Optimal. Leaf size=571 \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{81 b^3 d^3}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{54 b^3 d^2}+\frac{(b c-a d)^2 \log (a+b x) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{486 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{162 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{81 \sqrt{3} b^{11/3} d^{10/3}}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (-8 a d f-7 b c f+15 b d e)}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d} \]
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Rubi [A] time = 0.555393, antiderivative size = 571, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {90, 80, 50, 59} \[ \frac{\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{81 b^3 d^3}+\frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{54 b^3 d^2}+\frac{(b c-a d)^2 \log (a+b x) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right )}{486 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \log \left (\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{162 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \tan ^{-1}\left (\frac{2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt{3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac{1}{\sqrt{3}}\right )}{81 \sqrt{3} b^{11/3} d^{10/3}}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (-8 a d f-7 b c f+15 b d e)}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 50
Rule 59
Rubi steps
\begin{align*} \int \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2 \, dx &=\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d}+\frac{\int \sqrt [3]{a+b x} (c+d x)^{2/3} \left (\frac{1}{3} \left (12 b d e^2-f (4 b c e+5 a d e+3 a c f)\right )+\frac{1}{3} f (15 b d e-7 b c f-8 a d f) x\right ) \, dx}{4 b d}\\ &=\frac{f (15 b d e-7 b c f-8 a d f) (a+b x)^{4/3} (c+d x)^{5/3}}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d}+\frac{\left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \int \sqrt [3]{a+b x} (c+d x)^{2/3} \, dx}{27 b^2 d^2}\\ &=\frac{\left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{54 b^3 d^2}+\frac{f (15 b d e-7 b c f-8 a d f) (a+b x)^{4/3} (c+d x)^{5/3}}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d}+\frac{\left ((b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right )\right ) \int \frac{\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{81 b^3 d^2}\\ &=\frac{(b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{81 b^3 d^3}+\frac{\left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{54 b^3 d^2}+\frac{f (15 b d e-7 b c f-8 a d f) (a+b x)^{4/3} (c+d x)^{5/3}}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d}-\frac{\left ((b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right )\right ) \int \frac{1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{243 b^3 d^3}\\ &=\frac{(b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{81 b^3 d^3}+\frac{\left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{54 b^3 d^2}+\frac{f (15 b d e-7 b c f-8 a d f) (a+b x)^{4/3} (c+d x)^{5/3}}{36 b^2 d^2}+\frac{f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\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 )}{81 \sqrt{3} b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \log (a+b x)}{486 b^{11/3} d^{10/3}}+\frac{(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \log \left (-1+\frac{\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{162 b^{11/3} d^{10/3}}\\ \end{align*}
Mathematica [C] time = 0.254356, size = 179, normalized size = 0.31 \[ \frac{(a+b x)^{4/3} (c+d x)^{2/3} \left (\frac{\left (10 a^2 d^2 f^2+10 a b d f (c f-3 d e)+b^2 \left (7 c^2 f^2-24 c d e f+27 d^2 e^2\right )\right ) \, _2F_1\left (-\frac{2}{3},\frac{4}{3};\frac{7}{3};\frac{d (a+b x)}{a d-b c}\right )}{b^2 d \left (\frac{b (c+d x)}{b c-a d}\right )^{2/3}}+\frac{f (c+d x) (-8 a d f-7 b c f+15 b d e)}{b d}+9 f (c+d x) (e+f x)\right )}{36 b d} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \sqrt [3]{bx+a} \left ( dx+c \right ) ^{{\frac{2}{3}}} \left ( fx+e \right ) ^{2}\, 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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.08184, size = 4113, normalized size = 7.2 \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 )^{2}\, 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 )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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