Optimal. Leaf size=331 \[ \frac {(b c-a d) (9 b d e-4 b c f-5 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b^2 d^2}+\frac {(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac {(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{27 \sqrt {3} b^{8/3} d^{7/3}}+\frac {(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \log (a+b x)}{162 b^{8/3} d^{7/3}}+\frac {(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{54 b^{8/3} d^{7/3}} \]
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Rubi [A]
time = 0.15, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps
used = 4, 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)^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 ) (-5 a d f-4 b c f+9 b d e)}{27 \sqrt {3} b^{8/3} d^{7/3}}+\frac {(b c-a d)^2 \log (a+b x) (-5 a d f-4 b c f+9 b d e)}{162 b^{8/3} d^{7/3}}+\frac {(b c-a d)^2 (-5 a d f-4 b c f+9 b d e) \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^{7/3}}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3} (b c-a d) (-5 a d f-4 b c f+9 b d e)}{27 b^2 d^2}+\frac {(a+b x)^{4/3} (c+d x)^{2/3} (-5 a d f-4 b c f+9 b d e)}{18 b^2 d}+\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 61
Rule 81
Rubi steps
\begin {align*} \int \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x) \, dx &=\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac {\left (3 b d e-\left (\frac {4 b c}{3}+\frac {5 a d}{3}\right ) f\right ) \int \sqrt [3]{a+b x} (c+d x)^{2/3} \, dx}{3 b d}\\ &=\frac {(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac {((b c-a d) (9 b d e-4 b c f-5 a d f)) \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x}} \, dx}{27 b^2 d}\\ &=\frac {(b c-a d) (9 b d e-4 b c f-5 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b^2 d^2}+\frac {(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}-\frac {\left ((b c-a d)^2 (9 b d e-4 b c f-5 a d f)\right ) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{81 b^2 d^2}\\ &=\frac {(b c-a d) (9 b d e-4 b c f-5 a d f) \sqrt [3]{a+b x} (c+d x)^{2/3}}{27 b^2 d^2}+\frac {(9 b d e-4 b c f-5 a d f) (a+b x)^{4/3} (c+d x)^{2/3}}{18 b^2 d}+\frac {f (a+b x)^{4/3} (c+d x)^{5/3}}{3 b d}+\frac {(b c-a d)^2 (9 b d e-4 b c f-5 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 )}{27 \sqrt {3} b^{8/3} d^{7/3}}+\frac {(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \log (a+b x)}{162 b^{8/3} d^{7/3}}+\frac {(b c-a d)^2 (9 b d e-4 b c f-5 a d f) \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^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 1.10, size = 319, normalized size = 0.96 \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 (-5 a^2 d^2 f+a b d (9 d e+4 c f+3 d f x)+b^2 \left (-8 c^2 f+6 c d (3 e+f x)+9 d^2 x (3 e+2 f x)\right )\right )}{(b c-a d)^2}-2 \sqrt {3} (9 b d e-4 b c f-5 a d f) \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 )+2 (9 b d e-4 b c f-5 a d f) \log \left (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )-(9 b d e-4 b c f-5 a d f) \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 )}{162 b^{8/3} d^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 561 vs.
\(2 (280) = 560\).
time = 1.14, size = 1174, normalized size = 3.55 \begin {gather*} \left [-\frac {3 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} - 6 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} f - 9 \, {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\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 (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f - 9 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\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 (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f - 9 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\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 (18 \, b^{4} d^{3} f x^{2} + 3 \, {\left (2 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f x - {\left (8 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )} f + 9 \, {\left (3 \, b^{4} d^{3} x + 2 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{162 \, b^{4} d^{3}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left ({\left (4 \, b^{4} c^{3} d - 3 \, a b^{3} c^{2} d^{2} - 6 \, a^{2} b^{2} c d^{3} + 5 \, a^{3} b d^{4}\right )} f - 9 \, {\left (b^{4} c^{2} d^{2} - 2 \, a b^{3} c d^{3} + a^{2} b^{2} d^{4}\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 (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f - 9 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\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 (4 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} f - 9 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\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 (18 \, b^{4} d^{3} f x^{2} + 3 \, {\left (2 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} f x - {\left (8 \, b^{4} c^{2} d - 4 \, a b^{3} c d^{2} + 5 \, a^{2} b^{2} d^{3}\right )} f + 9 \, {\left (3 \, b^{4} d^{3} x + 2 \, b^{4} c d^{2} + a b^{3} d^{3}\right )} e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{162 \, b^{4} 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 \sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}} \left (e + f x\right )\, 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 )\,{\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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