Integrand size = 26, antiderivative size = 477 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}} \]
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Time = 0.44 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {105, 156, 12, 93} \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \arctan \left (\frac {2 \sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt {3} \sqrt [3]{c+d x} \sqrt [3]{b e-a f}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\log (e+f x) \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right )}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (2 c^2 f^2-6 c d e f+9 d^2 e^2\right )\right ) \log \left (\frac {\sqrt [3]{a+b x} \sqrt [3]{d e-c f}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x} (-5 a d f-4 b c f+9 b d e)}{6 (e+f x) (b e-a f)^2 (d e-c f)^2}-\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (e+f x)^2 (b e-a f) (d e-c f)} \]
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Rule 12
Rule 93
Rule 105
Rule 156
Rubi steps \begin{align*} \text {integral}& = -\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {\int \frac {\frac {1}{3} (-6 b d e+4 b c f+5 a d f)+b d f x}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^2} \, dx}{2 (b e-a f) (d e-c f)} \\ & = -\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\int \frac {2 \left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right )}{9 \sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx}{2 (b e-a f)^2 (d e-c f)^2} \\ & = -\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)} \, dx}{9 (b e-a f)^2 (d e-c f)^2} \\ & = -\frac {f (a+b x)^{2/3} \sqrt [3]{c+d x}}{2 (b e-a f) (d e-c f) (e+f x)^2}-\frac {f (9 b d e-4 b c f-5 a d f) (a+b x)^{2/3} \sqrt [3]{c+d x}}{6 (b e-a f)^2 (d e-c f)^2 (e+f x)}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt {3} \sqrt [3]{b e-a f} \sqrt [3]{c+d x}}\right )}{3 \sqrt {3} (b e-a f)^{7/3} (d e-c f)^{8/3}}+\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log (e+f x)}{18 (b e-a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2-2 a b d f (6 d e-c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\frac {\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}{\sqrt [3]{b e-a f}}-\sqrt [3]{c+d x}\right )}{6 (b e-a f)^{7/3} (d e-c f)^{8/3}} \\ \end{align*}
Time = 3.78 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.10 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\frac {1}{18} \left (\frac {3 f (a+b x)^{2/3} \sqrt [3]{c+d x} (-3 b d e (4 e+3 f x)+b c f (7 e+4 f x)+a f (8 d e-3 c f+5 d f x))}{(b e-a f)^2 (d e-c f)^2 (e+f x)^2}-\frac {2 \sqrt {3} \left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{d e-c f} \sqrt [3]{a+b x}}}{\sqrt {3}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}+\frac {2 \left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left (\sqrt [3]{d e-c f}+\frac {\sqrt [3]{-b e+a f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}-\frac {\left (5 a^2 d^2 f^2+2 a b d f (-6 d e+c f)+b^2 \left (9 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \log \left ((d e-c f)^{2/3}-\frac {\sqrt [3]{-b e+a f} \sqrt [3]{d e-c f} \sqrt [3]{c+d x}}{\sqrt [3]{a+b x}}+\frac {(-b e+a f)^{2/3} (c+d x)^{2/3}}{(a+b x)^{2/3}}\right )}{(-b e+a f)^{7/3} (d e-c f)^{8/3}}\right ) \]
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\[\int \frac {1}{\left (b x +a \right )^{\frac {1}{3}} \left (d x +c \right )^{\frac {2}{3}} \left (f x +e \right )^{3}}d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 2645 vs. \(2 (427) = 854\).
Time = 3.74 (sec) , antiderivative size = 5445, normalized size of antiderivative = 11.42 \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\int { \frac {1}{{\left (b x + a\right )}^{\frac {1}{3}} {\left (d x + c\right )}^{\frac {2}{3}} {\left (f x + e\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [3]{a+b x} (c+d x)^{2/3} (e+f x)^3} \, dx=\int \frac {1}{{\left (e+f\,x\right )}^3\,{\left (a+b\,x\right )}^{1/3}\,{\left (c+d\,x\right )}^{2/3}} \,d x \]
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