Optimal. Leaf size=30 \[ -\frac{3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (b c-a d)} \]
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Rubi [A] time = 0.0030281, antiderivative size = 30, 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} \[ -\frac{3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{4/3} (c+d x)^{2/3}} \, dx &=-\frac{3 \sqrt [3]{c+d x}}{(b c-a d) \sqrt [3]{a+b x}}\\ \end{align*}
Mathematica [A] time = 0.008687, size = 30, normalized size = 1. \[ \frac{3 \sqrt [3]{c+d x}}{\sqrt [3]{a+b x} (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 27, normalized size = 0.9 \begin{align*} 3\,{\frac{\sqrt [3]{dx+c}}{\sqrt [3]{bx+a} \left ( ad-bc \right ) }} \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{4}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47829, size = 97, normalized size = 3.23 \begin{align*} -\frac{3 \,{\left (b x + a\right )}^{\frac{2}{3}}{\left (d x + c\right )}^{\frac{1}{3}}}{a b c - a^{2} d +{\left (b^{2} c - a b d\right )} x} \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{4}{3}} \left (c + d x\right )^{\frac{2}{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{1}{{\left (b x + a\right )}^{\frac{4}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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