Optimal. Leaf size=339 \[ -\frac {\sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{\sqrt [3]{d} f}+\frac {\sqrt {3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{f \sqrt [3]{d e-c f}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f}-\frac {\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac {3 \sqrt [3]{b e-a f} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 \sqrt [3]{d} f} \]
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Rubi [A]
time = 0.07, antiderivative size = 339, 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 = {132, 61, 12, 93}
\begin {gather*} \frac {\sqrt {3} \sqrt [3]{b e-a f} \text {ArcTan}\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 {\sqrt {3} \sqrt [3]{b} \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 )}{\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 (\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 {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]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 61
Rule 93
Rule 132
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 [A]
time = 6.08, size = 491, normalized size = 1.45 \begin {gather*} \frac {\frac {2 \sqrt {3} \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{d} \sqrt [3]{a+b x}+2 \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt {3} \sqrt [3]{b e-a f} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}-2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{-d e+c f}}-\frac {2 \sqrt [3]{b} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{b e-a f} \log \left (\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}+\sqrt [3]{b e-a f} \sqrt [3]{c+d x}\right )}{\sqrt [3]{-d e+c f}}+\frac {\sqrt [3]{b} \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{\sqrt [3]{d}}+\frac {\sqrt [3]{b e-a f} \log \left ((-d e+c f)^{2/3} (a+b x)^{2/3}-\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+(b e-a f)^{2/3} (c+d x)^{2/3}\right )}{\sqrt [3]{-d e+c f}}}{2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, 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 \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 = 3.33, size = 524, normalized size = 1.55 \begin {gather*} -\frac {2 \, \sqrt {3} \left (\frac {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (c f - d e\right )} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} + \sqrt {3} {\left (a d f x + a c f - {\left (b d x + b c\right )} e\right )}}{3 \, {\left (a d f x + a c f - {\left (b d x + b c\right )} e\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 {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (\frac {a f - b e}{c f - d e}\right )^{\frac {2}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (\frac {a f - b e}{c f - d e}\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 {a f - b e}{c f - d e}\right )^{\frac {1}{3}} \log \left (-\frac {{\left (d x + c\right )} \left (\frac {a f - b e}{c f - d e}\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} \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}}{\sqrt [3]{c + d x} \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 \frac {{\left (a+b\,x\right )}^{1/3}}{\left (e+f\,x\right )\,{\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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