Optimal. Leaf size=475 \[ \frac {\left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (27 d^2 e^2-36 c d e f+14 c^2 f^2\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b^2 d^3}+\frac {f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac {(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (27 d^2 e^2-36 c d e f+14 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} b^{8/3} d^{10/3}}+\frac {(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (27 d^2 e^2-36 c d e f+14 c^2 f^2\right )\right ) \log (a+b x)}{162 b^{8/3} d^{10/3}}+\frac {(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (27 d^2 e^2-36 c d e f+14 c^2 f^2\right )\right ) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{54 b^{8/3} d^{10/3}} \]
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Rubi [A]
time = 0.25, antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps
used = 4, 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) \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 (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{27 \sqrt {3} b^{8/3} d^{10/3}}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{27 b^2 d^3}+\frac {(b c-a d) \log (a+b x) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 c d e f+27 d^2 e^2\right )\right )}{162 b^{8/3} d^{10/3}}+\frac {(b c-a d) \left (5 a^2 d^2 f^2-2 a b d f (9 d e-4 c f)+b^2 \left (14 c^2 f^2-36 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 )}{54 b^{8/3} d^{10/3}}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-7 b c f+12 b d e)}{18 b^2 d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 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 \frac {\sqrt [3]{a+b x} (e+f x)^2}{\sqrt [3]{c+d x}} \, dx &=\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac {\int \frac {\sqrt [3]{a+b x} \left (\frac {1}{3} \left (9 b d e^2-f (4 b c e+2 a d e+3 a c f)\right )+\frac {1}{3} f (12 b d e-7 b c f-5 a d f) x\right )}{\sqrt [3]{c+d x}} \, dx}{3 b d}\\ &=\frac {f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac {\left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\right )\right ) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{27 b d}\\ &=\frac {\left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^2}+\frac {f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}-\frac {\left ((b c-a d) \left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\right )\right )\right ) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{81 b d^2}\\ &=\frac {\left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\right )\right ) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b d^2}+\frac {f (12 b d e-7 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d^2}+\frac {f (a+b x)^{4/3} (c+d x)^{2/3} (e+f x)}{3 b d}+\frac {(b c-a d) \left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\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 )}{27 \sqrt {3} b^{5/3} d^{7/3}}+\frac {(b c-a d) \left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\right )\right ) \log (a+b x)}{162 b^{5/3} d^{7/3}}+\frac {(b c-a d) \left (\frac {5 a^2 d f^2}{b}-2 a f (9 d e-4 c f)+b \left (27 d e^2-36 c e f+\frac {14 c^2 f^2}{d}\right )\right ) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{54 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 = 10.15, size = 175, normalized size = 0.37 \begin {gather*} \frac {(a+b x)^{4/3} \left (-4 b f (-12 b d e+7 b c f+5 a d f) (c+d x)+24 b^2 d f (c+d x) (e+f x)+2 \left (5 a^2 d^2 f^2+2 a b d f (-9 d e+4 c f)+b^2 \left (27 d^2 e^2-36 c d e f+14 c^2 f^2\right )\right ) \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 )}{72 b^3 d^2 \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 \[\int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (f x +e \right )^{2}}{\left (d x +c \right )^{\frac {1}{3}}}\, dx\]
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 = 2.87, size = 1400, normalized size = 2.95 \begin {gather*} \left [-\frac {3 \, \sqrt {\frac {1}{3}} {\left ({\left (14 \, b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} f^{2} - 18 \, {\left (2 \, b^{4} c^{2} d^{2} - a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )} f e + 27 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} e^{2}\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 (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2} - 18 \, {\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} f e + 27 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2}\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 (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2} - 18 \, {\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} f e + 27 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2}\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 (18 \, b^{4} d^{3} f^{2} x^{2} + 54 \, b^{4} d^{3} e^{2} - 3 \, {\left (7 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} f^{2} x + {\left (28 \, b^{4} c^{2} d - 5 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} f^{2} + 18 \, {\left (3 \, b^{4} d^{3} f x - {\left (4 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} f\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{162 \, b^{4} d^{4}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left ({\left (14 \, b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} - 3 \, a^{2} b^{2} c d^{3} - 5 \, a^{3} b d^{4}\right )} f^{2} - 18 \, {\left (2 \, b^{4} c^{2} d^{2} - a b^{3} c d^{3} - a^{2} b^{2} d^{4}\right )} f e + 27 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} e^{2}\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 (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2} - 18 \, {\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} f e + 27 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2}\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 (14 \, b^{3} c^{3} - 6 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} f^{2} - 18 \, {\left (2 \, b^{3} c^{2} d - a b^{2} c d^{2} - a^{2} b d^{3}\right )} f e + 27 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e^{2}\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 (18 \, b^{4} d^{3} f^{2} x^{2} + 54 \, b^{4} d^{3} e^{2} - 3 \, {\left (7 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} f^{2} x + {\left (28 \, b^{4} c^{2} d - 5 \, a b^{3} c d^{2} - 5 \, a^{2} b^{2} d^{3}\right )} f^{2} + 18 \, {\left (3 \, b^{4} d^{3} f x - {\left (4 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} f\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{162 \, b^{4} d^{4}}\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 )^{2}}{\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 )}^2\,{\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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