Optimal. Leaf size=69 \[ \frac{2}{3} a^2 b^3 x^{15}+\frac{10}{13} a^3 b^2 x^{13}+\frac{5}{11} a^4 b x^{11}+\frac{a^5 x^9}{9}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{19}}{19} \]
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Rubi [A] time = 0.0248095, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {270} \[ \frac{2}{3} a^2 b^3 x^{15}+\frac{10}{13} a^3 b^2 x^{13}+\frac{5}{11} a^4 b x^{11}+\frac{a^5 x^9}{9}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{19}}{19} \]
Antiderivative was successfully verified.
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Rule 270
Rubi steps
\begin{align*} \int x^8 \left (a+b x^2\right )^5 \, dx &=\int \left (a^5 x^8+5 a^4 b x^{10}+10 a^3 b^2 x^{12}+10 a^2 b^3 x^{14}+5 a b^4 x^{16}+b^5 x^{18}\right ) \, dx\\ &=\frac{a^5 x^9}{9}+\frac{5}{11} a^4 b x^{11}+\frac{10}{13} a^3 b^2 x^{13}+\frac{2}{3} a^2 b^3 x^{15}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{19}}{19}\\ \end{align*}
Mathematica [A] time = 0.0021935, size = 69, normalized size = 1. \[ \frac{2}{3} a^2 b^3 x^{15}+\frac{10}{13} a^3 b^2 x^{13}+\frac{5}{11} a^4 b x^{11}+\frac{a^5 x^9}{9}+\frac{5}{17} a b^4 x^{17}+\frac{b^5 x^{19}}{19} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 58, normalized size = 0.8 \begin{align*}{\frac{{a}^{5}{x}^{9}}{9}}+{\frac{5\,{a}^{4}b{x}^{11}}{11}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{13}}{13}}+{\frac{2\,{a}^{2}{b}^{3}{x}^{15}}{3}}+{\frac{5\,a{b}^{4}{x}^{17}}{17}}+{\frac{{b}^{5}{x}^{19}}{19}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.21767, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{19} \, b^{5} x^{19} + \frac{5}{17} \, a b^{4} x^{17} + \frac{2}{3} \, a^{2} b^{3} x^{15} + \frac{10}{13} \, a^{3} b^{2} x^{13} + \frac{5}{11} \, a^{4} b x^{11} + \frac{1}{9} \, a^{5} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18403, size = 142, normalized size = 2.06 \begin{align*} \frac{1}{19} x^{19} b^{5} + \frac{5}{17} x^{17} b^{4} a + \frac{2}{3} x^{15} b^{3} a^{2} + \frac{10}{13} x^{13} b^{2} a^{3} + \frac{5}{11} x^{11} b a^{4} + \frac{1}{9} x^{9} a^{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.086702, size = 66, normalized size = 0.96 \begin{align*} \frac{a^{5} x^{9}}{9} + \frac{5 a^{4} b x^{11}}{11} + \frac{10 a^{3} b^{2} x^{13}}{13} + \frac{2 a^{2} b^{3} x^{15}}{3} + \frac{5 a b^{4} x^{17}}{17} + \frac{b^{5} x^{19}}{19} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.48448, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{19} \, b^{5} x^{19} + \frac{5}{17} \, a b^{4} x^{17} + \frac{2}{3} \, a^{2} b^{3} x^{15} + \frac{10}{13} \, a^{3} b^{2} x^{13} + \frac{5}{11} \, a^{4} b x^{11} + \frac{1}{9} \, a^{5} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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