Optimal. Leaf size=38 \[ \frac{3 (b x-a)^{5/3}}{5 b^2}+\frac{3 a (b x-a)^{2/3}}{2 b^2} \]
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Rubi [A] time = 0.0088401, antiderivative size = 38, 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 = {43} \[ \frac{3 (b x-a)^{5/3}}{5 b^2}+\frac{3 a (b x-a)^{2/3}}{2 b^2} \]
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 = 0.0316602, size = 26, normalized size = 0.68 \[ \frac{3 (b x-a)^{2/3} (3 a+2 b x)}{10 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 23, 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 = 1.05181, size = 41, normalized size = 1.08 \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 = 1.5218, size = 54, normalized size = 1.42 \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 [C] time = 1.54925, size = 488, normalized size = 12.84 \begin{align*} \begin{cases} - \frac{9 a^{\frac{11}{3}} \left (-1 + \frac{b x}{a}\right )^{\frac{2}{3}} e^{\frac{i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} - \frac{9 a^{\frac{11}{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} + \frac{3 a^{\frac{8}{3}} b x \left (-1 + \frac{b x}{a}\right )^{\frac{2}{3}} e^{\frac{i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} + \frac{9 a^{\frac{8}{3}} b x}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} + \frac{6 a^{\frac{5}{3}} b^{2} x^{2} \left (-1 + \frac{b x}{a}\right )^{\frac{2}{3}} e^{\frac{i \pi }{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\\frac{9 a^{\frac{11}{3}} \left (1 - \frac{b x}{a}\right )^{\frac{2}{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} - \frac{9 a^{\frac{11}{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} - \frac{3 a^{\frac{8}{3}} b x \left (1 - \frac{b x}{a}\right )^{\frac{2}{3}}}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} + \frac{9 a^{\frac{8}{3}} b x}{- 10 a^{2} b^{2} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} - \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} e^{\frac{i \pi }{3}} + 10 a b^{3} x e^{\frac{i \pi }{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17993, size = 39, normalized size = 1.03 \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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