Optimal. Leaf size=64 \[ -\frac{5 a^3 b^2}{2 x^4}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{8 x^8}+5 a b^4 \log (x)+\frac{b^5 x^2}{2} \]
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Rubi [A] time = 0.0328172, antiderivative size = 64, 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} \[ -\frac{5 a^3 b^2}{2 x^4}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{8 x^8}+5 a b^4 \log (x)+\frac{b^5 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^5}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b^5+\frac{a^5}{x^5}+\frac{5 a^4 b}{x^4}+\frac{10 a^3 b^2}{x^3}+\frac{10 a^2 b^3}{x^2}+\frac{5 a b^4}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^5}{8 x^8}-\frac{5 a^4 b}{6 x^6}-\frac{5 a^3 b^2}{2 x^4}-\frac{5 a^2 b^3}{x^2}+\frac{b^5 x^2}{2}+5 a b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0042871, size = 64, normalized size = 1. \[ -\frac{5 a^3 b^2}{2 x^4}-\frac{5 a^2 b^3}{x^2}-\frac{5 a^4 b}{6 x^6}-\frac{a^5}{8 x^8}+5 a b^4 \log (x)+\frac{b^5 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 57, normalized size = 0.9 \begin{align*} -{\frac{{a}^{5}}{8\,{x}^{8}}}-{\frac{5\,{a}^{4}b}{6\,{x}^{6}}}-{\frac{5\,{a}^{3}{b}^{2}}{2\,{x}^{4}}}-5\,{\frac{{a}^{2}{b}^{3}}{{x}^{2}}}+{\frac{{b}^{5}{x}^{2}}{2}}+5\,a{b}^{4}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58576, size = 82, normalized size = 1.28 \begin{align*} \frac{1}{2} \, b^{5} x^{2} + \frac{5}{2} \, a b^{4} \log \left (x^{2}\right ) - \frac{120 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 20 \, a^{4} b x^{2} + 3 \, a^{5}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.29682, size = 142, normalized size = 2.22 \begin{align*} \frac{12 \, b^{5} x^{10} + 120 \, a b^{4} x^{8} \log \left (x\right ) - 120 \, a^{2} b^{3} x^{6} - 60 \, a^{3} b^{2} x^{4} - 20 \, a^{4} b x^{2} - 3 \, a^{5}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.492037, size = 61, normalized size = 0.95 \begin{align*} 5 a b^{4} \log{\left (x \right )} + \frac{b^{5} x^{2}}{2} - \frac{3 a^{5} + 20 a^{4} b x^{2} + 60 a^{3} b^{2} x^{4} + 120 a^{2} b^{3} x^{6}}{24 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.11403, size = 95, normalized size = 1.48 \begin{align*} \frac{1}{2} \, b^{5} x^{2} + \frac{5}{2} \, a b^{4} \log \left (x^{2}\right ) - \frac{125 \, a b^{4} x^{8} + 120 \, a^{2} b^{3} x^{6} + 60 \, a^{3} b^{2} x^{4} + 20 \, a^{4} b x^{2} + 3 \, a^{5}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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