Optimal. Leaf size=73 \[ -\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac{a^5 b}{x^6}-\frac{a^6}{8 x^8}+3 a b^5 x^2+\frac{b^6 x^4}{4} \]
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Rubi [A] time = 0.0520787, antiderivative size = 73, 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} \[ -\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac{a^5 b}{x^6}-\frac{a^6}{8 x^8}+3 a b^5 x^2+\frac{b^6 x^4}{4} \]
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 )^3}{x^9} \, dx &=\frac{\int \frac{\left (a b+b^2 x^2\right )^6}{x^9} \, dx}{b^6}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (a b+b^2 x\right )^6}{x^5} \, dx,x,x^2\right )}{2 b^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (6 a b^{11}+\frac{a^6 b^6}{x^5}+\frac{6 a^5 b^7}{x^4}+\frac{15 a^4 b^8}{x^3}+\frac{20 a^3 b^9}{x^2}+\frac{15 a^2 b^{10}}{x}+b^{12} x\right ) \, dx,x,x^2\right )}{2 b^6}\\ &=-\frac{a^6}{8 x^8}-\frac{a^5 b}{x^6}-\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+3 a b^5 x^2+\frac{b^6 x^4}{4}+15 a^2 b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0080131, size = 73, normalized size = 1. \[ -\frac{15 a^4 b^2}{4 x^4}-\frac{10 a^3 b^3}{x^2}+15 a^2 b^4 \log (x)-\frac{a^5 b}{x^6}-\frac{a^6}{8 x^8}+3 a b^5 x^2+\frac{b^6 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 68, normalized size = 0.9 \begin{align*} -{\frac{{a}^{6}}{8\,{x}^{8}}}-{\frac{{a}^{5}b}{{x}^{6}}}-{\frac{15\,{a}^{4}{b}^{2}}{4\,{x}^{4}}}-10\,{\frac{{a}^{3}{b}^{3}}{{x}^{2}}}+3\,a{b}^{5}{x}^{2}+{\frac{{b}^{6}{x}^{4}}{4}}+15\,{a}^{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 = 0.983707, size = 95, normalized size = 1.3 \begin{align*} \frac{1}{4} \, b^{6} x^{4} + 3 \, a b^{5} x^{2} + \frac{15}{2} \, a^{2} b^{4} \log \left (x^{2}\right ) - \frac{80 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} + a^{6}}{8 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69756, size = 158, normalized size = 2.16 \begin{align*} \frac{2 \, b^{6} x^{12} + 24 \, a b^{5} x^{10} + 120 \, a^{2} b^{4} x^{8} \log \left (x\right ) - 80 \, a^{3} b^{3} x^{6} - 30 \, a^{4} b^{2} x^{4} - 8 \, a^{5} b x^{2} - a^{6}}{8 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.527583, size = 71, normalized size = 0.97 \begin{align*} 15 a^{2} b^{4} \log{\left (x \right )} + 3 a b^{5} x^{2} + \frac{b^{6} x^{4}}{4} - \frac{a^{6} + 8 a^{5} b x^{2} + 30 a^{4} b^{2} x^{4} + 80 a^{3} b^{3} x^{6}}{8 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16452, size = 109, normalized size = 1.49 \begin{align*} \frac{1}{4} \, b^{6} x^{4} + 3 \, a b^{5} x^{2} + \frac{15}{2} \, a^{2} b^{4} \log \left (x^{2}\right ) - \frac{125 \, a^{2} b^{4} x^{8} + 80 \, a^{3} b^{3} x^{6} + 30 \, a^{4} b^{2} x^{4} + 8 \, a^{5} b x^{2} + a^{6}}{8 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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