Optimal. Leaf size=94 \[ \frac{14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac{2 a^7 b}{x^4}-\frac{a^8}{6 x^6}+a b^7 x^8+\frac{b^8 x^{10}}{10} \]
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Rubi [A] time = 0.0570915, antiderivative size = 94, 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} \[ \frac{14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac{2 a^7 b}{x^4}-\frac{a^8}{6 x^6}+a b^7 x^8+\frac{b^8 x^{10}}{10} \]
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^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^8}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (70 a^4 b^4+\frac{a^8}{x^4}+\frac{8 a^7 b}{x^3}+\frac{28 a^6 b^2}{x^2}+\frac{56 a^5 b^3}{x}+56 a^3 b^5 x+28 a^2 b^6 x^2+8 a b^7 x^3+b^8 x^4\right ) \, dx,x,x^2\right )\\ &=-\frac{a^8}{6 x^6}-\frac{2 a^7 b}{x^4}-\frac{14 a^6 b^2}{x^2}+35 a^4 b^4 x^2+14 a^3 b^5 x^4+\frac{14}{3} a^2 b^6 x^6+a b^7 x^8+\frac{b^8 x^{10}}{10}+56 a^5 b^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0046497, size = 94, normalized size = 1. \[ \frac{14}{3} a^2 b^6 x^6+14 a^3 b^5 x^4+35 a^4 b^4 x^2-\frac{14 a^6 b^2}{x^2}+56 a^5 b^3 \log (x)-\frac{2 a^7 b}{x^4}-\frac{a^8}{6 x^6}+a b^7 x^8+\frac{b^8 x^{10}}{10} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 89, normalized size = 1. \begin{align*} -{\frac{{a}^{8}}{6\,{x}^{6}}}-2\,{\frac{{a}^{7}b}{{x}^{4}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}+35\,{a}^{4}{b}^{4}{x}^{2}+14\,{a}^{3}{b}^{5}{x}^{4}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+a{b}^{7}{x}^{8}+{\frac{{b}^{8}{x}^{10}}{10}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49857, size = 123, normalized size = 1.31 \begin{align*} \frac{1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac{14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac{84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36912, size = 220, normalized size = 2.34 \begin{align*} \frac{3 \, b^{8} x^{16} + 30 \, a b^{7} x^{14} + 140 \, a^{2} b^{6} x^{12} + 420 \, a^{3} b^{5} x^{10} + 1050 \, a^{4} b^{4} x^{8} + 1680 \, a^{5} b^{3} x^{6} \log \left (x\right ) - 420 \, a^{6} b^{2} x^{4} - 60 \, a^{7} b x^{2} - 5 \, a^{8}}{30 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.501642, size = 95, normalized size = 1.01 \begin{align*} 56 a^{5} b^{3} \log{\left (x \right )} + 35 a^{4} b^{4} x^{2} + 14 a^{3} b^{5} x^{4} + \frac{14 a^{2} b^{6} x^{6}}{3} + a b^{7} x^{8} + \frac{b^{8} x^{10}}{10} - \frac{a^{8} + 12 a^{7} b x^{2} + 84 a^{6} b^{2} x^{4}}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 3.88237, size = 138, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b^{8} x^{10} + a b^{7} x^{8} + \frac{14}{3} \, a^{2} b^{6} x^{6} + 14 \, a^{3} b^{5} x^{4} + 35 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} \log \left (x^{2}\right ) - \frac{308 \, a^{5} b^{3} x^{6} + 84 \, a^{6} b^{2} x^{4} + 12 \, a^{7} b x^{2} + a^{8}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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