Optimal. Leaf size=409 \[ \frac {\log (a+b x) (-a d f-2 b c f+3 b d e)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac {(-a d f-2 b c f+3 b d e) \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 b^{2/3} \sqrt [3]{d} f^2}+\frac {(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{d} f^2}+\frac {\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac {3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2}-\frac {\sqrt {3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{f^2}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f} \]
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Rubi [A] time = 0.40, antiderivative size = 409, 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 = {101, 157, 59, 91} \begin {gather*} \frac {\log (a+b x) (-a d f-2 b c f+3 b d e)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac {(-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 )}{2 b^{2/3} \sqrt [3]{d} f^2}+\frac {(-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 )}{\sqrt {3} b^{2/3} \sqrt [3]{d} f^2}+\frac {\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac {3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f^2}-\frac {\sqrt {3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{f^2}+\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 101
Rule 157
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{e+f x} \, dx &=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}-\frac {\int \frac {\frac {1}{3} (b c e+2 a d e-3 a c f)+\frac {1}{3} (3 b d e-2 b c f-a d f) x}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{f}\\ &=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}+\frac {((b e-a f) (d e-c f)) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{f^2}-\frac {(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}{3 f^2}\\ &=\frac {\sqrt [3]{a+b x} (c+d x)^{2/3}}{f}+\frac {(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 )}{\sqrt {3} b^{2/3} \sqrt [3]{d} f^2}-\frac {\sqrt {3} \sqrt [3]{b e-a f} (d e-c f)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{f^2}+\frac {(3 b d e-2 b c f-a d f) \log (a+b x)}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac {\sqrt [3]{b e-a f} (d e-c f)^{2/3} \log (e+f x)}{2 f^2}-\frac {3 \sqrt [3]{b e-a f} (d e-c f)^{2/3} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f^2}+\frac {(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 )}{2 b^{2/3} \sqrt [3]{d} f^2}\\ \end {align*}
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Mathematica [C] time = 0.21, size = 194, normalized size = 0.47 \begin {gather*} \frac {3 \sqrt [3]{a+b x} \left (-\frac {d (b e-a f) \sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {d (a+b x)}{a d-b c}\right )}{b f}+\frac {(d e-c f) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )}{f}+\frac {(c+d x) \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};\frac {d (a+b x)}{a d-b c}\right )}{\left (\frac {b (c+d x)}{b c-a d}\right )^{2/3}}\right )}{f \sqrt [3]{c+d x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.73, size = 586, normalized size = 1.43 \begin {gather*} \frac {(a d f+2 b c f-3 b d e) \log \left (\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}}+b^{2/3}\right )}{6 b^{2/3} \sqrt [3]{d} f^2}+\frac {(-a d f-2 b c f+3 b d e) \log \left (\sqrt [3]{b}-\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}\right )}{3 b^{2/3} \sqrt [3]{d} f^2}-\frac {(-a d f-2 b c f+3 b d e) \tan ^{-1}\left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} b^{2/3} \sqrt [3]{d} f^2}-\frac {\sqrt [3]{b e-a f} (c f-d e)^{2/3} \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{c f-d e}}{\sqrt [3]{c+d x}}+\sqrt [3]{b e-a f}\right )}{f^2}+\frac {\sqrt [3]{b e-a f} (c f-d e)^{2/3} \log \left (-\frac {\sqrt [3]{a+b x} \sqrt [3]{b e-a f} \sqrt [3]{c f-d e}}{\sqrt [3]{c+d x}}+\frac {(a+b x)^{2/3} (c f-d e)^{2/3}}{(c+d x)^{2/3}}+(b e-a f)^{2/3}\right )}{2 f^2}+\frac {\sqrt {3} \sqrt [3]{b e-a f} (c f-d e)^{2/3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{a+b x} \sqrt [3]{c f-d e}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}\right )}{f^2}+\frac {\frac {b c \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-\frac {a d \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}}{f \left (b-\frac {d (a+b x)}{c+d x}\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 12.64, size = 2014, normalized size = 4.92
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 \begin {gather*} \int \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{f x + e}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}{f x +e}\, dx \end {gather*}
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*} \int \frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{f x + e}\,{d x} \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 (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}}{e+f\,x} \,d x \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 (c + d x\right )^{\frac {2}{3}}}{e + f x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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