Optimal. Leaf size=95 \[ 7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]
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Rubi [A] time = 0.0389603, 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} \[ 7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]
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^5 \, dx &=\int \frac{(b+a x)^8}{x^3} \, dx\\ &=\int \left (56 a^3 b^5+\frac{b^8}{x^3}+\frac{8 a b^7}{x^2}+\frac{28 a^2 b^6}{x}+70 a^4 b^4 x+56 a^5 b^3 x^2+28 a^6 b^2 x^3+8 a^7 b x^4+a^8 x^5\right ) \, dx\\ &=-\frac{b^8}{2 x^2}-\frac{8 a b^7}{x}+56 a^3 b^5 x+35 a^4 b^4 x^2+\frac{56}{3} a^5 b^3 x^3+7 a^6 b^2 x^4+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}+28 a^2 b^6 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0045973, size = 95, normalized size = 1. \[ 7 a^6 b^2 x^4+\frac{56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac{8}{5} a^7 b x^5+\frac{a^8 x^6}{6}-\frac{8 a b^7}{x}-\frac{b^8}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{2\,{x}^{2}}}-8\,{\frac{{b}^{7}a}{x}}+56\,{a}^{3}{b}^{5}x+35\,{a}^{4}{b}^{4}{x}^{2}+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+7\,{a}^{6}{b}^{2}{x}^{4}+{\frac{8\,{a}^{7}b{x}^{5}}{5}}+{\frac{{a}^{8}{x}^{6}}{6}}+28\,{a}^{2}{b}^{6}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955887, size = 116, normalized size = 1.22 \begin{align*} \frac{1}{6} \, a^{8} x^{6} + \frac{8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \left (x\right ) - \frac{16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43349, size = 215, normalized size = 2.26 \begin{align*} \frac{5 \, a^{8} x^{8} + 48 \, a^{7} b x^{7} + 210 \, a^{6} b^{2} x^{6} + 560 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 1680 \, a^{3} b^{5} x^{3} + 840 \, a^{2} b^{6} x^{2} \log \left (x\right ) - 240 \, a b^{7} x - 15 \, b^{8}}{30 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.389573, size = 95, normalized size = 1. \begin{align*} \frac{a^{8} x^{6}}{6} + \frac{8 a^{7} b x^{5}}{5} + 7 a^{6} b^{2} x^{4} + \frac{56 a^{5} b^{3} x^{3}}{3} + 35 a^{4} b^{4} x^{2} + 56 a^{3} b^{5} x + 28 a^{2} b^{6} \log{\left (x \right )} - \frac{16 a b^{7} x + b^{8}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.56671, size = 117, normalized size = 1.23 \begin{align*} \frac{1}{6} \, a^{8} x^{6} + \frac{8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \left ({\left | x \right |}\right ) - \frac{16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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