Optimal. Leaf size=96 \[ -\frac{a^4 (a x+b)^9}{6435 b^5 x^9}+\frac{a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac{a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac{a (a x+b)^9}{39 b^2 x^{12}}-\frac{(a x+b)^9}{13 b x^{13}} \]
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Rubi [A] time = 0.0274715, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ -\frac{a^4 (a x+b)^9}{6435 b^5 x^9}+\frac{a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac{a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac{a (a x+b)^9}{39 b^2 x^{12}}-\frac{(a x+b)^9}{13 b x^{13}} \]
Antiderivative was successfully verified.
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Rule 263
Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^8}{x^6} \, dx &=\int \frac{(b+a x)^8}{x^{14}} \, dx\\ &=-\frac{(b+a x)^9}{13 b x^{13}}-\frac{(4 a) \int \frac{(b+a x)^8}{x^{13}} \, dx}{13 b}\\ &=-\frac{(b+a x)^9}{13 b x^{13}}+\frac{a (b+a x)^9}{39 b^2 x^{12}}+\frac{a^2 \int \frac{(b+a x)^8}{x^{12}} \, dx}{13 b^2}\\ &=-\frac{(b+a x)^9}{13 b x^{13}}+\frac{a (b+a x)^9}{39 b^2 x^{12}}-\frac{a^2 (b+a x)^9}{143 b^3 x^{11}}-\frac{\left (2 a^3\right ) \int \frac{(b+a x)^8}{x^{11}} \, dx}{143 b^3}\\ &=-\frac{(b+a x)^9}{13 b x^{13}}+\frac{a (b+a x)^9}{39 b^2 x^{12}}-\frac{a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac{a^3 (b+a x)^9}{715 b^4 x^{10}}+\frac{a^4 \int \frac{(b+a x)^8}{x^{10}} \, dx}{715 b^4}\\ &=-\frac{(b+a x)^9}{13 b x^{13}}+\frac{a (b+a x)^9}{39 b^2 x^{12}}-\frac{a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac{a^3 (b+a x)^9}{715 b^4 x^{10}}-\frac{a^4 (b+a x)^9}{6435 b^5 x^9}\\ \end{align*}
Mathematica [A] time = 0.0084798, size = 104, normalized size = 1.08 \[ -\frac{4 a^6 b^2}{x^7}-\frac{7 a^5 b^3}{x^8}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{5 x^{10}}-\frac{28 a^2 b^6}{11 x^{11}}-\frac{4 a^7 b}{3 x^6}-\frac{a^8}{5 x^5}-\frac{2 a b^7}{3 x^{12}}-\frac{b^8}{13 x^{13}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 91, normalized size = 1. \begin{align*} -{\frac{{a}^{8}}{5\,{x}^{5}}}-{\frac{2\,{b}^{7}a}{3\,{x}^{12}}}-{\frac{28\,{a}^{2}{b}^{6}}{11\,{x}^{11}}}-{\frac{28\,{a}^{3}{b}^{5}}{5\,{x}^{10}}}-7\,{\frac{{a}^{5}{b}^{3}}{{x}^{8}}}-{\frac{{b}^{8}}{13\,{x}^{13}}}-{\frac{4\,{a}^{7}b}{3\,{x}^{6}}}-4\,{\frac{{a}^{6}{b}^{2}}{{x}^{7}}}-{\frac{70\,{a}^{4}{b}^{4}}{9\,{x}^{9}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.98433, size = 122, normalized size = 1.27 \begin{align*} -\frac{1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36975, size = 231, normalized size = 2.41 \begin{align*} -\frac{1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.923319, size = 97, normalized size = 1.01 \begin{align*} - \frac{1287 a^{8} x^{8} + 8580 a^{7} b x^{7} + 25740 a^{6} b^{2} x^{6} + 45045 a^{5} b^{3} x^{5} + 50050 a^{4} b^{4} x^{4} + 36036 a^{3} b^{5} x^{3} + 16380 a^{2} b^{6} x^{2} + 4290 a b^{7} x + 495 b^{8}}{6435 x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0964, size = 122, normalized size = 1.27 \begin{align*} -\frac{1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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