Integrand size = 19, antiderivative size = 149 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=-\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{d^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 d^{4/3}}-\frac {3 \sqrt [3]{b} \log \left (-1+\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}\right )}{2 d^{4/3}} \]
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Time = 0.02 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {49, 61} \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=-\frac {\sqrt {3} \sqrt [3]{b} \arctan \left (\frac {2 \sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{a+b x}}+\frac {1}{\sqrt {3}}\right )}{d^{4/3}}-\frac {3 \sqrt [3]{b} \log \left (\frac {\sqrt [3]{b} \sqrt [3]{c+d x}}{\sqrt [3]{d} \sqrt [3]{a+b x}}-1\right )}{2 d^{4/3}}-\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 d^{4/3}} \]
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Rule 49
Rule 61
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}+\frac {b \int \frac {1}{(a+b x)^{2/3} \sqrt [3]{c+d x}} \, dx}{d} \\ & = -\frac {3 \sqrt [3]{a+b x}}{d \sqrt [3]{c+d x}}-\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 )}{d^{4/3}}-\frac {\sqrt [3]{b} \log (a+b x)}{2 d^{4/3}}-\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 d^{4/3}} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\frac {-\frac {6 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{c+d x}}-2 \sqrt {3} \sqrt [3]{b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )-2 \sqrt [3]{b} \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )+\sqrt [3]{b} \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{2 d^{4/3}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {1}{3}}}{\left (d x +c \right )^{\frac {4}{3}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (109) = 218\).
Time = 0.24 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.56 \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=-\frac {2 \, \sqrt {3} {\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} d \left (-\frac {b}{d}\right )^{\frac {2}{3}} + \sqrt {3} {\left (b d x + b c\right )}}{3 \, {\left (b d x + b c\right )}}\right ) + {\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {2}{3}} - {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{d x + c}\right ) - 2 \, {\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} \log \left (\frac {{\left (d x + c\right )} \left (-\frac {b}{d}\right )^{\frac {1}{3}} + {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{d x + c}\right ) + 6 \, {\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}}}{2 \, {\left (d^{2} x + c d\right )}} \]
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\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int \frac {\sqrt [3]{a + b x}}{\left (c + d x\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {1}{3}}}{{\left (d x + c\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{a+b x}}{(c+d x)^{4/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^{1/3}}{{\left (c+d\,x\right )}^{4/3}} \,d x \]
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