Optimal. Leaf size=66 \[ \frac{9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \]
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Rubi [A] time = 0.0090983, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{9 d (c+d x)^{2/3}}{10 (a+b x)^{2/3} (b c-a d)^2}-\frac{3 (c+d x)^{2/3}}{5 (a+b x)^{5/3} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a+b x)^{8/3} \sqrt [3]{c+d x}} \, dx &=-\frac{3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}-\frac{(3 d) \int \frac{1}{(a+b x)^{5/3} \sqrt [3]{c+d x}} \, dx}{5 (b c-a d)}\\ &=-\frac{3 (c+d x)^{2/3}}{5 (b c-a d) (a+b x)^{5/3}}+\frac{9 d (c+d x)^{2/3}}{10 (b c-a d)^2 (a+b x)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0171238, size = 46, normalized size = 0.7 \[ \frac{3 (c+d x)^{2/3} (5 a d-2 b c+3 b d x)}{10 (a+b x)^{5/3} (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 54, normalized size = 0.8 \begin{align*}{\frac{9\,bdx+15\,ad-6\,bc}{10\,{a}^{2}{d}^{2}-20\,abcd+10\,{b}^{2}{c}^{2}} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( bx+a \right ) ^{-{\frac{5}{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{8}{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 [B] time = 1.8712, size = 259, normalized size = 3.92 \begin{align*} \frac{3 \,{\left (3 \, b d x - 2 \, b c + 5 \, a d\right )}{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}{10 \,{\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2} +{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\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{8}{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{8}{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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