2*x**10+x**8-4*x**5+2)/x**10/(x**5-1),x)

[Out]

Timed out

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