Optimal. Leaf size=81 \[ \frac{6 b^2 (a x+b)^{11}}{11 a^5}-\frac{2 b^3 (a x+b)^{10}}{5 a^5}+\frac{b^4 (a x+b)^9}{9 a^5}+\frac{(a x+b)^{13}}{13 a^5}-\frac{b (a x+b)^{12}}{3 a^5} \]
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Rubi [A] time = 0.0376675, antiderivative size = 81, 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} \[ \frac{6 b^2 (a x+b)^{11}}{11 a^5}-\frac{2 b^3 (a x+b)^{10}}{5 a^5}+\frac{b^4 (a x+b)^9}{9 a^5}+\frac{(a x+b)^{13}}{13 a^5}-\frac{b (a x+b)^{12}}{3 a^5} \]
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^{12} \, dx &=\int x^4 (b+a x)^8 \, dx\\ &=\int \left (\frac{b^4 (b+a x)^8}{a^4}-\frac{4 b^3 (b+a x)^9}{a^4}+\frac{6 b^2 (b+a x)^{10}}{a^4}-\frac{4 b (b+a x)^{11}}{a^4}+\frac{(b+a x)^{12}}{a^4}\right ) \, dx\\ &=\frac{b^4 (b+a x)^9}{9 a^5}-\frac{2 b^3 (b+a x)^{10}}{5 a^5}+\frac{6 b^2 (b+a x)^{11}}{11 a^5}-\frac{b (b+a x)^{12}}{3 a^5}+\frac{(b+a x)^{13}}{13 a^5}\\ \end{align*}
Mathematica [A] time = 0.0027473, size = 104, normalized size = 1.28 \[ \frac{28}{11} a^6 b^2 x^{11}+\frac{28}{5} a^5 b^3 x^{10}+\frac{70}{9} a^4 b^4 x^9+7 a^3 b^5 x^8+4 a^2 b^6 x^7+\frac{2}{3} a^7 b x^{12}+\frac{a^8 x^{13}}{13}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^5}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 91, normalized size = 1.1 \begin{align*}{\frac{{a}^{8}{x}^{13}}{13}}+{\frac{2\,{a}^{7}b{x}^{12}}{3}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{11}}{11}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{10}}{5}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+7\,{a}^{3}{b}^{5}{x}^{8}+4\,{b}^{6}{a}^{2}{x}^{7}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{5}}{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970348, size = 122, normalized size = 1.51 \begin{align*} \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37681, size = 208, normalized size = 2.57 \begin{align*} \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.080558, size = 104, normalized size = 1.28 \begin{align*} \frac{a^{8} x^{13}}{13} + \frac{2 a^{7} b x^{12}}{3} + \frac{28 a^{6} b^{2} x^{11}}{11} + \frac{28 a^{5} b^{3} x^{10}}{5} + \frac{70 a^{4} b^{4} x^{9}}{9} + 7 a^{3} b^{5} x^{8} + 4 a^{2} b^{6} x^{7} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17509, size = 122, normalized size = 1.51 \begin{align*} \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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