Optimal. Leaf size=331 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-5 a d f-4 b c f+9 b d e)}{27 b^2 d^2}+\frac{(b c-a d)^2 \log (a+b x) (-5 a d f-4 b c f+9 b d e)}{162 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{7/3}}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-4 b c f+9 b d e)}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d} \]
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Rubi [A] time = 0.226755, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {80, 50, 59} \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-5 a d f-4 b c f+9 b d e)}{27 b^2 d^2}+\frac{(b c-a d)^2 \log (a+b x) (-5 a d f-4 b c f+9 b d e)}{162 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{7/3}}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-4 b c f+9 b d e)}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 80
Rule 50
Rule 59
Rubi steps
\begin{align*} \int \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x) \, dx &=\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac{\left (3 b d e-\left (\frac{4 b c}{3}+\frac{5 a d}{3}\right ) f\right ) \int \sqrt [3]{a+b x} (c+d x)^{2/3} \, dx}{3 b d}\\ &=\frac{(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac{((b c-a d) (9 b d e-4 b c f-5 a d f)) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{27 b^2 d}\\ &=\frac{(b c-a d) (9 b d e-4 b c f-5 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b^2 d^2}+\frac{(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}-\frac{\left ((b c-a d)^2 (9 b d e-4 b c f-5 a d f)\right ) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{81 b^2 d^2}\\ &=\frac{(b c-a d) (9 b d e-4 b c f-5 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b^2 d^2}+\frac{(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac{(b c-a d)^2 (9 b d e-4 b c f-5 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 )}{27 \sqrt{3} b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \log (a+b x)}{162 b^{8/3} d^{7/3}}+\frac{(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{54 b^{8/3} d^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.165665, size = 103, normalized size = 0.31 \[ \frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (\frac{(-5 a d f-4 b c f+9 b d e) \, _2F_1\left (-\frac{2}{3},\frac{4}{3};\frac{7}{3};\frac{d (a+b x)}{a d-b c}\right )}{\left (\frac{b (c+d x)}{b c-a d}\right )^{2/3}}+4 b f (c+d x)\right )}{12 b^2 d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.02, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( fx+e \right ) \, 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{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.6755, size = 2743, normalized size = 8.29 \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 \sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}} \left (e + f x\right )\, 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{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}{\left (f x + e\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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