Optimal. Leaf size=66 \[ \frac{a^2 x^4}{4 b^3}-\frac{a^3 x^2}{2 b^4}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b} \]
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Rubi [A] time = 0.0443748, antiderivative size = 66, 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} \[ \frac{a^2 x^4}{4 b^3}-\frac{a^3 x^2}{2 b^4}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^9}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a^3}{b^4}+\frac{a^2 x}{b^3}-\frac{a x^2}{b^2}+\frac{x^3}{b}+\frac{a^4}{b^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^3 x^2}{2 b^4}+\frac{a^2 x^4}{4 b^3}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}\\ \end{align*}
Mathematica [A] time = 0.0055564, size = 66, normalized size = 1. \[ \frac{a^2 x^4}{4 b^3}-\frac{a^3 x^2}{2 b^4}+\frac{a^4 \log \left (a+b x^2\right )}{2 b^5}-\frac{a x^6}{6 b^2}+\frac{x^8}{8 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{3}{x}^{2}}{2\,{b}^{4}}}+{\frac{{a}^{2}{x}^{4}}{4\,{b}^{3}}}-{\frac{a{x}^{6}}{6\,{b}^{2}}}+{\frac{{x}^{8}}{8\,b}}+{\frac{{a}^{4}\ln \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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Maxima [A] time = 2.28757, size = 77, normalized size = 1.17 \begin{align*} \frac{a^{4} \log \left (b x^{2} + a\right )}{2 \, b^{5}} + \frac{3 \, b^{3} x^{8} - 4 \, a b^{2} x^{6} + 6 \, a^{2} b x^{4} - 12 \, a^{3} x^{2}}{24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28839, size = 123, normalized size = 1.86 \begin{align*} \frac{3 \, b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 12 \, a^{3} b x^{2} + 12 \, a^{4} \log \left (b x^{2} + a\right )}{24 \, b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.309818, size = 56, normalized size = 0.85 \begin{align*} \frac{a^{4} \log{\left (a + b x^{2} \right )}}{2 b^{5}} - \frac{a^{3} x^{2}}{2 b^{4}} + \frac{a^{2} x^{4}}{4 b^{3}} - \frac{a x^{6}}{6 b^{2}} + \frac{x^{8}}{8 b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.39218, size = 78, normalized size = 1.18 \begin{align*} \frac{a^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{5}} + \frac{3 \, b^{3} x^{8} - 4 \, a b^{2} x^{6} + 6 \, a^{2} b x^{4} - 12 \, a^{3} x^{2}}{24 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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