Optimal. Leaf size=50 \[ -\frac{3 a^2 b^2}{2 x^4}-\frac{2 a^3 b}{3 x^6}-\frac{a^4}{8 x^8}-\frac{2 a b^3}{x^2}+b^4 \log (x) \]
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Rubi [A] time = 0.0326616, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {28, 266, 43} \[ -\frac{3 a^2 b^2}{2 x^4}-\frac{2 a^3 b}{3 x^6}-\frac{a^4}{8 x^8}-\frac{2 a b^3}{x^2}+b^4 \log (x) \]
Antiderivative was successfully verified.
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Rule 28
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^9} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^4}{x^9} \, dx}{b^4}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^4}{x^5} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^4 b^4}{x^5}+\frac{4 a^3 b^5}{x^4}+\frac{6 a^2 b^6}{x^3}+\frac{4 a b^7}{x^2}+\frac{b^8}{x}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac{a^4}{8 x^8}-\frac{2 a^3 b}{3 x^6}-\frac{3 a^2 b^2}{2 x^4}-\frac{2 a b^3}{x^2}+b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0047331, size = 50, normalized size = 1. \[ -\frac{3 a^2 b^2}{2 x^4}-\frac{2 a^3 b}{3 x^6}-\frac{a^4}{8 x^8}-\frac{2 a b^3}{x^2}+b^4 \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 45, normalized size = 0.9 \begin{align*} -{\frac{{a}^{4}}{8\,{x}^{8}}}-{\frac{2\,{a}^{3}b}{3\,{x}^{6}}}-{\frac{3\,{b}^{2}{a}^{2}}{2\,{x}^{4}}}-2\,{\frac{a{b}^{3}}{{x}^{2}}}+{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 = 0.992852, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{2} \, b^{4} \log \left (x^{2}\right ) - \frac{48 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 16 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70528, size = 115, normalized size = 2.3 \begin{align*} \frac{24 \, b^{4} x^{8} \log \left (x\right ) - 48 \, a b^{3} x^{6} - 36 \, a^{2} b^{2} x^{4} - 16 \, a^{3} b x^{2} - 3 \, a^{4}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.489087, size = 48, normalized size = 0.96 \begin{align*} b^{4} \log{\left (x \right )} - \frac{3 a^{4} + 16 a^{3} b x^{2} + 36 a^{2} b^{2} x^{4} + 48 a 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 = 1.11652, size = 78, normalized size = 1.56 \begin{align*} \frac{1}{2} \, b^{4} \log \left (x^{2}\right ) - \frac{25 \, b^{4} x^{8} + 48 \, a b^{3} x^{6} + 36 \, a^{2} b^{2} x^{4} + 16 \, a^{3} b x^{2} + 3 \, a^{4}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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