Optimal. Leaf size=32 \[ \frac{3 a}{b^2 \sqrt [3]{a+b x}}+\frac{3 (a+b x)^{2/3}}{2 b^2} \]
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Rubi [A] time = 0.0085798, antiderivative size = 32, 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} \[ \frac{3 a}{b^2 \sqrt [3]{a+b x}}+\frac{3 (a+b x)^{2/3}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int \frac{x}{(a+b x)^{4/3}} \, dx &=\int \left (-\frac{a}{b (a+b x)^{4/3}}+\frac{1}{b \sqrt [3]{a+b x}}\right ) \, dx\\ &=\frac{3 a}{b^2 \sqrt [3]{a+b x}}+\frac{3 (a+b x)^{2/3}}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.04836, size = 23, normalized size = 0.72 \[ \frac{3 (3 a+b x)}{2 b^2 \sqrt [3]{a+b x}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 20, normalized size = 0.6 \begin{align*}{\frac{3\,bx+9\,a}{2\,{b}^{2}}{\frac{1}{\sqrt [3]{bx+a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02825, size = 35, normalized size = 1.09 \begin{align*} \frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}}}{2 \, b^{2}} + \frac{3 \, a}{{\left (b x + a\right )}^{\frac{1}{3}} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60112, size = 66, normalized size = 2.06 \begin{align*} \frac{3 \,{\left (b x + 3 \, a\right )}{\left (b x + a\right )}^{\frac{2}{3}}}{2 \,{\left (b^{3} x + a b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.04138, size = 41, normalized size = 1.28 \begin{align*} \begin{cases} \frac{9 a}{2 b^{2} \sqrt [3]{a + b x}} + \frac{3 x}{2 b \sqrt [3]{a + b x}} & \text{for}\: b \neq 0 \\\frac{x^{2}}{2 a^{\frac{4}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21717, size = 41, normalized size = 1.28 \begin{align*} \frac{3 \,{\left (\frac{{\left (b x + a\right )}^{\frac{2}{3}}}{b} + \frac{2 \, a}{{\left (b x + a\right )}^{\frac{1}{3}} b}\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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