Integrand size = 19, antiderivative size = 169 \[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{d}+\frac {2 (b c-a d) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt {3} \sqrt [3]{b} d^{5/3}}+\frac {(b c-a d) \log (c+d x)}{3 \sqrt [3]{b} d^{5/3}}+\frac {(b c-a d) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b} d^{5/3}} \]
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Time = 0.03 (sec) , antiderivative size = 169, 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 = {52, 61} \[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\frac {2 (b c-a d) \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{b} d^{5/3}}+\frac {(b c-a d) \log (c+d x)}{3 \sqrt [3]{b} d^{5/3}}+\frac {(b c-a d) \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{\sqrt [3]{b} d^{5/3}}+\frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{d} \]
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Rule 52
Rule 61
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{d}-\frac {(2 (b c-a d)) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{3 d} \\ & = \frac {(a+b x)^{2/3} \sqrt [3]{c+d x}}{d}+\frac {2 (b c-a d) \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} \sqrt [3]{b} d^{5/3}}+\frac {(b c-a d) \log (c+d x)}{3 \sqrt [3]{b} d^{5/3}}+\frac {(b c-a d) \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{\sqrt [3]{b} d^{5/3}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.36 \[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\frac {3 \sqrt [3]{b} d^{2/3} (a+b x)^{2/3} \sqrt [3]{c+d x}-2 \sqrt {3} (b c-a d) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )+2 (b c-a d) \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )-(b c-a d) \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{3 \sqrt [3]{b} d^{5/3}} \]
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\[\int \frac {\left (b x +a \right )^{\frac {2}{3}}}{\left (d x +c \right )^{\frac {2}{3}}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 286 vs. \(2 (131) = 262\).
Time = 0.25 (sec) , antiderivative size = 619, normalized size of antiderivative = 3.66 \[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\left [\frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{2} c d - a b d^{2}\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 (b c - a d\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 (b c - a d\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 )}{3 \, b d^{3}}, \frac {3 \, {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} b d^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (b^{2} c d - a b d^{2}\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 (b c - a d\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 (b c - a d\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 )}{3 \, b d^{3}}\right ] \]
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\[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\int \frac {\left (a + b x\right )^{\frac {2}{3}}}{\left (c + d x\right )^{\frac {2}{3}}}\, dx \]
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\[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {2}{3}}}{{\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
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\[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\int { \frac {{\left (b x + a\right )}^{\frac {2}{3}}}{{\left (d x + c\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx=\int \frac {{\left (a+b\,x\right )}^{2/3}}{{\left (c+d\,x\right )}^{2/3}} \,d x \]
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