Optimal. Leaf size=75 \[ \frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8} \]
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Rubi [A] time = 0.0408999, antiderivative size = 75, 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, 44} \[ \frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^9 \left (a+b x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^5 (a+b x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a x^5}-\frac{b}{a^2 x^4}+\frac{b^2}{a^3 x^3}-\frac{b^3}{a^4 x^2}+\frac{b^4}{a^5 x}-\frac{b^5}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{8 a x^8}+\frac{b}{6 a^2 x^6}-\frac{b^2}{4 a^3 x^4}+\frac{b^3}{2 a^4 x^2}+\frac{b^4 \log (x)}{a^5}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}\\ \end{align*}
Mathematica [A] time = 0.0065807, size = 75, normalized size = 1. \[ \frac{b^3}{2 a^4 x^2}-\frac{b^2}{4 a^3 x^4}-\frac{b^4 \log \left (a+b x^2\right )}{2 a^5}+\frac{b^4 \log (x)}{a^5}+\frac{b}{6 a^2 x^6}-\frac{1}{8 a x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 66, normalized size = 0.9 \begin{align*} -{\frac{1}{8\,a{x}^{8}}}+{\frac{b}{6\,{a}^{2}{x}^{6}}}-{\frac{{b}^{2}}{4\,{a}^{3}{x}^{4}}}+{\frac{{b}^{3}}{2\,{a}^{4}{x}^{2}}}+{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{5}}}-{\frac{{b}^{4}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.47016, size = 93, normalized size = 1.24 \begin{align*} -\frac{b^{4} \log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac{b^{4} \log \left (x^{2}\right )}{2 \, a^{5}} + \frac{12 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 4 \, a^{2} b x^{2} - 3 \, a^{3}}{24 \, a^{4} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25234, size = 159, normalized size = 2.12 \begin{align*} -\frac{12 \, b^{4} x^{8} \log \left (b x^{2} + a\right ) - 24 \, b^{4} x^{8} \log \left (x\right ) - 12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{5} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.683919, size = 68, normalized size = 0.91 \begin{align*} \frac{- 3 a^{3} + 4 a^{2} b x^{2} - 6 a b^{2} x^{4} + 12 b^{3} x^{6}}{24 a^{4} x^{8}} + \frac{b^{4} \log{\left (x \right )}}{a^{5}} - \frac{b^{4} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.74214, size = 109, normalized size = 1.45 \begin{align*} \frac{b^{4} \log \left (x^{2}\right )}{2 \, a^{5}} - \frac{b^{4} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5}} - \frac{25 \, b^{4} x^{8} - 12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, a^{5} x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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