Optimal. Leaf size=100 \[ -\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{x}+a^8 \log (x)-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]
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Rubi [A] time = 0.0420873, antiderivative size = 100, 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} \[ -\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{x}+a^8 \log (x)-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^8}{x} \, dx &=\int \frac{(b+a x)^8}{x^9} \, dx\\ &=\int \left (\frac{b^8}{x^9}+\frac{8 a b^7}{x^8}+\frac{28 a^2 b^6}{x^7}+\frac{56 a^3 b^5}{x^6}+\frac{70 a^4 b^4}{x^5}+\frac{56 a^5 b^3}{x^4}+\frac{28 a^6 b^2}{x^3}+\frac{8 a^7 b}{x^2}+\frac{a^8}{x}\right ) \, dx\\ &=-\frac{b^8}{8 x^8}-\frac{8 a b^7}{7 x^7}-\frac{14 a^2 b^6}{3 x^6}-\frac{56 a^3 b^5}{5 x^5}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^5 b^3}{3 x^3}-\frac{14 a^6 b^2}{x^2}-\frac{8 a^7 b}{x}+a^8 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0041736, size = 100, normalized size = 1. \[ -\frac{14 a^6 b^2}{x^2}-\frac{56 a^5 b^3}{3 x^3}-\frac{35 a^4 b^4}{2 x^4}-\frac{56 a^3 b^5}{5 x^5}-\frac{14 a^2 b^6}{3 x^6}-\frac{8 a^7 b}{x}+a^8 \log (x)-\frac{8 a b^7}{7 x^7}-\frac{b^8}{8 x^8} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{8\,{x}^{8}}}-{\frac{8\,{b}^{7}a}{7\,{x}^{7}}}-{\frac{14\,{a}^{2}{b}^{6}}{3\,{x}^{6}}}-{\frac{56\,{a}^{3}{b}^{5}}{5\,{x}^{5}}}-{\frac{35\,{a}^{4}{b}^{4}}{2\,{x}^{4}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,{x}^{3}}}-14\,{\frac{{a}^{6}{b}^{2}}{{x}^{2}}}-8\,{\frac{{a}^{7}b}{x}}+{a}^{8}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990892, size = 120, normalized size = 1.2 \begin{align*} a^{8} \log \left (x\right ) - \frac{6720 \, a^{7} b x^{7} + 11760 \, a^{6} b^{2} x^{6} + 15680 \, a^{5} b^{3} x^{5} + 14700 \, a^{4} b^{4} x^{4} + 9408 \, a^{3} b^{5} x^{3} + 3920 \, a^{2} b^{6} x^{2} + 960 \, a b^{7} x + 105 \, b^{8}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42111, size = 231, normalized size = 2.31 \begin{align*} \frac{840 \, a^{8} x^{8} \log \left (x\right ) - 6720 \, a^{7} b x^{7} - 11760 \, a^{6} b^{2} x^{6} - 15680 \, a^{5} b^{3} x^{5} - 14700 \, a^{4} b^{4} x^{4} - 9408 \, a^{3} b^{5} x^{3} - 3920 \, a^{2} b^{6} x^{2} - 960 \, a b^{7} x - 105 \, b^{8}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.71769, size = 94, normalized size = 0.94 \begin{align*} a^{8} \log{\left (x \right )} - \frac{6720 a^{7} b x^{7} + 11760 a^{6} b^{2} x^{6} + 15680 a^{5} b^{3} x^{5} + 14700 a^{4} b^{4} x^{4} + 9408 a^{3} b^{5} x^{3} + 3920 a^{2} b^{6} x^{2} + 960 a b^{7} x + 105 b^{8}}{840 x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09003, size = 122, normalized size = 1.22 \begin{align*} a^{8} \log \left ({\left | x \right |}\right ) - \frac{6720 \, a^{7} b x^{7} + 11760 \, a^{6} b^{2} x^{6} + 15680 \, a^{5} b^{3} x^{5} + 14700 \, a^{4} b^{4} x^{4} + 9408 \, a^{3} b^{5} x^{3} + 3920 \, a^{2} b^{6} x^{2} + 960 \, a b^{7} x + 105 \, b^{8}}{840 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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