3.17.53 \(\int \frac {(1+x^5)^{2/3} (-3+2 x^5) (2+x^3+2 x^5)}{x^6 (2-x^3+2 x^5)} \, dx\)

Optimal. Leaf size=140 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1}-x\right )}{2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^5+1}+x}\right )}{2^{2/3}}+\frac {3 \left (x^5+1\right )^{2/3} \left (2 x^5+5 x^3+2\right )}{10 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1} x+2^{2/3} \left (x^5+1\right )^{2/3}+x^2\right )}{2\ 2^{2/3}} \]

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Rubi [F]  time = 1.29, 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 )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(3*(1 + x^5)^(2/3))/5 + (3*(1 + x^5)^(2/3))/(5*x^5) + (3*Hypergeometric2F1[-2/3, -2/5, 3/5, -x^5])/(2*x^2) - 3
*Defer[Int][(1 + x^5)^(2/3)/(2 - x^3 + 2*x^5), x] + 10*Defer[Int][(x^2*(1 + x^5)^(2/3))/(2 - x^3 + 2*x^5), x]

Rubi steps

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

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Mathematica [F]  time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right ) \left (2+x^3+2 x^5\right )}{x^6 \left (2-x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.57, size = 140, normalized size = 1.00 \begin {gather*} \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1}-x\right )}{2^{2/3}}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^5+1}+x}\right )}{2^{2/3}}+\frac {3 \left (x^5+1\right )^{2/3} \left (2 x^5+5 x^3+2\right )}{10 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^5+1} x+2^{2/3} \left (x^5+1\right )^{2/3}+x^2\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1 + x^5)^(2/3)*(2 + 5*x^3 + 2*x^5))/(10*x^5) - (Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(1 + x^5)^(1/3))
])/2^(2/3) + Log[-x + 2^(1/3)*(1 + x^5)^(1/3)]/2^(2/3) - Log[x^2 + 2^(1/3)*x*(1 + x^5)^(1/3) + 2^(2/3)*(1 + x^
5)^(2/3)]/(2*2^(2/3))

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fricas [B]  time = 174.90, size = 399, normalized size = 2.85 \begin {gather*} -\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{11} + x^{9} - x^{7} + 4 \, x^{6} + x^{4} + 2 \, x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{15} + 60 \, x^{13} + 24 \, x^{11} + 24 \, x^{10} - x^{9} + 120 \, x^{8} + 24 \, x^{6} + 24 \, x^{5} + 60 \, x^{3} + 8\right )} + 12 \, {\left (4 \, x^{12} + 14 \, x^{10} + x^{8} + 8 \, x^{7} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{15} - 12 \, x^{13} - 48 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 48 \, x^{6} + 24 \, x^{5} - 12 \, x^{3} + 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{5} - x^{3} + 2\right )} - 12 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{5} - x^{3} + 2}\right ) + 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{6} + x^{4} + x\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{10} + 14 \, x^{8} + x^{6} + 8 \, x^{5} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{7} + x^{5} + 4 \, x^{2}\right )} {\left (x^{5} + 1\right )}^{\frac {1}{3}}}{4 \, x^{10} - 4 \, x^{8} + x^{6} + 8 \, x^{5} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (2 \, x^{5} + 5 \, x^{3} + 2\right )} {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{120 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/120*(20*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(2*x^11 + x^9 - x^7 + 4*x^6 + x^4 + 2*x)
*(x^5 + 1)^(2/3) + 4^(1/3)*(8*x^15 + 60*x^13 + 24*x^11 + 24*x^10 - x^9 + 120*x^8 + 24*x^6 + 24*x^5 + 60*x^3 +
8) + 12*(4*x^12 + 14*x^10 + x^8 + 8*x^7 + 14*x^5 + 4*x^2)*(x^5 + 1)^(1/3))/(8*x^15 - 12*x^13 - 48*x^11 + 24*x^
10 - x^9 - 24*x^8 - 48*x^6 + 24*x^5 - 12*x^3 + 8)) - 10*4^(2/3)*x^5*log((6*4^(1/3)*(x^5 + 1)^(1/3)*x^2 + 4^(2/
3)*(2*x^5 - x^3 + 2) - 12*(x^5 + 1)^(2/3)*x)/(2*x^5 - x^3 + 2)) + 5*4^(2/3)*x^5*log((6*4^(2/3)*(x^6 + x^4 + x)
*(x^5 + 1)^(2/3) + 4^(1/3)*(4*x^10 + 14*x^8 + x^6 + 8*x^5 + 14*x^3 + 4) + 6*(4*x^7 + x^5 + 4*x^2)*(x^5 + 1)^(1
/3))/(4*x^10 - 4*x^8 + x^6 + 8*x^5 - 4*x^3 + 4)) - 36*(2*x^5 + 5*x^3 + 2)*(x^5 + 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-1)]  time = 180.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{5}+1\right )^{\frac {2}{3}} \left (2 x^{5}-3\right ) \left (2 x^{5}+x^{3}+2\right )}{x^{6} \left (2 x^{5}-x^{3}+2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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