Optimal. Leaf size=95 \[ 14 a^6 b^2 x^2-\frac{14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac{56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]
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Rubi [A] time = 0.0392652, antiderivative size = 95, 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 = {263, 43} \[ 14 a^6 b^2 x^2-\frac{14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac{56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^8 x^3 \, dx &=\int \frac{(b+a x)^8}{x^5} \, dx\\ &=\int \left (56 a^5 b^3+\frac{b^8}{x^5}+\frac{8 a b^7}{x^4}+\frac{28 a^2 b^6}{x^3}+\frac{56 a^3 b^5}{x^2}+\frac{70 a^4 b^4}{x}+28 a^6 b^2 x+8 a^7 b x^2+a^8 x^3\right ) \, dx\\ &=-\frac{b^8}{4 x^4}-\frac{8 a b^7}{3 x^3}-\frac{14 a^2 b^6}{x^2}-\frac{56 a^3 b^5}{x}+56 a^5 b^3 x+14 a^6 b^2 x^2+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}+70 a^4 b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0047304, size = 95, normalized size = 1. \[ 14 a^6 b^2 x^2-\frac{14 a^2 b^6}{x^2}+56 a^5 b^3 x-\frac{56 a^3 b^5}{x}+70 a^4 b^4 \log (x)+\frac{8}{3} a^7 b x^3+\frac{a^8 x^4}{4}-\frac{8 a b^7}{3 x^3}-\frac{b^8}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{4\,{x}^{4}}}-{\frac{8\,{b}^{7}a}{3\,{x}^{3}}}-14\,{\frac{{a}^{2}{b}^{6}}{{x}^{2}}}-56\,{\frac{{a}^{3}{b}^{5}}{x}}+56\,{a}^{5}{b}^{3}x+14\,{a}^{6}{b}^{2}{x}^{2}+{\frac{8\,{a}^{7}b{x}^{3}}{3}}+{\frac{{a}^{8}{x}^{4}}{4}}+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 = 1.00472, size = 119, normalized size = 1.25 \begin{align*} \frac{1}{4} \, a^{8} x^{4} + \frac{8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \left (x\right ) - \frac{672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43669, size = 209, normalized size = 2.2 \begin{align*} \frac{3 \, a^{8} x^{8} + 32 \, a^{7} b x^{7} + 168 \, a^{6} b^{2} x^{6} + 672 \, a^{5} b^{3} x^{5} + 840 \, a^{4} b^{4} x^{4} \log \left (x\right ) - 672 \, a^{3} b^{5} x^{3} - 168 \, a^{2} b^{6} x^{2} - 32 \, a b^{7} x - 3 \, b^{8}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.479598, size = 95, normalized size = 1. \begin{align*} \frac{a^{8} x^{4}}{4} + \frac{8 a^{7} b x^{3}}{3} + 14 a^{6} b^{2} x^{2} + 56 a^{5} b^{3} x + 70 a^{4} b^{4} \log{\left (x \right )} - \frac{672 a^{3} b^{5} x^{3} + 168 a^{2} b^{6} x^{2} + 32 a b^{7} x + 3 b^{8}}{12 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11281, size = 120, normalized size = 1.26 \begin{align*} \frac{1}{4} \, a^{8} x^{4} + \frac{8}{3} \, a^{7} b x^{3} + 14 \, a^{6} b^{2} x^{2} + 56 \, a^{5} b^{3} x + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac{672 \, a^{3} b^{5} x^{3} + 168 \, a^{2} b^{6} x^{2} + 32 \, a b^{7} x + 3 \, b^{8}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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