Integrand size = 24, antiderivative size = 132 \[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=-\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {4 b (e f-d g)^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g} \]
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Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2442, 52, 65, 214} \[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {4 b n (e f-d g)^{3/2} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}-\frac {4 b n \sqrt {f+g x} (e f-d g)}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g} \]
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Rule 52
Rule 65
Rule 214
Rule 2442
Rubi steps \begin{align*} \text {integral}& = \frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {(2 b e n) \int \frac {(f+g x)^{3/2}}{d+e x} \, dx}{3 g} \\ & = -\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {(2 b (e f-d g) n) \int \frac {\sqrt {f+g x}}{d+e x} \, dx}{3 g} \\ & = -\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {\left (2 b (e f-d g)^2 n\right ) \int \frac {1}{(d+e x) \sqrt {f+g x}} \, dx}{3 e g} \\ & = -\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac {\left (4 b (e f-d g)^2 n\right ) \text {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{3 e g^2} \\ & = -\frac {4 b (e f-d g) n \sqrt {f+g x}}{3 e g}-\frac {4 b n (f+g x)^{3/2}}{9 g}+\frac {4 b (e f-d g)^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{3 e^{3/2} g}+\frac {2 (f+g x)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.89 \[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\frac {2 \left (6 b (e f-d g)^{3/2} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+\sqrt {e} \sqrt {f+g x} \left (3 a e (f+g x)-2 b n (4 e f-3 d g+e g x)+3 b e (f+g x) \log \left (c (d+e x)^n\right )\right )\right )}{9 e^{3/2} g} \]
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\[\int \sqrt {g x +f}\, \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )d x\]
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none
Time = 0.34 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.36 \[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\left [-\frac {2 \, {\left (3 \, {\left (b e f - b d g\right )} n \sqrt {\frac {e f - d g}{e}} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, \sqrt {g x + f} e \sqrt {\frac {e f - d g}{e}}}{e x + d}\right ) - {\left (3 \, a e f - 2 \, {\left (4 \, b e f - 3 \, b d g\right )} n - {\left (2 \, b e g n - 3 \, a e g\right )} x + 3 \, {\left (b e g n x + b e f n\right )} \log \left (e x + d\right ) + 3 \, {\left (b e g x + b e f\right )} \log \left (c\right )\right )} \sqrt {g x + f}\right )}}{9 \, e g}, \frac {2 \, {\left (6 \, {\left (b e f - b d g\right )} n \sqrt {-\frac {e f - d g}{e}} \arctan \left (-\frac {\sqrt {g x + f} e \sqrt {-\frac {e f - d g}{e}}}{e f - d g}\right ) + {\left (3 \, a e f - 2 \, {\left (4 \, b e f - 3 \, b d g\right )} n - {\left (2 \, b e g n - 3 \, a e g\right )} x + 3 \, {\left (b e g n x + b e f n\right )} \log \left (e x + d\right ) + 3 \, {\left (b e g x + b e f\right )} \log \left (c\right )\right )} \sqrt {g x + f}\right )}}{9 \, e g}\right ] \]
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\[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \sqrt {f + g x}\, dx \]
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Exception generated. \[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\text {Exception raised: ValueError} \]
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\[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int { \sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx=\int \sqrt {f+g\,x}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right ) \,d x \]
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