Optimal. Leaf size=74 \[ -\frac{a^4}{4 b^5 \left (a+b x^2\right )^2}+\frac{2 a^3}{b^5 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{x^4}{4 b^3} \]
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Rubi [A] time = 0.0581404, antiderivative size = 74, 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} \[ -\frac{a^4}{4 b^5 \left (a+b x^2\right )^2}+\frac{2 a^3}{b^5 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{x^4}{4 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+b x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(a+b x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{3 a}{b^4}+\frac{x}{b^3}+\frac{a^4}{b^4 (a+b x)^3}-\frac{4 a^3}{b^4 (a+b x)^2}+\frac{6 a^2}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{3 a x^2}{2 b^4}+\frac{x^4}{4 b^3}-\frac{a^4}{4 b^5 \left (a+b x^2\right )^2}+\frac{2 a^3}{b^5 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{b^5}\\ \end{align*}
Mathematica [A] time = 0.0229814, size = 63, normalized size = 0.85 \[ \frac{-\frac{a^4}{\left (a+b x^2\right )^2}+\frac{8 a^3}{a+b x^2}+12 a^2 \log \left (a+b x^2\right )-6 a b x^2+b^2 x^4}{4 b^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 69, normalized size = 0.9 \begin{align*} -{\frac{3\,a{x}^{2}}{2\,{b}^{4}}}+{\frac{{x}^{4}}{4\,{b}^{3}}}-{\frac{{a}^{4}}{4\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+2\,{\frac{{a}^{3}}{{b}^{5} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{{b}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.06585, size = 104, normalized size = 1.41 \begin{align*} \frac{8 \, a^{3} b x^{2} + 7 \, a^{4}}{4 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} + \frac{3 \, a^{2} \log \left (b x^{2} + a\right )}{b^{5}} + \frac{b x^{4} - 6 \, a x^{2}}{4 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.13331, size = 211, normalized size = 2.85 \begin{align*} \frac{b^{4} x^{8} - 4 \, a b^{3} x^{6} - 11 \, a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + 7 \, a^{4} + 12 \,{\left (a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{7} x^{4} + 2 \, a b^{6} x^{2} + a^{2} b^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.561601, size = 78, normalized size = 1.05 \begin{align*} \frac{3 a^{2} \log{\left (a + b x^{2} \right )}}{b^{5}} - \frac{3 a x^{2}}{2 b^{4}} + \frac{7 a^{4} + 8 a^{3} b x^{2}}{4 a^{2} b^{5} + 8 a b^{6} x^{2} + 4 b^{7} x^{4}} + \frac{x^{4}}{4 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.63261, size = 108, normalized size = 1.46 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{b^{5}} + \frac{b^{3} x^{4} - 6 \, a b^{2} x^{2}}{4 \, b^{6}} - \frac{18 \, a^{2} b^{2} x^{4} + 28 \, a^{3} b x^{2} + 11 \, a^{4}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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