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

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

________________________________________________________________________________________

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*}

________________________________________________________________________________________

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]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 3.36, size = 140, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^5\right )^{2/3} \left (2+5 x^3+2 x^5\right )}{10 x^5}-\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^5}}\right )}{2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^5}\right )}{2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^5}+2^{2/3} \left (1+x^5\right )^{2/3}\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))

________________________________________________________________________________________

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+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]

Timed out

________________________________________________________________________________________

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)

________________________________________________________________________________________

maple [C]  time = 279.84, size = 725, normalized size = 5.18

method	result	size
risch	\(\frac {\frac {3}{5} x^{10}+\frac {3}{2} x^{8}+\frac {6}{5} x^{5}+\frac {3}{2} x^{3}+\frac {3}{5}}{x^{5} \left (x^{5}+1\right )^{\frac {1}{3}}}+\RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{5}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{5}+3 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{3}-2\right ) x^{3}-2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+6 \left (x^{5}+1\right )^{\frac {2}{3}} x +2 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{5}-x^{3}+2}\right )-\frac {\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{5}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{5}-3 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-6 \left (x^{5}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{5}-x^{3}+2}\right ) \RootOf \left (\textit {\_Z}^{3}-2\right )}{2}-\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{3}-2\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{3}-4 \RootOf \left (\textit {\_Z}^{3}-2\right ) x^{5}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right ) x^{5}-3 \left (x^{5}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}-2\right )^{2} x^{2}-6 \left (x^{5}+1\right )^{\frac {2}{3}} x -4 \RootOf \left (\textit {\_Z}^{3}-2\right )-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )}{2 x^{5}-x^{3}+2}\right ) \RootOf \left (\RootOf \left (\textit {\_Z}^{3}-2\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-2\right )+4 \textit {\_Z}^{2}\right )\)	\(725\)
trager	\(\text {Expression too large to display}\)	\(1470\)

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,method=_RETURNVERBOSE)

[Out]

3/10*(2*x^10+5*x^8+4*x^5+5*x^3+2)/x^5/(x^5+1)^(1/3)+RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*ln(-(R
ootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(
_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3+2*RootOf(_Z^3-2)*x^5-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z
^2)*x^5+3*(x^5+1)^(1/3)*RootOf(_Z^3-2)^2*x^2+RootOf(_Z^3-2)*x^3-2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+
4*_Z^2)*x^3+6*(x^5+1)^(2/3)*x+2*RootOf(_Z^3-2)-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(2*x^5-x
^3+2))-1/2*ln((2*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+2*RootOf(RootOf(_Z^3
-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-4*RootOf(_Z^3-2)*x^5-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*R
ootOf(_Z^3-2)+4*_Z^2)*x^5-3*(x^5+1)^(1/3)*RootOf(_Z^3-2)^2*x^2-6*(x^5+1)^(2/3)*x-4*RootOf(_Z^3-2)-4*RootOf(Roo
tOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(2*x^5-x^3+2))*RootOf(_Z^3-2)-ln((2*RootOf(_Z^3-2)^3*RootOf(RootOf(
_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+2*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-
2)^2*x^3-4*RootOf(_Z^3-2)*x^5-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^5-3*(x^5+1)^(1/3)*RootOf
(_Z^3-2)^2*x^2-6*(x^5+1)^(2/3)*x-4*RootOf(_Z^3-2)-4*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2))/(2*x^
5-x^3+2))*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________

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)

________________________________________________________________________________________