Integrand size = 19, antiderivative size = 126 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 \sqrt [3]{b} d^{2/3}} \]
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Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {61} \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{2 \sqrt [3]{b} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}} \]
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Rule 61
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {3} \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} d^{2/3}}-\frac {\log (c+d x)}{2 \sqrt [3]{b} d^{2/3}}-\frac {3 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{2 \sqrt [3]{b} d^{2/3}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-2 \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 \sqrt [3]{b} d^{2/3}} \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (90) = 180\).
Time = 0.25 (sec) , antiderivative size = 521, normalized size of antiderivative = 4.13 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\left [\frac {\sqrt {3} b d \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 - \sqrt {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}} \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}} \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 )}{2 \, b d^{2}}, \frac {2 \, \sqrt {3} b d \sqrt {-\frac {\left (-b d^{2}\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {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}}}{3 \, {\left (b d^{2} x + a d^{2}\right )}}\right ) - 2 \, \left (-b d^{2}\right )^{\frac {2}{3}} \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}} \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 )}{2 \, b d^{2}}\right ] \]
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\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {1}{\sqrt [3]{a + b x} \left (c + d x\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx=\int \frac {1}{{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \]
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