Optimal. Leaf size=30 \[ \frac{(a+b x)^5}{5 b^2}-\frac{a (a+b x)^4}{4 b^2} \]
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Rubi [A] time = 0.008693, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{(a+b x)^5}{5 b^2}-\frac{a (a+b x)^4}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x (a+b x)^3 \, dx &=\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx\\ &=-\frac{a (a+b x)^4}{4 b^2}+\frac{(a+b x)^5}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0015622, size = 40, normalized size = 1.33 \[ a^2 b x^3+\frac{a^3 x^2}{2}+\frac{3}{4} a b^2 x^4+\frac{b^3 x^5}{5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 35, normalized size = 1.2 \begin{align*}{\frac{{b}^{3}{x}^{5}}{5}}+{\frac{3\,{b}^{2}a{x}^{4}}{4}}+{a}^{2}b{x}^{3}+{\frac{{a}^{3}{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03847, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36675, size = 74, normalized size = 2.47 \begin{align*} \frac{1}{5} x^{5} b^{3} + \frac{3}{4} x^{4} b^{2} a + x^{3} b a^{2} + \frac{1}{2} x^{2} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.065796, size = 36, normalized size = 1.2 \begin{align*} \frac{a^{3} x^{2}}{2} + a^{2} b x^{3} + \frac{3 a b^{2} x^{4}}{4} + \frac{b^{3} x^{5}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14779, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{5} \, b^{3} x^{5} + \frac{3}{4} \, a b^{2} x^{4} + a^{2} b x^{3} + \frac{1}{2} \, a^{3} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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