Integrand size = 34, antiderivative size = 510 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \operatorname {PolyLog}\left (2,-\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \operatorname {PolyLog}\left (2,-\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}} \]
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Time = 0.45 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {2454, 222, 2451, 12, 4825, 4617, 2221, 2317, 2438} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \operatorname {PolyLog}\left (2,-\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \operatorname {PolyLog}\left (2,-\frac {e e^{i \arcsin \left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \log \left (1+\frac {e f e^{i \arcsin \left (\frac {g x}{f}\right )}}{-\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right ) \log \left (1+\frac {e f e^{i \arcsin \left (\frac {g x}{f}\right )}}{\sqrt {e^2 f^2-d^2 g^2}+i d g}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \arcsin \left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}} \]
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Rule 12
Rule 222
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 2454
Rule 4617
Rule 4825
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-\frac {g^2 x^2}{f^2}} \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {1-\frac {g^2 x^2}{f^2}}} \, dx}{\sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (b e n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \int \frac {f \sin ^{-1}\left (\frac {g x}{f}\right )}{d g+e g x} \, dx}{\sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \int \frac {\sin ^{-1}\left (\frac {g x}{f}\right )}{d g+e g x} \, dx}{\sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \frac {x \cos (x)}{\frac {d g^2}{f}+e g \sin (x)} \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{\sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{e e^{i x} g+\frac {i d g^2}{f}-\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}} \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{\sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b e f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \frac {e^{i x} x}{e e^{i x} g+\frac {i d g^2}{f}+\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}} \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{\sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {\left (b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \log \left (1+\frac {e e^{i x} g}{\frac {i d g^2}{f}-\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right ) \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {\left (b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \log \left (1+\frac {e e^{i x} g}{\frac {i d g^2}{f}+\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right ) \, dx,x,\sin ^{-1}\left (\frac {g x}{f}\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e g x}{\frac {i d g^2}{f}-\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {\left (i b f n \sqrt {1-\frac {g^2 x^2}{f^2}}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {e g x}{\frac {i d g^2}{f}+\frac {g \sqrt {e^2 f^2-d^2 g^2}}{f}}\right )}{x} \, dx,x,e^{i \sin ^{-1}\left (\frac {g x}{f}\right )}\right )}{g \sqrt {f-g x} \sqrt {f+g x}} \\ & = \frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right )^2}{2 g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}-\frac {b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \log \left (1+\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {f \sqrt {1-\frac {g^2 x^2}{f^2}} \sin ^{-1}\left (\frac {g x}{f}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {Li}_2\left (-\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g-\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}}+\frac {i b f n \sqrt {1-\frac {g^2 x^2}{f^2}} \text {Li}_2\left (-\frac {e e^{i \sin ^{-1}\left (\frac {g x}{f}\right )} f}{i d g+\sqrt {e^2 f^2-d^2 g^2}}\right )}{g \sqrt {f-g x} \sqrt {f+g x}} \\ \end{align*}
Time = 10.18 (sec) , antiderivative size = 631, normalized size of antiderivative = 1.24 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\frac {\arctan \left (\frac {g x}{\sqrt {f-g x} \sqrt {f+g x}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{g}-\frac {b n \sqrt {f-g x} \left (2 f g (d+e x) \sqrt {\frac {f+g x}{f-g x}} \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right ) \log (d+e x)+(f+g x) \left (d g+e f \cos \left (2 \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right )\right ) \csc \left (2 \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right ) \left (2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )^2-4 i \arcsin \left (\frac {\sqrt {1+\frac {d g}{e f}}}{\sqrt {2}}\right ) \arctan \left (\frac {-e f+d g}{\sqrt {-e^2 f^2+d^2 g^2} \sqrt {\frac {f+g x}{f-g x}}}\right )-2 \left (\arcsin \left (\frac {\sqrt {1+\frac {d g}{e f}}}{\sqrt {2}}\right )+\arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (d g-\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )+2 \left (\arcsin \left (\frac {\sqrt {1+\frac {d g}{e f}}}{\sqrt {2}}\right )-\arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )\right ) \log \left (1+\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (d g+\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (-d g+\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )+\operatorname {PolyLog}\left (2,-\frac {e^{2 i \arctan \left (\frac {1}{\sqrt {\frac {f+g x}{f-g x}}}\right )} \left (d g+\sqrt {-e^2 f^2+d^2 g^2}\right )}{e f}\right )\right )\right )\right )}{f g^2 (d+e x) \sqrt {f+g x}} \]
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\[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\sqrt {-g x +f}\, \sqrt {g x +f}}d x\]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x + f} \sqrt {-g x + f}} \,d x } \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\sqrt {f - g x} \sqrt {f + g x}}\, dx \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x + f} \sqrt {-g x + f}} \,d x } \]
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\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\int { \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{\sqrt {g x + f} \sqrt {-g x + f}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f-g x} \sqrt {f+g x}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {f+g\,x}\,\sqrt {f-g\,x}} \,d x \]
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