Optimal. Leaf size=49 \[ -\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3} \]
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Rubi [A] time = 0.0377695, antiderivative size = 49, 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} \[ -\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^3}-\frac{2 a}{b^2 (a+b x)^2}+\frac{1}{b^2 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.0183525, size = 39, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+4 b x^2\right )}{\left (a+b x^2\right )^2}+2 \log \left (a+b x^2\right )}{4 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 46, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{a}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.11304, size = 74, normalized size = 1.51 \begin{align*} \frac{4 \, a b x^{2} + 3 \, a^{2}}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15257, size = 143, normalized size = 2.92 \begin{align*} \frac{4 \, a b x^{2} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.468551, size = 53, normalized size = 1.08 \begin{align*} \frac{3 a^{2} + 4 a b x^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.02583, size = 57, normalized size = 1.16 \begin{align*} \frac{\log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} - \frac{3 \, b x^{4} + 2 \, a x^{2}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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