Optimal. Leaf size=273 \[ \frac {(b c-a d) \log (a+b x) (-a d f-2 b c f+3 b d e)}{18 b^{5/3} d^{7/3}}+\frac {(b c-a d) (-a d f-2 b c f+3 b d e) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{7/3}}+\frac {(b c-a d) (-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} d^{7/3}}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-a d f-2 b c f+3 b d e)}{3 b d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d} \]
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Rubi [A] time = 0.17, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {80, 50, 59} \[ \frac {(b c-a d) \log (a+b x) (-a d f-2 b c f+3 b d e)}{18 b^{5/3} d^{7/3}}+\frac {(b c-a d) (-a d f-2 b c f+3 b d e) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{6 b^{5/3} d^{7/3}}+\frac {(b c-a d) (-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} d^{7/3}}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (-a d f-2 b c f+3 b d e)}{3 b d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d} \]
Antiderivative was successfully verified.
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Rule 50
Rule 59
Rule 80
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x} (e+f x)}{\sqrt [3]{c+d x}} \, dx &=\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}+\frac {\left (2 b d e-\left (\frac {4 b c}{3}+\frac {2 a d}{3}\right ) f\right ) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{2 b d}\\ &=\frac {(3 b d e-2 b c f-a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 b d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}-\frac {((b c-a d) (3 b d e-2 b c f-a d f)) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{9 b d^2}\\ &=\frac {(3 b d e-2 b c f-a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 b d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}+\frac {(b c-a d) (3 b d e-2 b c f-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 )}{3 \sqrt {3} b^{5/3} d^{7/3}}+\frac {(b c-a d) (3 b d e-2 b c f-a d f) \log (a+b x)}{18 b^{5/3} d^{7/3}}+\frac {(b c-a d) (3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{6 b^{5/3} d^{7/3}}\\ \end {align*}
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Mathematica [C] time = 0.17, size = 103, normalized size = 0.38 \[ \frac {(a+b x)^{4/3} \left (\sqrt [3]{\frac {b (c+d x)}{b c-a d}} (-a d f-2 b c f+3 b d e) \, _2F_1\left (\frac {1}{3},\frac {4}{3};\frac {7}{3};\frac {d (a+b x)}{a d-b c}\right )+2 b f (c+d x)\right )}{4 b^2 d \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.13, size = 892, normalized size = 3.27 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (f x + e\right )}}{{\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (f x +e \right )}{\left (d x +c \right )^{\frac {1}{3}}}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (f x + e\right )}}{{\left (d x + c\right )}^{\frac {1}{3}}}\,{d x} \]
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 \[ \int \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^{1/3}}{{\left (c+d\,x\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\sqrt [3]{a + b x} \left (e + f x\right )}{\sqrt [3]{c + d x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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