3.14.66 \(\int \frac {(4+x^5) (1-x^4-2 x^5+x^8+x^9+x^{10})}{x^2 (-1+x^5)^{3/4} (1+x^4-2 x^5-x^8-x^9+x^{10})} \, dx\)

Optimal. Leaf size=98 \[ \frac {4 \sqrt [4]{x^5-1}}{x}-2 \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4-1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^5-1}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-\log \left (\sqrt [4]{x^5-1}-\text {$\#$1} x\right )+\log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ] \]

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Rubi [F]  time = 3.95, 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 ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10
)),x]

[Out]

(-4*(1 - x^5)^(3/4)*Hypergeometric2F1[-1/5, 3/4, 4/5, x^5])/(x*(-1 + x^5)^(3/4)) + (6*x*(1 - x^5)^(3/4)*Hyperg
eometric2F1[1/5, 3/4, 6/5, x^5])/(-1 + x^5)^(3/4) + (2*x^2*(1 - x^5)^(3/4)*Hypergeometric2F1[2/5, 3/4, 7/5, x^
5])/(-1 + x^5)^(3/4) + (2*x^3*(1 - x^5)^(3/4)*Hypergeometric2F1[3/5, 3/4, 8/5, x^5])/(3*(-1 + x^5)^(3/4)) + (x
^4*(1 - x^5)^(3/4)*Hypergeometric2F1[3/4, 4/5, 9/5, x^5])/(4*(-1 + x^5)^(3/4)) - 6*Defer[Int][1/((-1 + x^5)^(3
/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] - 4*Defer[Int][x/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9
+ x^10)), x] - 10*Defer[Int][x^2/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] - 6*Defer[Int][x^
4/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] + 8*Defer[Int][x^5/((-1 + x^5)^(3/4)*(1 + x^4 -
2*x^5 - x^8 - x^9 + x^10)), x] + 14*Defer[Int][x^6/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x]
+ 10*Defer[Int][x^7/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] + 6*Defer[Int][x^8/((-1 + x^5
)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9 + x^10)), x] + 10*Defer[Int][x^9/((-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8
- x^9 + x^10)), x]

Rubi steps

\begin {align*} \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx &=\int \left (\frac {6}{\left (-1+x^5\right )^{3/4}}+\frac {4}{x^2 \left (-1+x^5\right )^{3/4}}+\frac {4 x}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^2}{\left (-1+x^5\right )^{3/4}}+\frac {x^3}{\left (-1+x^5\right )^{3/4}}+\frac {2 \left (-3-2 x-5 x^2-3 x^4+4 x^5+7 x^6+5 x^7+3 x^8+5 x^9\right )}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}\right ) \, dx\\ &=2 \int \frac {x^2}{\left (-1+x^5\right )^{3/4}} \, dx+2 \int \frac {-3-2 x-5 x^2-3 x^4+4 x^5+7 x^6+5 x^7+3 x^8+5 x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+4 \int \frac {1}{x^2 \left (-1+x^5\right )^{3/4}} \, dx+4 \int \frac {x}{\left (-1+x^5\right )^{3/4}} \, dx+6 \int \frac {1}{\left (-1+x^5\right )^{3/4}} \, dx+\int \frac {x^3}{\left (-1+x^5\right )^{3/4}} \, dx\\ &=2 \int \left (-\frac {3}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {2 x}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {5 x^2}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}-\frac {3 x^4}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {4 x^5}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {7 x^6}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {5 x^7}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {3 x^8}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}+\frac {5 x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )}\right ) \, dx+\frac {\left (1-x^5\right )^{3/4} \int \frac {x^3}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (2 \left (1-x^5\right )^{3/4}\right ) \int \frac {x^2}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (4 \left (1-x^5\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (4 \left (1-x^5\right )^{3/4}\right ) \int \frac {x}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}+\frac {\left (6 \left (1-x^5\right )^{3/4}\right ) \int \frac {1}{\left (1-x^5\right )^{3/4}} \, dx}{\left (-1+x^5\right )^{3/4}}\\ &=-\frac {4 \left (1-x^5\right )^{3/4} \, _2F_1\left (-\frac {1}{5},\frac {3}{4};\frac {4}{5};x^5\right )}{x \left (-1+x^5\right )^{3/4}}+\frac {6 x \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {1}{5},\frac {3}{4};\frac {6}{5};x^5\right )}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^2 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {2}{5},\frac {3}{4};\frac {7}{5};x^5\right )}{\left (-1+x^5\right )^{3/4}}+\frac {2 x^3 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {3}{5},\frac {3}{4};\frac {8}{5};x^5\right )}{3 \left (-1+x^5\right )^{3/4}}+\frac {x^4 \left (1-x^5\right )^{3/4} \, _2F_1\left (\frac {3}{4},\frac {4}{5};\frac {9}{5};x^5\right )}{4 \left (-1+x^5\right )^{3/4}}-4 \int \frac {x}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-6 \int \frac {1}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-6 \int \frac {x^4}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+6 \int \frac {x^8}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+8 \int \frac {x^5}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx-10 \int \frac {x^2}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+10 \int \frac {x^7}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+10 \int \frac {x^9}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx+14 \int \frac {x^6}{\left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx\\ \end {align*}

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Mathematica [F]  time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (4+x^5\right ) \left (1-x^4-2 x^5+x^8+x^9+x^{10}\right )}{x^2 \left (-1+x^5\right )^{3/4} \left (1+x^4-2 x^5-x^8-x^9+x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9
+ x^10)),x]

[Out]

Integrate[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x^8 - x^9
+ x^10)), x]

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IntegrateAlgebraic [A]  time = 12.87, size = 98, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{-1+x^5}}{x}-2 \text {RootSum}\left [-1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{-1+x^5}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((4 + x^5)*(1 - x^4 - 2*x^5 + x^8 + x^9 + x^10))/(x^2*(-1 + x^5)^(3/4)*(1 + x^4 - 2*x^5 - x
^8 - x^9 + x^10)),x]

[Out]

(4*(-1 + x^5)^(1/4))/x - 2*RootSum[-1 - #1^4 + #1^8 & , (Log[x] - Log[(-1 + x^5)^(1/4) - x*#1] + Log[x]*#1^4 -
Log[(-1 + x^5)^(1/4) - x*#1]*#1^4)/(-#1^3 + 2*#1^7) & ]

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fricas [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)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x, algorithm="fricas
")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{10} + x^{9} + x^{8} - 2 \, x^{5} - x^{4} + 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{10} - x^{9} - x^{8} - 2 \, x^{5} + x^{4} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x, algorithm="giac")

[Out]

integrate((x^10 + x^9 + x^8 - 2*x^5 - x^4 + 1)*(x^5 + 4)/((x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)*(x^5 - 1)^(3/4)
*x^2), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{5}+4\right ) \left (x^{10}+x^{9}+x^{8}-2 x^{5}-x^{4}+1\right )}{x^{2} \left (x^{5}-1\right )^{\frac {3}{4}} \left (x^{10}-x^{9}-x^{8}-2 x^{5}+x^{4}+1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x)

[Out]

int((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{10} + x^{9} + x^{8} - 2 \, x^{5} - x^{4} + 1\right )} {\left (x^{5} + 4\right )}}{{\left (x^{10} - x^{9} - x^{8} - 2 \, x^{5} + x^{4} + 1\right )} {\left (x^{5} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^5+4)*(x^10+x^9+x^8-2*x^5-x^4+1)/x^2/(x^5-1)^(3/4)/(x^10-x^9-x^8-2*x^5+x^4+1),x, algorithm="maxima
")

[Out]

integrate((x^10 + x^9 + x^8 - 2*x^5 - x^4 + 1)*(x^5 + 4)/((x^10 - x^9 - x^8 - 2*x^5 + x^4 + 1)*(x^5 - 1)^(3/4)
*x^2), 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 (x^{10}+x^9+x^8-2\,x^5-x^4+1\right )}{x^2\,{\left (x^5-1\right )}^{3/4}\,\left (x^{10}-x^9-x^8-2\,x^5+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^5 + 4)*(x^8 - 2*x^5 - x^4 + x^9 + x^10 + 1))/(x^2*(x^5 - 1)^(3/4)*(x^4 - 2*x^5 - x^8 - x^9 + x^10 + 1)
),x)

[Out]

int(((x^5 + 4)*(x^8 - 2*x^5 - x^4 + x^9 + x^10 + 1))/(x^2*(x^5 - 1)^(3/4)*(x^4 - 2*x^5 - x^8 - x^9 + x^10 + 1)
), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{5} + 4\right ) \left (x^{10} + x^{9} + x^{8} - 2 x^{5} - x^{4} + 1\right )}{x^{2} \left (\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )\right )^{\frac {3}{4}} \left (x^{10} - x^{9} - x^{8} - 2 x^{5} + x^{4} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**5+4)*(x**10+x**9+x**8-2*x**5-x**4+1)/x**2/(x**5-1)**(3/4)/(x**10-x**9-x**8-2*x**5+x**4+1),x)

[Out]

Integral((x**5 + 4)*(x**10 + x**9 + x**8 - 2*x**5 - x**4 + 1)/(x**2*((x - 1)*(x**4 + x**3 + x**2 + x + 1))**(3
/4)*(x**10 - x**9 - x**8 - 2*x**5 + x**4 + 1)), x)

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