Optimal. Leaf size=34 \[ \frac{\left (a+b x^3\right )^5}{15 b^2}-\frac{a \left (a+b x^3\right )^4}{12 b^2} \]
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Rubi [A] time = 0.0304055, antiderivative size = 34, 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 = {266, 43} \[ \frac{\left (a+b x^3\right )^5}{15 b^2}-\frac{a \left (a+b x^3\right )^4}{12 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \left (a+b x^3\right )^3 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (a+b x^3\right )^4}{12 b^2}+\frac{\left (a+b x^3\right )^5}{15 b^2}\\ \end{align*}
Mathematica [A] time = 0.0016564, size = 43, normalized size = 1.26 \[ \frac{1}{3} a^2 b x^9+\frac{a^3 x^6}{6}+\frac{1}{4} a b^2 x^{12}+\frac{b^3 x^{15}}{15} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 1.1 \begin{align*}{\frac{{b}^{3}{x}^{15}}{15}}+{\frac{a{b}^{2}{x}^{12}}{4}}+{\frac{{a}^{2}b{x}^{9}}{3}}+{\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 = 0.949532, size = 47, normalized size = 1.38 \begin{align*} \frac{1}{15} \, b^{3} x^{15} + \frac{1}{4} \, a b^{2} x^{12} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{6} \, a^{3} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.536, size = 84, normalized size = 2.47 \begin{align*} \frac{1}{15} x^{15} b^{3} + \frac{1}{4} x^{12} b^{2} a + \frac{1}{3} x^{9} b a^{2} + \frac{1}{6} x^{6} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.073353, size = 36, normalized size = 1.06 \begin{align*} \frac{a^{3} x^{6}}{6} + \frac{a^{2} b x^{9}}{3} + \frac{a b^{2} x^{12}}{4} + \frac{b^{3} x^{15}}{15} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15908, size = 47, normalized size = 1.38 \begin{align*} \frac{1}{15} \, b^{3} x^{15} + \frac{1}{4} \, a b^{2} x^{12} + \frac{1}{3} \, a^{2} b x^{9} + \frac{1}{6} \, a^{3} x^{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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