Optimal. Leaf size=34 \[ \frac{3 (a+b x)^{5/3}}{5 b^2}-\frac{3 a (a+b x)^{2/3}}{2 b^2} \]
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Rubi [A] time = 0.0078093, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{3 (a+b x)^{5/3}}{5 b^2}-\frac{3 a (a+b x)^{2/3}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{\sqrt [3]{a+b x}} \, dx &=\int \left (-\frac{a}{b \sqrt [3]{a+b x}}+\frac{(a+b x)^{2/3}}{b}\right ) \, dx\\ &=-\frac{3 a (a+b x)^{2/3}}{2 b^2}+\frac{3 (a+b x)^{5/3}}{5 b^2}\\ \end{align*}
Mathematica [A] time = 0.0273163, size = 24, normalized size = 0.71 \[ \frac{3 (a+b x)^{2/3} (2 b x-3 a)}{10 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-6\,bx+9\,a}{10\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08437, size = 35, normalized size = 1.03 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{5}{3}}}{5 \, b^{2}} - \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}} a}{2 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64726, size = 54, normalized size = 1.59 \begin{align*} \frac{3 \,{\left (2 \, b x - 3 \, a\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{10 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.44533, size = 162, normalized size = 4.76 \begin{align*} - \frac{9 a^{\frac{11}{3}} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac{9 a^{\frac{11}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} - \frac{3 a^{\frac{8}{3}} b x \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac{9 a^{\frac{8}{3}} b x}{10 a^{2} b^{2} + 10 a b^{3} x} + \frac{6 a^{\frac{5}{3}} b^{2} x^{2} \left (1 + \frac{b x}{a}\right )^{\frac{2}{3}}}{10 a^{2} b^{2} + 10 a b^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15929, size = 34, normalized size = 1. \begin{align*} \frac{3 \,{\left (2 \,{\left (b x + a\right )}^{\frac{5}{3}} - 5 \,{\left (b x + a\right )}^{\frac{2}{3}} a\right )}}{10 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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