Integrand size = 24, antiderivative size = 97 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=-\frac {4 b n \sqrt {f+g x}}{g}+\frac {4 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} g}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]
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Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2442, 52, 65, 214} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {4 b n \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} g}-\frac {4 b n \sqrt {f+g x}}{g} \]
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Rule 52
Rule 65
Rule 214
Rule 2442
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {(2 b e n) \int \frac {\sqrt {f+g x}}{d+e x} \, dx}{g} \\ & = -\frac {4 b n \sqrt {f+g x}}{g}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {(2 b (e f-d g) n) \int \frac {1}{(d+e x) \sqrt {f+g x}} \, dx}{g} \\ & = -\frac {4 b n \sqrt {f+g x}}{g}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {(4 b (e f-d g) n) \text {Subst}\left (\int \frac {1}{d-\frac {e f}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{g^2} \\ & = -\frac {4 b n \sqrt {f+g x}}{g}+\frac {4 b \sqrt {e f-d g} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} g}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\frac {2 \left (\frac {2 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e}}+\sqrt {f+g x} \left (a-2 b n+b \log \left (c (d+e x)^n\right )\right )\right )}{g} \]
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Time = 0.86 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.16
method | result | size |
derivativedivides | \(\frac {2 \sqrt {g x +f}\, a +2 b \left (\ln \left (c \left (\frac {\left (g x +f \right ) e +d g -e f}{g}\right )^{n}\right ) \sqrt {g x +f}-2 e n \left (\frac {\sqrt {g x +f}}{e}+\frac {\left (-d g +e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \sqrt {\left (d g -e f \right ) e}}\right )\right )}{g}\) | \(113\) |
default | \(\frac {2 \sqrt {g x +f}\, a +2 b \left (\ln \left (c \left (\frac {\left (g x +f \right ) e +d g -e f}{g}\right )^{n}\right ) \sqrt {g x +f}-2 e n \left (\frac {\sqrt {g x +f}}{e}+\frac {\left (-d g +e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \sqrt {\left (d g -e f \right ) e}}\right )\right )}{g}\) | \(113\) |
parts | \(\frac {2 a \sqrt {g x +f}}{g}+\frac {2 b \left (\ln \left (c \left (\frac {\left (g x +f \right ) e +d g -e f}{g}\right )^{n}\right ) \sqrt {g x +f}-2 e n \left (\frac {\sqrt {g x +f}}{e}+\frac {\left (-d g +e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \sqrt {\left (d g -e f \right ) e}}\right )\right )}{g}\) | \(116\) |
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Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.91 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\left [\frac {2 \, {\left (b n \sqrt {\frac {e f - d g}{e}} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {g x + f} e \sqrt {\frac {e f - d g}{e}}}{e x + d}\right ) + {\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g}, \frac {2 \, {\left (2 \, b n \sqrt {-\frac {e f - d g}{e}} \arctan \left (-\frac {\sqrt {g x + f} e \sqrt {-\frac {e f - d g}{e}}}{e f - d g}\right ) + {\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g}\right ] \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\sqrt {f + g x}}\, dx \]
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Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=-\frac {2 \, {\left ({\left (2 \, e {\left (\frac {{\left (e f - d g\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{\sqrt {-e^{2} f + d e g} e} + \frac {\sqrt {g x + f}}{e}\right )} - \sqrt {g x + f} \log \left (e x + d\right )\right )} b n - \sqrt {g x + f} b \log \left (c\right ) - \sqrt {g x + f} a\right )}}{g} \]
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Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {f+g\,x}} \,d x \]
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