Optimal. Leaf size=99 \[ -\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac{b^8 x^2}{2} \]
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Rubi [A] time = 0.0514795, antiderivative size = 99, 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} \[ -\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac{b^8 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^8}{x^{15}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^8} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (b^8+\frac{a^8}{x^8}+\frac{8 a^7 b}{x^7}+\frac{28 a^6 b^2}{x^6}+\frac{56 a^5 b^3}{x^5}+\frac{70 a^4 b^4}{x^4}+\frac{56 a^3 b^5}{x^3}+\frac{28 a^2 b^6}{x^2}+\frac{8 a b^7}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{14 x^{14}}-\frac{2 a^7 b}{3 x^{12}}-\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}+\frac{b^8 x^2}{2}+8 a b^7 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0045511, size = 99, normalized size = 1. \[ -\frac{14 a^6 b^2}{5 x^{10}}-\frac{7 a^5 b^3}{x^8}-\frac{35 a^4 b^4}{3 x^6}-\frac{14 a^3 b^5}{x^4}-\frac{14 a^2 b^6}{x^2}-\frac{2 a^7 b}{3 x^{12}}-\frac{a^8}{14 x^{14}}+8 a b^7 \log (x)+\frac{b^8 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 90, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{14\,{x}^{14}}}-{\frac{2\,{a}^{7}b}{3\,{x}^{12}}}-{\frac{14\,{a}^{6}{b}^{2}}{5\,{x}^{10}}}-7\,{\frac{{a}^{5}{b}^{3}}{{x}^{8}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,{x}^{6}}}-14\,{\frac{{a}^{3}{b}^{5}}{{x}^{4}}}-14\,{\frac{{a}^{2}{b}^{6}}{{x}^{2}}}+{\frac{{b}^{8}{x}^{2}}{2}}+8\,a{b}^{7}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.33088, size = 127, normalized size = 1.28 \begin{align*} \frac{1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac{2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27335, size = 234, normalized size = 2.36 \begin{align*} \frac{105 \, b^{8} x^{16} + 1680 \, a b^{7} x^{14} \log \left (x\right ) - 2940 \, a^{2} b^{6} x^{12} - 2940 \, a^{3} b^{5} x^{10} - 2450 \, a^{4} b^{4} x^{8} - 1470 \, a^{5} b^{3} x^{6} - 588 \, a^{6} b^{2} x^{4} - 140 \, a^{7} b x^{2} - 15 \, a^{8}}{210 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.955497, size = 97, normalized size = 0.98 \begin{align*} 8 a b^{7} \log{\left (x \right )} + \frac{b^{8} x^{2}}{2} - \frac{15 a^{8} + 140 a^{7} b x^{2} + 588 a^{6} b^{2} x^{4} + 1470 a^{5} b^{3} x^{6} + 2450 a^{4} b^{4} x^{8} + 2940 a^{3} b^{5} x^{10} + 2940 a^{2} b^{6} x^{12}}{210 x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.27729, size = 139, normalized size = 1.4 \begin{align*} \frac{1}{2} \, b^{8} x^{2} + 4 \, a b^{7} \log \left (x^{2}\right ) - \frac{2178 \, a b^{7} x^{14} + 2940 \, a^{2} b^{6} x^{12} + 2940 \, a^{3} b^{5} x^{10} + 2450 \, a^{4} b^{4} x^{8} + 1470 \, a^{5} b^{3} x^{6} + 588 \, a^{6} b^{2} x^{4} + 140 \, a^{7} b x^{2} + 15 \, a^{8}}{210 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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