Optimal. Leaf size=571 \[ \frac {(b c-a d) \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{81 b^3 d^3}+\frac {\left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) (a+b x)^{4/3} (c+d x)^{2/3}}{54 b^3 d^2}+\frac {f (15 b d e-7 b c f-8 a d f) (a+b x)^{4/3} (c+d x)^{5/3}}{36 b^2 d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{5/3} (e+f x)}{4 b d}+\frac {(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{81 \sqrt {3} b^{11/3} d^{10/3}}+\frac {(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \log (a+b x)}{486 b^{11/3} d^{10/3}}+\frac {(b c-a d)^2 \left (10 a^2 d^2 f^2-10 a b d f (3 d e-c f)+b^2 \left (27 d^2 e^2-24 c d e f+7 c^2 f^2\right )\right ) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{162 b^{11/3} d^{10/3}} \]
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Rubi [A]
time = 0.36, antiderivative size = 571, normalized size of antiderivative = 1.00, 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 = {92, 81, 52, 61}
\begin {gather*} \frac {(b c-a d)^2 \text {ArcTan}\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 ) \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 \sqrt {3} b^{11/3} d^{10/3}}+\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 {(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 {\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 {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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rule 81
Rule 92
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*}
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Mathematica [A]
time = 1.79, size = 528, normalized size = 0.92 \begin {gather*} \frac {(b c-a d)^2 \left (\frac {3 b^{2/3} \sqrt [3]{d} \sqrt [3]{a+b x} (c+d x)^{2/3} \left (20 a^3 d^3 f^2-12 a^2 b d^2 f (5 d e+c f+d f x)+3 a b^2 d \left (-3 c^2 f^2+2 c d f (8 e+f x)+3 d^2 \left (6 e^2+4 e f x+f^2 x^2\right )\right )+b^3 \left (28 c^3 f^2-3 c^2 d f (32 e+7 f x)+18 c d^2 \left (6 e^2+4 e f x+f^2 x^2\right )+27 d^3 x \left (6 e^2+8 e f x+3 f^2 x^2\right )\right )\right )}{(b c-a d)^2}-4 \sqrt {3} \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+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}}{\sqrt {3}}\right )+4 \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 (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )-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 (b^{2/3}+\frac {d^{2/3} (a+b x)^{2/3}}{(c+d x)^{2/3}}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )\right )}{972 b^{11/3} d^{10/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.88, size = 1853, normalized size = 3.25 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}} \left (e + f x\right )^{2}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (e+f\,x\right )}^2\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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