Optimal. Leaf size=339 \[ -\frac {\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac {3 \sqrt [3]{b e-a f} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}+\frac {\sqrt {3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{f \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} f}-\frac {\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]
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Rubi [A] time = 0.10, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {105, 59, 91} \[ -\frac {\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac {3 \sqrt [3]{b e-a f} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}+\frac {\sqrt {3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{f \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} f}-\frac {\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]
Antiderivative was successfully verified.
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Rule 59
Rule 91
Rule 105
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx &=\frac {b \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{f}-\frac {(b e-a f) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx}{f}\\ &=-\frac {\sqrt {3} \sqrt [3]{b} \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]{d} f}+\frac {\sqrt {3} \sqrt [3]{b e-a f} \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 \sqrt [3]{d e-c f}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f}-\frac {\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac {3 \sqrt [3]{b e-a f} \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 \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 \sqrt [3]{d} f}\\ \end {align*}
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Mathematica [C] time = 0.09, size = 114, normalized size = 0.34 \[ \frac {3 \sqrt [3]{a+b x} \left (\sqrt [3]{\frac {b (c+d x)}{b c-a d}} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};\frac {d (a+b x)}{a d-b c}\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}\right )\right )}{f \sqrt [3]{c+d x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 509, normalized size = 1.50 \[ -\frac {2 \, \sqrt {3} \left (\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (d e - c f\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {b e - a f}{d e - c f}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b c e - a c f + {\left (b d e - a d f\right )} x\right )}}{3 \, {\left (b c e - a c f + {\left (b d e - a d f\right )} x\right )}}\right ) + 2 \, \sqrt {3} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} d \left (-\frac {b}{d}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) + \left (\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {b e - a f}{d e - c f}\right )^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) + \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 2 \, \left (\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {b e - a f}{d e - c f}\right )^{\frac {1}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) - 2 \, \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right )}{2 \, f} \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )}\, dx \]
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 \[ \int \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x\right )}^{1/3}}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
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 \[ \int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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