Optimal. Leaf size=587 \[ \frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2+3 b d f x (-7 a d f-8 b c f+15 b d e)-a b d f (108 d e-31 c f)+b^2 \left (40 c^2 f^2-135 c d e f+144 d^2 e^2\right )\right )}{54 b^3 d^3}+\frac{\log (c+d x) \left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right )}{162 b^{10/3} d^{11/3}}+\frac{\left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{54 b^{10/3} d^{11/3}}+\frac{\left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{10/3} d^{11/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d} \]
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Rubi [A] time = 0.481127, antiderivative size = 587, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {100, 147, 59} \[ \frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2+3 b d f x (-7 a d f-8 b c f+15 b d e)-a b d f (108 d e-31 c f)+b^2 \left (40 c^2 f^2-135 c d e f+144 d^2 e^2\right )\right )}{54 b^3 d^3}+\frac{\log (c+d x) \left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right )}{162 b^{10/3} d^{11/3}}+\frac{\left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \log \left (\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{54 b^{10/3} d^{11/3}}+\frac{\left (-6 a^2 b d^2 f^2 (9 d e-2 c f)+14 a^3 d^3 f^3+3 a b^2 d f \left (5 c^2 f^2-18 c d e f+27 d^2 e^2\right )+b^3 \left (-\left (135 c^2 d e f^2-40 c^3 f^3-162 c d^2 e^2 f+81 d^3 e^3\right )\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac{1}{\sqrt{3}}\right )}{27 \sqrt{3} b^{10/3} d^{11/3}}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d} \]
Antiderivative was successfully verified.
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Rule 100
Rule 147
Rule 59
Rubi steps
\begin{align*} \int \frac{(e+f x)^3}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx &=\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d}+\frac{\int \frac{(e+f x) \left (\frac{1}{3} \left (9 b d e^2-f (2 b c e+a d e+6 a c f)\right )+\frac{1}{3} f (15 b d e-8 b c f-7 a d f) x\right )}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{3 b d}\\ &=\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2-a b d f (108 d e-31 c f)+b^2 \left (144 d^2 e^2-135 c d e f+40 c^2 f^2\right )+3 b d f (15 b d e-8 b c f-7 a d f) x\right )}{54 b^3 d^3}-\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \int \frac{1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{81 b^3 d^3}\\ &=\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)^2}{3 b d}+\frac{f (a+b x)^{2/3} \sqrt [3]{c+d x} \left (28 a^2 d^2 f^2-a b d f (108 d e-31 c f)+b^2 \left (144 d^2 e^2-135 c d e f+40 c^2 f^2\right )+3 b d f (15 b d e-8 b c f-7 a d f) x\right )}{54 b^3 d^3}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt{3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{27 \sqrt{3} b^{10/3} d^{11/3}}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \log (c+d x)}{162 b^{10/3} d^{11/3}}+\frac{\left (14 a^3 d^3 f^3-6 a^2 b d^2 f^2 (9 d e-2 c f)+3 a b^2 d f \left (27 d^2 e^2-18 c d e f+5 c^2 f^2\right )-b^3 \left (81 d^3 e^3-162 c d^2 e^2 f+135 c^2 d e f^2-40 c^3 f^3\right )\right ) \log \left (-1+\frac{\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{54 b^{10/3} d^{11/3}}\\ \end{align*}
Mathematica [C] time = 0.782037, size = 291, normalized size = 0.5 \[ \frac{(a+b x)^{2/3} \left (\frac{f^2 (c+d x)^2 (-7 a d f-8 b c f+15 b d e) \, _2F_1\left (-\frac{4}{3},\frac{2}{3};\frac{5}{3};\frac{d (a+b x)}{a d-b c}\right )}{b d^2 \left (\frac{b (c+d x)}{b c-a d}\right )^{4/3}}+\frac{(d e-c f)^2 \left (\frac{b (c+d x)}{b c-a d}\right )^{2/3} (-a d f-8 b c f+9 b d e) \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{d (a+b x)}{a d-b c}\right )}{b d^2}+\frac{8 f (c+d x) (c f-d e) (a d f+2 b c f-3 b d e) \, _2F_1\left (-\frac{1}{3},\frac{2}{3};\frac{5}{3};\frac{d (a+b x)}{a d-b c}\right )}{b d^2 \sqrt [3]{\frac{b (c+d x)}{b c-a d}}}+2 f (c+d x) (e+f x)^2\right )}{6 b d (c+d x)^{2/3}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{ \left ( fx+e \right ) ^{3}{\frac{1}{\sqrt [3]{bx+a}}} \left ( dx+c \right ) ^{-{\frac{2}{3}}}}\, 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 (f x + e\right )}^{3}}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.86081, size = 3393, normalized size = 5.78 \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 (e + f x\right )^{3}}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac{2}{3}}}\, 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 (f x + e\right )}^{3}}{{\left (b x + a\right )}^{\frac{1}{3}}{\left (d x + c\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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