Optimal. Leaf size=216 \[ -\frac{(b c-a d)^2 \log (a+b x)}{9 b^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{7/3}}-\frac{2 (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3} d^{7/3}}-\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 d^2}+\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 d} \]
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Rubi [A] time = 0.0903724, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {50, 59} \[ -\frac{(b c-a d)^2 \log (a+b x)}{9 b^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{3 b^{2/3} d^{7/3}}-\frac{2 (b c-a d)^2 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{2/3} d^{7/3}}-\frac{2 \sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d)}{3 d^2}+\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{(a+b x)^{4/3}}{\sqrt [3]{c+d x}} \, dx &=\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 d}-\frac{(2 (b c-a d)) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{3 d}\\ &=-\frac{2 (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^2}+\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 d}+\frac{\left (2 (b c-a d)^2\right ) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 d^2}\\ &=-\frac{2 (b c-a d) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 d^2}+\frac{(a+b x)^{4/3} (c+d x)^{2/3}}{2 d}-\frac{2 (b c-a d)^2 \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt{3} b^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \log (a+b x)}{9 b^{2/3} d^{7/3}}-\frac{(b c-a d)^2 \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 b^{2/3} d^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.0299672, size = 73, normalized size = 0.34 \[ \frac{3 (a+b x)^{7/3} \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \, _2F_1\left (\frac{1}{3},\frac{7}{3};\frac{10}{3};\frac{d (a+b x)}{a d-b c}\right )}{7 b \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bx+a \right ) ^{{\frac{4}{3}}}{\frac{1}{\sqrt [3]{dx+c}}}}\, dx \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{{\left (b x + a\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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Fricas [B] time = 2.00156, size = 1835, normalized size = 8.5 \begin{align*} \text{result too large to display} \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{\left (a + b x\right )^{\frac{4}{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{{\left (b x + a\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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