Integrand size = 31, antiderivative size = 349 \[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=-\frac {4 b n \sqrt {f+g x}}{e}+\frac {4 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2}}+\frac {2 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}-\frac {4 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}}-\frac {2 b \sqrt {e f-d g} n \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}} \]
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Time = 0.66 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2458, 2388, 65, 214, 2390, 12, 1601, 6873, 6131, 6055, 2449, 2352, 2356, 52} \[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=-\frac {2 \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}+\frac {2 b n \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{3/2}}+\frac {4 b n \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{e^{3/2}}-\frac {4 b n \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}}-\frac {2 b n \sqrt {e f-d g} \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}}-\frac {4 b n \sqrt {f+g x}}{e} \]
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Rule 12
Rule 52
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2356
Rule 2388
Rule 2390
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{e} \\ & = \frac {g \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^2}+\frac {(e f-d g) \text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^2} \\ & = \frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}-\frac {(2 b n) \text {Subst}\left (\int \frac {\sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}}{x} \, dx,x,d+e x\right )}{e}-\frac {(b (e f-d g) n) \text {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{e^2} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}+\frac {\left (2 b \sqrt {e f-d g} n\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{e^{3/2}}-\frac {(2 b (e f-d g) n) \text {Subst}\left (\int \frac {1}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e^2} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}+\frac {\left (4 b \sqrt {e f-d g} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{\sqrt {e}}-\frac {(4 b (e f-d g) n) \text {Subst}\left (\int \frac {1}{-\frac {e f-d g}{g}+\frac {e x^2}{g}} \, dx,x,\sqrt {f+g x}\right )}{e g} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\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 )}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}+\frac {\left (4 b \sqrt {e f-d g} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{\sqrt {e}} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\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 )}{e^{3/2}}+\frac {2 b \sqrt {e f-d g} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}-\frac {(4 b n) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{e} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\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 )}{e^{3/2}}+\frac {2 b \sqrt {e f-d g} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}-\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 ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}}+\frac {(4 b n) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{e} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\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 )}{e^{3/2}}+\frac {2 b \sqrt {e f-d g} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}-\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 ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}}-\frac {\left (4 b \sqrt {e f-d g} n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}} \\ & = -\frac {4 b n \sqrt {f+g x}}{e}+\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 )}{e^{3/2}}+\frac {2 b \sqrt {e f-d g} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{e^{3/2}}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{e}-\frac {2 \sqrt {e f-d g} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^{3/2}}-\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 ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}}-\frac {2 b \sqrt {e f-d g} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{e^{3/2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=\frac {4 a \sqrt {e} \sqrt {f+g x}-8 b \sqrt {e} n \sqrt {f+g x}+8 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )+4 b \sqrt {e} \sqrt {f+g x} \log \left (c (d+e x)^n\right )+2 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )-2 \sqrt {e f-d g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )-b \sqrt {e f-d g} n \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}-\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+b \sqrt {e f-d g} n \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right ) \left (\log \left (\sqrt {e f-d g}+\sqrt {e} \sqrt {f+g x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {e} \sqrt {f+g x}}{2 \sqrt {e f-d g}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )\right )\right )}{2 e^{3/2}} \]
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\[\int \frac {\sqrt {g x +f}\, \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}{e x +d}d x\]
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\[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=\int { \frac {\sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}{e x + d} \,d x } \]
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\[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \sqrt {f + g x}}{d + e x}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=\int { \frac {\sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}}{e x + d} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx=\int \frac {\sqrt {f+g\,x}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{d+e\,x} \,d x \]
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