Optimal. Leaf size=72 \[ \frac{3 a^2 x^5}{5 b^4}-\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0583653, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{3 a^2 x^5}{5 b^4}-\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{24}}{\left (a+b x^5\right )^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (\frac{3 a^2}{b^4}-\frac{2 a x}{b^3}+\frac{x^2}{b^2}+\frac{a^4}{b^4 (a+b x)^2}-\frac{4 a^3}{b^4 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=\frac{3 a^2 x^5}{5 b^4}-\frac{a x^{10}}{5 b^3}+\frac{x^{15}}{15 b^2}-\frac{a^4}{5 b^5 \left (a+b x^5\right )}-\frac{4 a^3 \log \left (a+b x^5\right )}{5 b^5}\\ \end{align*}
Mathematica [A] time = 0.0233569, size = 60, normalized size = 0.83 \[ \frac{9 a^2 b x^5-\frac{3 a^4}{a+b x^5}-12 a^3 \log \left (a+b x^5\right )-3 a b^2 x^{10}+b^3 x^{15}}{15 b^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.01, size = 63, normalized size = 0.9 \begin{align*}{\frac{3\,{x}^{5}{a}^{2}}{5\,{b}^{4}}}-{\frac{a{x}^{10}}{5\,{b}^{3}}}+{\frac{{x}^{15}}{15\,{b}^{2}}}-{\frac{{a}^{4}}{5\,{b}^{5} \left ( b{x}^{5}+a \right ) }}-{\frac{4\,{a}^{3}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00645, size = 88, normalized size = 1.22 \begin{align*} -\frac{a^{4}}{5 \,{\left (b^{6} x^{5} + a b^{5}\right )}} - \frac{4 \, a^{3} \log \left (b x^{5} + a\right )}{5 \, b^{5}} + \frac{b^{2} x^{15} - 3 \, a b x^{10} + 9 \, a^{2} x^{5}}{15 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.72371, size = 171, normalized size = 2.38 \begin{align*} \frac{b^{4} x^{20} - 2 \, a b^{3} x^{15} + 6 \, a^{2} b^{2} x^{10} + 9 \, a^{3} b x^{5} - 3 \, a^{4} - 12 \,{\left (a^{3} b x^{5} + a^{4}\right )} \log \left (b x^{5} + a\right )}{15 \,{\left (b^{6} x^{5} + a b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 1.01986, size = 68, normalized size = 0.94 \begin{align*} - \frac{a^{4}}{5 a b^{5} + 5 b^{6} x^{5}} - \frac{4 a^{3} \log{\left (a + b x^{5} \right )}}{5 b^{5}} + \frac{3 a^{2} x^{5}}{5 b^{4}} - \frac{a x^{10}}{5 b^{3}} + \frac{x^{15}}{15 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.19263, size = 108, normalized size = 1.5 \begin{align*} -\frac{4 \, a^{3} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{5}} + \frac{b^{4} x^{15} - 3 \, a b^{3} x^{10} + 9 \, a^{2} b^{2} x^{5}}{15 \, b^{6}} + \frac{4 \, a^{3} b x^{5} + 3 \, a^{4}}{5 \,{\left (b x^{5} + a\right )} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]