Optimal. Leaf size=106 \[ -\frac{7 a^6 b^2}{2 x^8}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a^7 b}{7 x^7}-\frac{a^8}{6 x^6}-\frac{8 a b^7}{13 x^{13}}-\frac{b^8}{14 x^{14}} \]
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Rubi [A] time = 0.0430039, antiderivative size = 106, 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} \[ -\frac{7 a^6 b^2}{2 x^8}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a^7 b}{7 x^7}-\frac{a^8}{6 x^6}-\frac{8 a b^7}{13 x^{13}}-\frac{b^8}{14 x^{14}} \]
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^7} \, dx &=\int \frac{(b+a x)^8}{x^{15}} \, dx\\ &=\int \left (\frac{b^8}{x^{15}}+\frac{8 a b^7}{x^{14}}+\frac{28 a^2 b^6}{x^{13}}+\frac{56 a^3 b^5}{x^{12}}+\frac{70 a^4 b^4}{x^{11}}+\frac{56 a^5 b^3}{x^{10}}+\frac{28 a^6 b^2}{x^9}+\frac{8 a^7 b}{x^8}+\frac{a^8}{x^7}\right ) \, dx\\ &=-\frac{b^8}{14 x^{14}}-\frac{8 a b^7}{13 x^{13}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^6 b^2}{2 x^8}-\frac{8 a^7 b}{7 x^7}-\frac{a^8}{6 x^6}\\ \end{align*}
Mathematica [A] time = 0.0039482, size = 106, normalized size = 1. \[ -\frac{7 a^6 b^2}{2 x^8}-\frac{56 a^5 b^3}{9 x^9}-\frac{7 a^4 b^4}{x^{10}}-\frac{56 a^3 b^5}{11 x^{11}}-\frac{7 a^2 b^6}{3 x^{12}}-\frac{8 a^7 b}{7 x^7}-\frac{a^8}{6 x^6}-\frac{8 a b^7}{13 x^{13}}-\frac{b^8}{14 x^{14}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{14\,{x}^{14}}}-{\frac{8\,{b}^{7}a}{13\,{x}^{13}}}-{\frac{7\,{a}^{2}{b}^{6}}{3\,{x}^{12}}}-{\frac{56\,{a}^{3}{b}^{5}}{11\,{x}^{11}}}-7\,{\frac{{a}^{4}{b}^{4}}{{x}^{10}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,{x}^{9}}}-{\frac{7\,{a}^{6}{b}^{2}}{2\,{x}^{8}}}-{\frac{8\,{a}^{7}b}{7\,{x}^{7}}}-{\frac{{a}^{8}}{6\,{x}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00521, size = 122, normalized size = 1.15 \begin{align*} -\frac{3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37608, size = 239, normalized size = 2.25 \begin{align*} -\frac{3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.964314, size = 97, normalized size = 0.92 \begin{align*} - \frac{3003 a^{8} x^{8} + 20592 a^{7} b x^{7} + 63063 a^{6} b^{2} x^{6} + 112112 a^{5} b^{3} x^{5} + 126126 a^{4} b^{4} x^{4} + 91728 a^{3} b^{5} x^{3} + 42042 a^{2} b^{6} x^{2} + 11088 a b^{7} x + 1287 b^{8}}{18018 x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12183, size = 122, normalized size = 1.15 \begin{align*} -\frac{3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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