Optimal. Leaf size=273 \[ \frac {(3 b d e-2 b c f-a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{3 b d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3}}{2 b d}+\frac {(b c-a d) (3 b d e-2 b c f-a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{3 \sqrt {3} b^{5/3} d^{7/3}}+\frac {(b c-a d) (3 b d e-2 b c f-a d f) \log (a+b x)}{18 b^{5/3} d^{7/3}}+\frac {(b c-a d) (3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{6 b^{5/3} d^{7/3}} \]
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Rubi [A]
time = 0.11, 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 = {81, 52, 61}
\begin {gather*} \frac {(b c-a d) \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 ) (-a d f-2 b c f+3 b d e)}{3 \sqrt {3} b^{5/3} d^{7/3}}+\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 {\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rule 81
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] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 10.09, size = 103, normalized size = 0.38 \begin {gather*} \frac {(a+b x)^{4/3} \left (2 b f (c+d x)+(3 b d e-2 b c f-a d f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{3},\frac {4}{3};\frac {7}{3};\frac {d (a+b x)}{-b c+a d}\right )\right )}{4 b^2 d \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \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 = 2.52, size = 914, normalized size = 3.35 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} f - 3 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (3 \, b^{2} d x + b^{2} c + 2 \, a b d + 3 \, \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} + \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}\right ) + \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f - 3 \, {\left (b^{2} c d - a b d^{2}\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f - 3 \, {\left (b^{2} c d - a b d^{2}\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{2} f x + 6 \, b^{3} d^{2} e - {\left (4 \, b^{3} c d - a b^{2} d^{2}\right )} f\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, b^{3} d^{3}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} f - 3 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}\right )} \sqrt {-\frac {\left (-b^{2} d\right )^{\frac {1}{3}}}{d}}}{b^{2} d x + b^{2} c}\right ) + \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f - 3 \, {\left (b^{2} c d - a b d^{2}\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d + \left (-b^{2} d\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} - \left (-b^{2} d\right )^{\frac {1}{3}} {\left (b d x + b c\right )}}{d x + c}\right ) - 2 \, \left (-b^{2} d\right )^{\frac {2}{3}} {\left ({\left (2 \, b^{2} c^{2} - a b c d - a^{2} d^{2}\right )} f - 3 \, {\left (b^{2} c d - a b d^{2}\right )} e\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (-b^{2} d\right )^{\frac {2}{3}} {\left (d x + c\right )}}{d x + c}\right ) + 3 \, {\left (3 \, b^{3} d^{2} f x + 6 \, b^{3} d^{2} e - {\left (4 \, b^{3} c d - a b^{2} d^{2}\right )} f\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{18 \, b^{3} d^{3}}\right ] \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 \frac {\sqrt [3]{a + b x} \left (e + f x\right )}{\sqrt [3]{c + d x}}\, 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 \frac {\left (e+f\,x\right )\,{\left (a+b\,x\right )}^{1/3}}{{\left (c+d\,x\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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