Integrand size = 19, antiderivative size = 219 \[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\frac {(b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}+\frac {(b c-a d)^2 \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{6 b^{4/3} d^{5/3}} \]
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Time = 0.06 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {52, 61} \[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\frac {(b c-a d)^2 \arctan \left (\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log \left (\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}-1\right )}{6 b^{4/3} d^{5/3}}+\frac {(a+b x)^{2/3} \sqrt [3]{c+d x} (b c-a d)}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b} \]
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Rule 52
Rule 61
Rubi steps \begin{align*} \text {integral}& = \frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}+\frac {(b c-a d) \int \frac {(a+b x)^{2/3}}{(c+d x)^{2/3}} \, dx}{6 b} \\ & = \frac {(b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}-\frac {(b c-a d)^2 \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3}} \, dx}{9 b d} \\ & = \frac {(b c-a d) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 b d}+\frac {(a+b x)^{5/3} \sqrt [3]{c+d x}}{2 b}+\frac {(b c-a d)^2 \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 )}{3 \sqrt {3} b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log (c+d x)}{18 b^{4/3} d^{5/3}}+\frac {(b c-a d)^2 \log \left (-1+\frac {\sqrt [3]{d} \sqrt [3]{a+b x}}{\sqrt [3]{b} \sqrt [3]{c+d x}}\right )}{6 b^{4/3} d^{5/3}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.13 \[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\frac {3 \sqrt [3]{b} d^{2/3} (a+b x)^{2/3} \sqrt [3]{c+d x} (2 a d+b (c+3 d x))-2 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} \sqrt [3]{c+d x}}{2 \sqrt [3]{d} \sqrt [3]{a+b x}+\sqrt [3]{b} \sqrt [3]{c+d x}}\right )+2 (b c-a d)^2 \log \left (\sqrt [3]{d} \sqrt [3]{a+b x}-\sqrt [3]{b} \sqrt [3]{c+d x}\right )-(b c-a d)^2 \log \left (d^{2/3} (a+b x)^{2/3}+\sqrt [3]{b} \sqrt [3]{d} \sqrt [3]{a+b x} \sqrt [3]{c+d x}+b^{2/3} (c+d x)^{2/3}\right )}{18 b^{4/3} d^{5/3}} \]
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\[\int \left (b x +a \right )^{\frac {2}{3}} \left (d x +c \right )^{\frac {1}{3}}d x\]
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Time = 0.26 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.27 \[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 ) + 3 \, {\left (3 \, b^{2} d^{3} x + b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b^{2} d^{3}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \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^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\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 ) - 3 \, {\left (3 \, b^{2} d^{3} x + b^{2} c d^{2} + 2 \, a b d^{3}\right )} {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}}}{18 \, b^{2} d^{3}}\right ] \]
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\[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\int \left (a + b x\right )^{\frac {2}{3}} \sqrt [3]{c + d x}\, dx \]
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\[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
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\[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\int { {\left (b x + a\right )}^{\frac {2}{3}} {\left (d x + c\right )}^{\frac {1}{3}} \,d x } \]
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Timed out. \[ \int (a+b x)^{2/3} \sqrt [3]{c+d x} \, dx=\int {\left (a+b\,x\right )}^{2/3}\,{\left (c+d\,x\right )}^{1/3} \,d x \]
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