Optimal. Leaf size=106 \[ \frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]
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Rubi [A] time = 0.0569334, 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} \[ \frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]
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^{16} \, dx &=\int x^8 (b+a x)^8 \, dx\\ &=\int \left (b^8 x^8+8 a b^7 x^9+28 a^2 b^6 x^{10}+56 a^3 b^5 x^{11}+70 a^4 b^4 x^{12}+56 a^5 b^3 x^{13}+28 a^6 b^2 x^{14}+8 a^7 b x^{15}+a^8 x^{16}\right ) \, dx\\ &=\frac{b^8 x^9}{9}+\frac{4}{5} a b^7 x^{10}+\frac{28}{11} a^2 b^6 x^{11}+\frac{14}{3} a^3 b^5 x^{12}+\frac{70}{13} a^4 b^4 x^{13}+4 a^5 b^3 x^{14}+\frac{28}{15} a^6 b^2 x^{15}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}\\ \end{align*}
Mathematica [A] time = 0.0027548, size = 106, normalized size = 1. \[ \frac{28}{15} a^6 b^2 x^{15}+4 a^5 b^3 x^{14}+\frac{70}{13} a^4 b^4 x^{13}+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{11} a^2 b^6 x^{11}+\frac{1}{2} a^7 b x^{16}+\frac{a^8 x^{17}}{17}+\frac{4}{5} a b^7 x^{10}+\frac{b^8 x^9}{9} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 91, normalized size = 0.9 \begin{align*}{\frac{{b}^{8}{x}^{9}}{9}}+{\frac{4\,a{b}^{7}{x}^{10}}{5}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{11}}{11}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{13}}{13}}+4\,{a}^{5}{b}^{3}{x}^{14}+{\frac{28\,{a}^{6}{b}^{2}{x}^{15}}{15}}+{\frac{{a}^{7}b{x}^{16}}{2}}+{\frac{{a}^{8}{x}^{17}}{17}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959201, size = 122, normalized size = 1.15 \begin{align*} \frac{1}{17} \, a^{8} x^{17} + \frac{1}{2} \, a^{7} b x^{16} + \frac{28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{4}{5} \, a b^{7} x^{10} + \frac{1}{9} \, b^{8} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41615, size = 220, normalized size = 2.08 \begin{align*} \frac{1}{17} \, a^{8} x^{17} + \frac{1}{2} \, a^{7} b x^{16} + \frac{28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{4}{5} \, a b^{7} x^{10} + \frac{1}{9} \, b^{8} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.081245, size = 104, normalized size = 0.98 \begin{align*} \frac{a^{8} x^{17}}{17} + \frac{a^{7} b x^{16}}{2} + \frac{28 a^{6} b^{2} x^{15}}{15} + 4 a^{5} b^{3} x^{14} + \frac{70 a^{4} b^{4} x^{13}}{13} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{11}}{11} + \frac{4 a b^{7} x^{10}}{5} + \frac{b^{8} x^{9}}{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1983, size = 122, normalized size = 1.15 \begin{align*} \frac{1}{17} \, a^{8} x^{17} + \frac{1}{2} \, a^{7} b x^{16} + \frac{28}{15} \, a^{6} b^{2} x^{15} + 4 \, a^{5} b^{3} x^{14} + \frac{70}{13} \, a^{4} b^{4} x^{13} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{28}{11} \, a^{2} b^{6} x^{11} + \frac{4}{5} \, a b^{7} x^{10} + \frac{1}{9} \, b^{8} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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