Optimal. Leaf size=59 \[ \frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2} \]
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Rubi [A] time = 0.0453078, antiderivative size = 59, 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} \[ \frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{19}}{\left (a+b x^5\right )^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x^3}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (-\frac{2 a}{b^3}+\frac{x}{b^2}-\frac{a^3}{b^3 (a+b x)^2}+\frac{3 a^2}{b^3 (a+b x)}\right ) \, dx,x,x^5\right )\\ &=-\frac{2 a x^5}{5 b^3}+\frac{x^{10}}{10 b^2}+\frac{a^3}{5 b^4 \left (a+b x^5\right )}+\frac{3 a^2 \log \left (a+b x^5\right )}{5 b^4}\\ \end{align*}
Mathematica [A] time = 0.0156562, size = 49, normalized size = 0.83 \[ \frac{\frac{2 a^3}{a+b x^5}+6 a^2 \log \left (a+b x^5\right )-4 a b x^5+b^2 x^{10}}{10 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 52, normalized size = 0.9 \begin{align*} -{\frac{2\,a{x}^{5}}{5\,{b}^{3}}}+{\frac{{x}^{10}}{10\,{b}^{2}}}+{\frac{{a}^{3}}{5\,{b}^{4} \left ( b{x}^{5}+a \right ) }}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04403, size = 73, normalized size = 1.24 \begin{align*} \frac{a^{3}}{5 \,{\left (b^{5} x^{5} + a b^{4}\right )}} + \frac{3 \, a^{2} \log \left (b x^{5} + a\right )}{5 \, b^{4}} + \frac{b x^{10} - 4 \, a x^{5}}{10 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5793, size = 147, normalized size = 2.49 \begin{align*} \frac{b^{3} x^{15} - 3 \, a b^{2} x^{10} - 4 \, a^{2} b x^{5} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{5} + a^{3}\right )} \log \left (b x^{5} + a\right )}{10 \,{\left (b^{5} x^{5} + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.949049, size = 56, normalized size = 0.95 \begin{align*} \frac{a^{3}}{5 a b^{4} + 5 b^{5} x^{5}} + \frac{3 a^{2} \log{\left (a + b x^{5} \right )}}{5 b^{4}} - \frac{2 a x^{5}}{5 b^{3}} + \frac{x^{10}}{10 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19982, size = 90, normalized size = 1.53 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, b^{4}} + \frac{b^{2} x^{10} - 4 \, a b x^{5}}{10 \, b^{4}} - \frac{3 \, a^{2} b x^{5} + 2 \, a^{3}}{5 \,{\left (b x^{5} + a\right )} b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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