Optimal. Leaf size=43 \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^6}{6}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^3}{3} \]
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Rubi [A] time = 0.0176521, antiderivative size = 43, 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} \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^6}{6}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 263
Rule 43
Rubi steps
\begin{align*} \int \left (a+\frac{b}{x}\right )^3 x^5 \, dx &=\int x^2 (b+a x)^3 \, dx\\ &=\int \left (b^3 x^2+3 a b^2 x^3+3 a^2 b x^4+a^3 x^5\right ) \, dx\\ &=\frac{b^3 x^3}{3}+\frac{3}{4} a b^2 x^4+\frac{3}{5} a^2 b x^5+\frac{a^3 x^6}{6}\\ \end{align*}
Mathematica [A] time = 0.0016893, size = 43, normalized size = 1. \[ \frac{3}{5} a^2 b x^5+\frac{a^3 x^6}{6}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3}}+{\frac{3\,a{b}^{2}{x}^{4}}{4}}+{\frac{3\,{a}^{2}b{x}^{5}}{5}}+{\frac{{a}^{3}{x}^{6}}{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973878, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{6} \, a^{3} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.40607, size = 80, normalized size = 1.86 \begin{align*} \frac{1}{6} \, a^{3} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.063416, size = 39, normalized size = 0.91 \begin{align*} \frac{a^{3} x^{6}}{6} + \frac{3 a^{2} b x^{5}}{5} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17773, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{6} \, a^{3} x^{6} + \frac{3}{5} \, a^{2} b x^{5} + \frac{3}{4} \, a b^{2} x^{4} + \frac{1}{3} \, b^{3} x^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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