Optimal. Leaf size=475 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{27 b^2 d^3}+\frac{(b c-a d) \log (a+b x) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{162 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{10/3}}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-7 b c f+12 b d e)}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d} \]
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Rubi [A] time = 0.399542, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {90, 80, 50, 59} \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{27 b^2 d^3}+\frac{(b c-a d) \log (a+b x) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{162 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{54 b^{8/3} d^{10/3}}+\frac{(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{8/3} d^{10/3}}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-7 b c f+12 b d e)}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 50
Rule 59
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x} (e+f x)^2}{\sqrt [3]{c+d x}} \, dx &=\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac{\int \frac{\sqrt [3]{a+b x} \left (\frac{1}{3} \left (9 b d e^2-f (4 b c e+2 a d e+3 a c f)\right )+\frac{1}{3} f (12 b d e-7 b c f-5 a d f) x\right )}{\sqrt [3]{c+d x}} \, dx}{3 b d}\\ &=\frac{f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac{\left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right ) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{27 b d}\\ &=\frac{\left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^2}+\frac{f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}-\frac{\left ((b c-a d) \left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right )\right ) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{81 b d^2}\\ &=\frac{\left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^2}+\frac{f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac{(b c-a d) \left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right ) \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^{5/3} d^{7/3}}+\frac{(b c-a d) \left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right ) \log (a+b x)}{162 b^{5/3} d^{7/3}}+\frac{(b c-a d) \left (\frac{5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac{14 c^2 f^2}{d}\right )\right ) \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{54 b^{5/3} d^{7/3}}\\ \end{align*}
Mathematica [C] time = 0.191793, size = 175, normalized size = 0.37 \[ \frac{(a+b x)^{4/3} \left (2 \sqrt [3]{\frac{b (c+d x)}{b c-a d}} \left (5 a^2 d^2 f^2+2 a b d f (4 c f-9 d e)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right ) \, _2F_1\left (\frac{1}{3},\frac{4}{3};\frac{7}{3};\frac{d (a+b x)}{a d-b c}\right )-4 b f (c+d x) (5 a d f+7 b c f-12 b d e)+24 b^2 d f (c+d x) (e+f x)\right )}{72 b^3 d^2 \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx+e \right ) ^{2}\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 )}^{2}}{{\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 = 3.18811, size = 3140, normalized size = 6.61 \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 )^{2}}{\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 )}^{2}}{{\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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