Optimal. Leaf size=33 \[ \frac{a}{5 b^2 \left (a+b x^5\right )}+\frac{\log \left (a+b x^5\right )}{5 b^2} \]
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Rubi [A] time = 0.0247409, antiderivative size = 33, 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} \[ \frac{a}{5 b^2 \left (a+b x^5\right )}+\frac{\log \left (a+b x^5\right )}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+b x^5\right )^2} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{x}{(a+b x)^2} \, dx,x,x^5\right )\\ &=\frac{1}{5} \operatorname{Subst}\left (\int \left (-\frac{a}{b (a+b x)^2}+\frac{1}{b (a+b x)}\right ) \, dx,x,x^5\right )\\ &=\frac{a}{5 b^2 \left (a+b x^5\right )}+\frac{\log \left (a+b x^5\right )}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.008115, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+b x^5}+\log \left (a+b x^5\right )}{5 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 30, normalized size = 0.9 \begin{align*}{\frac{a}{5\,{b}^{2} \left ( b{x}^{5}+a \right ) }}+{\frac{\ln \left ( b{x}^{5}+a \right ) }{5\,{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00683, size = 43, normalized size = 1.3 \begin{align*} \frac{a}{5 \,{\left (b^{3} x^{5} + a b^{2}\right )}} + \frac{\log \left (b x^{5} + a\right )}{5 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6439, size = 76, normalized size = 2.3 \begin{align*} \frac{{\left (b x^{5} + a\right )} \log \left (b x^{5} + a\right ) + a}{5 \,{\left (b^{3} x^{5} + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.876122, size = 29, normalized size = 0.88 \begin{align*} \frac{a}{5 a b^{2} + 5 b^{3} x^{5}} + \frac{\log{\left (a + b x^{5} \right )}}{5 b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18154, size = 65, normalized size = 1.97 \begin{align*} -\frac{\frac{\log \left (\frac{{\left | b x^{5} + a \right |}}{{\left (b x^{5} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{5} + a\right )} b}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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