Optimal. Leaf size=32 \[ \frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \]
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Rubi [A] time = 0.0048773, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {74} \[ \frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b c+2 b d x}} \]
Antiderivative was successfully verified.
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Rule 74
Rubi steps
\begin{align*} \int \frac{a+b x}{\sqrt [3]{c+d x} (b c+a d+2 b d x)^{4/3}} \, dx &=\frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{b c+a d+2 b d x}}\\ \end{align*}
Mathematica [A] time = 0.0255556, size = 32, normalized size = 1. \[ \frac{3 (c+d x)^{2/3}}{2 d^2 \sqrt [3]{a d+b (c+2 d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{3}{2\,{d}^{2}} \left ( dx+c \right ) ^{{\frac{2}{3}}}{\frac{1}{\sqrt [3]{2\,bdx+ad+bc}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.53377, size = 35, normalized size = 1.09 \begin{align*} \frac{3 \,{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (2 \, b d x + b c + a d\right )}^{\frac{1}{3}} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.38088, size = 107, normalized size = 3.34 \begin{align*} \frac{3 \,{\left (2 \, b d x + b c + a d\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (2 \, b d^{3} x + b c d^{2} + a d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b x}{\sqrt [3]{c + d x} \left (a d + b c + 2 b d x\right )^{\frac{4}{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{{\left (2 \, b d x + b c + a d\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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