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

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 = 2.67, size = 85, normalized size = 1.00 \begin {gather*} 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} \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 = 47.99, 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 = 4.35, size = 1334, normalized size = 15.69

result too large to display

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)

[Out]

4/45*(20*x^15+12*x^14+87*x^13+43*x^12-60*x^10-24*x^9-87*x^8+60*x^5+12*x^4-20)/x^9/(2*x^5+x^4-2)^(3/4)+(ln((-4*
x^15-8*x^14-5*x^13+4*(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)^(1/4)*x^11-x^12+4*(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)^(1/4)*x^10+(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)^(1/4)*x^9+12*x^10-2*(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)^(1/2)*x^7+16*x^9-(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)^(1/2)*x^6+5*x^8-8*(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)^(1/4)*x^6+(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)^(3/4)*x^3-4*(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)^(1/4)
*x^5-12*x^5+2*(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)^(1/2)*x^2-8*x^4+4*(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)^(1/4)*x+4)/(-1+x)/(x^4+x^3+x^2+x+1)/(2*x^5+x^4-2)^2)+Roo
tOf(_Z^2+1)*ln((4*x^15+8*x^14-4*RootOf(_Z^2+1)*(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)^(1/4)*x^11+5*x^13-4*RootOf(_Z^2+1)*(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)^(1/4)*
x^10+x^12-RootOf(_Z^2+1)*(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)^(1/4)*x^9-12*x^10-2
*(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)^(1/2)*x^7-16*x^9-(8*x^15+12*x^14+6*x^13+x^1
2-24*x^10-24*x^9-6*x^8+24*x^5+12*x^4-8)^(1/2)*x^6+8*RootOf(_Z^2+1)*(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)^(1/4)*x^6-5*x^8+(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)^(3/4)
*RootOf(_Z^2+1)*x^3+4*RootOf(_Z^2+1)*(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)^(1/4)*x
^5+12*x^5+2*(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)^(1/2)*x^2+8*x^4-4*RootOf(_Z^2+1)
*(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)^(1/4)*x-4)/(-1+x)/(x^4+x^3+x^2+x+1)/(2*x^5+
x^4-2)^2))/(2*x^5+x^4-2)^(3/4)*((2*x^5+x^4-2)^3)^(1/4)

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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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