Optimal. Leaf size=24 \[ -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {x^5-1}}\right )}{\sqrt {a}} \]
________________________________________________________________________________________
Rubi [F] time = 0.68, 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 {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx &=\int \left (\frac {3}{a \sqrt {-1+x^5}}+\frac {5 a+3 x^2}{a \sqrt {-1+x^5} \left (-a-x^2+a x^5\right )}\right ) \, dx\\ &=\frac {\int \frac {5 a+3 x^2}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx}{a}+\frac {3 \int \frac {1}{\sqrt {-1+x^5}} \, dx}{a}\\ &=\frac {\int \left (\frac {5 a}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )}+\frac {3 x^2}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )}\right ) \, dx}{a}+\frac {\left (3 \sqrt {1-x^5}\right ) \int \frac {1}{\sqrt {1-x^5}} \, dx}{a \sqrt {-1+x^5}}\\ &=\frac {3 x \sqrt {1-x^5} \, _2F_1\left (\frac {1}{5},\frac {1}{2};\frac {6}{5};x^5\right )}{a \sqrt {-1+x^5}}+5 \int \frac {1}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx+\frac {3 \int \frac {x^2}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx}{a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x^5}{\sqrt {-1+x^5} \left (-a-x^2+a x^5\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 2.25, size = 24, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {x}{\sqrt {a} \sqrt {-1+x^5}}\right )}{\sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.51, size = 151, normalized size = 6.29 \begin {gather*} \left [\frac {\log \left (\frac {a^{2} x^{10} + 6 \, a x^{7} - 2 \, a^{2} x^{5} + x^{4} - 6 \, a x^{2} - 4 \, {\left (a x^{6} + x^{3} - a x\right )} \sqrt {x^{5} - 1} \sqrt {a} + a^{2}}{a^{2} x^{10} - 2 \, a x^{7} - 2 \, a^{2} x^{5} + x^{4} + 2 \, a x^{2} + a^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (a x^{5} + x^{2} - a\right )} \sqrt {x^{5} - 1} \sqrt {-a}}{2 \, {\left (a x^{6} - a x\right )}}\right )}{a}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + 2}{{\left (a x^{5} - x^{2} - a\right )} \sqrt {x^{5} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {3 x^{5}+2}{\sqrt {x^{5}-1}\, \left (a \,x^{5}-x^{2}-a \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \, x^{5} + 2}{{\left (a x^{5} - x^{2} - a\right )} \sqrt {x^{5} - 1}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.86, size = 52, normalized size = 2.17 \begin {gather*} \frac {\ln \left (\frac {a^4\,\left (x^5-1\right )+a^3\,x^2-2\,a^{7/2}\,x\,\sqrt {x^5-1}}{4\,x^2-4\,a\,\left (x^5-1\right )}\right )}{\sqrt {a}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 x^{5} + 2}{\sqrt {\left (x - 1\right ) \left (x^{4} + x^{3} + x^{2} + x + 1\right )} \left (a x^{5} - a - x^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________