Integrand size = 24, antiderivative size = 200 \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}-\frac {(3 b d e-2 b c f-a d f) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log (c+d x)}{6 b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{5/3}} \]
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Time = 0.08 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {81, 61} \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=-\frac {\arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right ) (-a d f-2 b c f+3 b d e)}{\sqrt {3} b^{4/3} d^{5/3}}-\frac {\log (c+d x) (-a d f-2 b c f+3 b d e)}{6 b^{4/3} d^{5/3}}-\frac {(-a d f-2 b c f+3 b d e) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 b^{4/3} d^{5/3}}+\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d} \]
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Rule 61
Rule 81
Rubi steps \begin{align*} \text {integral}& = \frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}+\frac {\left (b d e-\left (\frac {2 b c}{3}+\frac {a d}{3}\right ) f\right ) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{b d} \\ & = \frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{b d}-\frac {(3 b d e-2 b c f-a d f) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log (c+d x)}{6 b^{4/3} d^{5/3}}-\frac {(3 b d e-2 b c f-a d f) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 b^{4/3} d^{5/3}} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.17 \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\frac {6 \sqrt [3]{b} d^{2/3} f (a+b x)^{2/3} \sqrt [3]{c+d x}+2 \sqrt {3} (3 b d e-2 b c f-a d f) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )+2 (-3 b d e+2 b c f+a d f) \log \left (\sqrt [3]{d}-\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )+(3 b d e-2 b c f-a d f) \log \left (d^{2/3}+\frac {\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {b^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{6 b^{4/3} d^{5/3}} \]
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\[\int \frac {f x +e}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}d x\]
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Time = 0.26 (sec) , antiderivative size = 660, normalized size of antiderivative = 3.30 \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\left [\frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} f - 3 \, \sqrt {\frac {1}{3}} {\left (3 \, b^{2} d^{2} e - {\left (2 \, b^{2} c d + a b d^{2}\right )} f\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \log \left (-3 \, b d^{2} x - 2 \, b c d - a d^{2} + 3 \, \left (b d^{2}\right )^{\frac {1}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} d + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} - \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {-\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}\right ) - 2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{6 \, b^{2} d^{3}}, \frac {6 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} f + 6 \, \sqrt {\frac {1}{3}} {\left (3 \, b^{2} d^{2} e - {\left (2 \, b^{2} c d + a b d^{2}\right )} f\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}\right )} \sqrt {\frac {\left (b d^{2}\right )^{\frac {1}{3}}}{b}}}{b d^{2} x + a d^{2}}\right ) - 2 \, \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d - \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}}{b x + a}\right ) + \left (b d^{2}\right )^{\frac {2}{3}} {\left (3 \, b d e - {\left (2 \, b c + a d\right )} f\right )} \log \left (\frac {{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} b d + \left (b d^{2}\right )^{\frac {2}{3}} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} + \left (b d^{2}\right )^{\frac {1}{3}} {\left (b d x + a d\right )}}{b x + a}\right )}{6 \, b^{2} d^{3}}\right ] \]
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\[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {e + f x}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {f x + e}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {f x + e}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {e+f x}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {e+f\,x}{{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \]
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