Integrand size = 26, antiderivative size = 339 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{\sqrt [3]{d} f}+\frac {\sqrt {3} \sqrt [3]{b e-a f} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}\right )}{f \sqrt [3]{d e-c f}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f}-\frac {\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac {3 \sqrt [3]{b e-a f} \log \left (-\sqrt [3]{a+b x}+\frac {\sqrt [3]{b e-a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f}}\right )}{2 f \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 \sqrt [3]{d} f} \]
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Time = 0.09 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {132, 61, 12, 93} \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=\frac {\sqrt {3} \sqrt [3]{b e-a f} \arctan \left (\frac {2 \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt {3} \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}+\frac {1}{\sqrt {3}}\right )}{f \sqrt [3]{d e-c f}}-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{d} f}-\frac {\sqrt [3]{b e-a f} \log (e+f x)}{2 f \sqrt [3]{d e-c f}}+\frac {3 \sqrt [3]{b e-a f} \log \left (\frac {\sqrt [3]{c+d x} \sqrt [3]{b e-a f}}{\sqrt [3]{d e-c f}}-\sqrt [3]{a+b x}\right )}{2 f \sqrt [3]{d e-c f}}-\frac {3 \sqrt [3]{b} \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 \sqrt [3]{d} f}-\frac {\sqrt [3]{b} \log (a+b x)}{2 \sqrt [3]{d} f} \]
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Rule 12
Rule 61
Rule 93
Rule 132
Rubi steps \begin{align*} \text {integral}& = \frac {b \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{f}+\int \frac {a-\frac {b e}{f}}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx \\ & = -\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]{b} \log (a+b x)}{2 \sqrt [3]{d} 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}+\left (a-\frac {b e}{f}\right ) \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x} (e+f x)} \, dx \\ & = -\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*}
Time = 5.61 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=\frac {\frac {2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{d} \sqrt [3]{a+b x}+2 \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{d}}+\frac {2 \sqrt {3} \sqrt [3]{b e-a f} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x}}{\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}-2 \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{-d e+c f}}-\frac {2 \sqrt [3]{b} \log \left (-\sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{b e-a f} \log \left (\sqrt [3]{-d e+c f} \sqrt [3]{a+b x}+\sqrt [3]{b e-a f} \sqrt [3]{c+d x}\right )}{\sqrt [3]{-d e+c f}}+\frac {\sqrt [3]{b} \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{\sqrt [3]{d}}+\frac {\sqrt [3]{b e-a f} \log \left ((-d e+c f)^{2/3} (a+b x)^{2/3}-\sqrt [3]{b e-a f} \sqrt [3]{-d e+c f} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+(b e-a f)^{2/3} (c+d x)^{2/3}\right )}{\sqrt [3]{-d e+c f}}}{2 f} \]
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\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {1}{3}} \left (f x +e \right )}d x\]
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Time = 0.28 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=-\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} \]
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\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=\int \frac {\sqrt [3]{a + b x}}{\sqrt [3]{c + d x} \left (e + f x\right )}\, dx \]
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\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}} \,d x } \]
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\[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {1}{3}} {\left (f x + e\right )}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{\sqrt [3]{c+d x} (e+f x)} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{\left (e+f\,x\right )\,{\left (c+d\,x\right )}^{1/3}} \,d x \]
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