Optimal. Leaf size=32 \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]
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Rubi [A] time = 0.0029671, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx &=-\frac{3 (c+d x)^{2/3}}{2 (b c-a d) (a+b x)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0121046, size = 32, normalized size = 1. \[ -\frac{3 (c+d x)^{2/3}}{2 (a+b x)^{2/3} (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{3}{2\,ad-2\,bc} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( bx+a \right ) ^{-{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x + a\right )}^{\frac{5}{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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Fricas [A] time = 1.76267, size = 100, normalized size = 3.12 \begin{align*} -\frac{3 \,{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{2 \,{\left (a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x\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{1}{\left (a + b x\right )^{\frac{5}{3}} \sqrt [3]{c + d x}}\, 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{1}{{\left (b x + a\right )}^{\frac{5}{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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