Optimal. Leaf size=273 \[ \frac{(b c-a d) \log (a+b x) (-a d f-2 b c f+3 b d e)}{18 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d^{7/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-a d f-2 b c f+3 b d e)}{3 b d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d} \]
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Rubi [A] time = 0.17093, antiderivative size = 273, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 50, 59} \[ \frac{(b c-a d) \log (a+b x) (-a d f-2 b c f+3 b d e)}{18 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{7/3}}+\frac{(b c-a d) (-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} b^{5/3} d^{7/3}}+\frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (-a d f-2 b c f+3 b d e)}{3 b d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x} (e+f x)}{\sqrt [3]{c+d x}} \, dx &=\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}+\frac{\left (2 b d e-\left (\frac{4 b c}{3}+\frac{2 a d}{3}\right ) f\right ) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{2 b d}\\ &=\frac{(3 b d e-2 b c f-a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 b d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}-\frac{((b c-a d) (3 b d e-2 b c f-a d f)) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 b d^2}\\ &=\frac{(3 b d e-2 b c f-a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 b d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}+\frac{(b c-a d) (3 b d e-2 b c f-a d f) \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^{5/3} d^{7/3}}+\frac{(b c-a d) (3 b d e-2 b c f-a d f) \log (a+b x)}{18 b^{5/3} d^{7/3}}+\frac{(b c-a d) (3 b d e-2 b c f-a d f) \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{6 b^{5/3} d^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.118787, size = 103, normalized size = 0.38 \[ \frac{(a+b x)^{4/3} \left (\sqrt [3]{\frac{b (c+d x)}{b c-a d}} (-a d f-2 b c f+3 b d e) \, _2F_1\left (\frac{1}{3},\frac{4}{3};\frac{7}{3};\frac{d (a+b x)}{a d-b c}\right )+2 b f (c+d x)\right )}{4 b^2 d \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{(fx+e)\sqrt [3]{bx+a}{\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{1}{3}}{\left (f x + e\right )}}{{\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 = 2.47636, size = 2106, normalized size = 7.71 \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{\sqrt [3]{a + b x} \left (e + f x\right )}{\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{1}{3}}{\left (f x + e\right )}}{{\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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