Optimal. Leaf size=97 \[ -\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{8 x^8}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8} \]
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Rubi [A] time = 0.0568999, antiderivative size = 97, 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} \[ -\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{8 x^8}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (56 a^3 b^5+\frac{a^8}{x^5}+\frac{8 a^7 b}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{56 a^5 b^3}{x^2}+\frac{70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{8 x^8}-\frac{4 a^7 b}{3 x^6}-\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8}+70 a^4 b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0046682, size = 97, normalized size = 1. \[ -\frac{7 a^6 b^2}{x^4}-\frac{28 a^5 b^3}{x^2}+28 a^3 b^5 x^2+7 a^2 b^6 x^4+70 a^4 b^4 \log (x)-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{8 x^8}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^8}{8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{8\,{x}^{8}}}-{\frac{4\,{a}^{7}b}{3\,{x}^{6}}}-7\,{\frac{{a}^{6}{b}^{2}}{{x}^{4}}}-28\,{\frac{{a}^{5}{b}^{3}}{{x}^{2}}}+28\,{a}^{3}{b}^{5}{x}^{2}+7\,{a}^{2}{b}^{6}{x}^{4}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{8}}{8}}+70\,{a}^{4}{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 = 2.33408, size = 127, normalized size = 1.31 \begin{align*} \frac{1}{8} \, b^{8} x^{8} + \frac{4}{3} \, a b^{7} x^{6} + 7 \, a^{2} b^{6} x^{4} + 28 \, a^{3} b^{5} x^{2} + 35 \, a^{4} b^{4} \log \left (x^{2}\right ) - \frac{672 \, a^{5} b^{3} x^{6} + 168 \, a^{6} b^{2} x^{4} + 32 \, a^{7} b x^{2} + 3 \, a^{8}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17767, size = 219, normalized size = 2.26 \begin{align*} \frac{3 \, b^{8} x^{16} + 32 \, a b^{7} x^{14} + 168 \, a^{2} b^{6} x^{12} + 672 \, a^{3} b^{5} x^{10} + 1680 \, a^{4} b^{4} x^{8} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{6} - 168 \, a^{6} b^{2} x^{4} - 32 \, a^{7} b x^{2} - 3 \, a^{8}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.596116, size = 99, normalized size = 1.02 \begin{align*} 70 a^{4} b^{4} \log{\left (x \right )} + 28 a^{3} b^{5} x^{2} + 7 a^{2} b^{6} x^{4} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{8}}{8} - \frac{3 a^{8} + 32 a^{7} b x^{2} + 168 a^{6} b^{2} x^{4} + 672 a^{5} 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 = 2.47352, size = 142, normalized size = 1.46 \begin{align*} \frac{1}{8} \, b^{8} x^{8} + \frac{4}{3} \, a b^{7} x^{6} + 7 \, a^{2} b^{6} x^{4} + 28 \, a^{3} b^{5} x^{2} + 35 \, a^{4} b^{4} \log \left (x^{2}\right ) - \frac{1750 \, a^{4} b^{4} x^{8} + 672 \, a^{5} b^{3} x^{6} + 168 \, a^{6} b^{2} x^{4} + 32 \, a^{7} b x^{2} + 3 \, a^{8}}{24 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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