Integrand size = 25, antiderivative size = 372 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
[Out]
Time = 0.27 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2465, 2443, 2481, 2421, 6724} \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (-\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (\sqrt {e^2-4 d f}+e+2 f x\right )}{2 a f-b \left (\sqrt {e^2-4 d f}+e\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n^2 \operatorname {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \]
[In]
[Out]
Rule 2421
Rule 2443
Rule 2465
Rule 2481
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 f \log ^2\left (c (a+b x)^n\right )}{\sqrt {e^2-4 d f} \left (e-\sqrt {e^2-4 d f}+2 f x\right )}-\frac {2 f \log ^2\left (c (a+b x)^n\right )}{\sqrt {e^2-4 d f} \left (e+\sqrt {e^2-4 d f}+2 f x\right )}\right ) \, dx \\ & = \frac {(2 f) \int \frac {\log ^2\left (c (a+b x)^n\right )}{e-\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}}-\frac {(2 f) \int \frac {\log ^2\left (c (a+b x)^n\right )}{e+\sqrt {e^2-4 d f}+2 f x} \, dx}{\sqrt {e^2-4 d f}} \\ & = \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {(2 b n) \int \frac {\log \left (c (a+b x)^n\right ) \log \left (\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{a+b x} \, dx}{\sqrt {e^2-4 d f}}+\frac {(2 b n) \int \frac {\log \left (c (a+b x)^n\right ) \log \left (\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{a+b x} \, dx}{\sqrt {e^2-4 d f}} \\ & = \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {(2 n) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \log \left (\frac {b \left (\frac {-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}{b}+\frac {2 f x}{b}\right )}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}+\frac {(2 n) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \log \left (\frac {b \left (\frac {-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}{b}+\frac {2 f x}{b}\right )}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}} \\ & = \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\left (2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 f x}{-2 a f+b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}}+\frac {\left (2 n^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {2 f x}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {e^2-4 d f}} \\ & = \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e-\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {\log ^2\left (c (a+b x)^n\right ) \log \left (-\frac {b \left (e+\sqrt {e^2-4 d f}+2 f x\right )}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n \log \left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}-\frac {2 n^2 \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e-\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}}+\frac {2 n^2 \text {Li}_3\left (\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {e^2-4 d f}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.76 \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\frac {2 \sqrt {e^2-4 d f} n^2 \arctan \left (\frac {e+2 f x}{\sqrt {-e^2+4 d f}}\right ) \log ^2(a+b x)-4 \sqrt {e^2-4 d f} n \arctan \left (\frac {e+2 f x}{\sqrt {-e^2+4 d f}}\right ) \log (a+b x) \log \left (c (a+b x)^n\right )+2 \sqrt {e^2-4 d f} \arctan \left (\frac {e+2 f x}{\sqrt {-e^2+4 d f}}\right ) \log ^2\left (c (a+b x)^n\right )-\sqrt {-e^2+4 d f} n^2 \log ^2(a+b x) \log \left (1-\frac {2 f (a+b x)}{-b e+2 a f+b \sqrt {e^2-4 d f}}\right )+2 \sqrt {-e^2+4 d f} n \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {2 f (a+b x)}{-b e+2 a f+b \sqrt {e^2-4 d f}}\right )+\sqrt {-e^2+4 d f} n^2 \log ^2(a+b x) \log \left (1+\frac {2 f (a+b x)}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )-2 \sqrt {-e^2+4 d f} n \log (a+b x) \log \left (c (a+b x)^n\right ) \log \left (1+\frac {2 f (a+b x)}{-2 a f+b \left (e+\sqrt {e^2-4 d f}\right )}\right )+2 \sqrt {-e^2+4 d f} n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f+b \left (-e+\sqrt {e^2-4 d f}\right )}\right )-2 \sqrt {-e^2+4 d f} n \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )-2 \sqrt {-e^2+4 d f} n^2 \operatorname {PolyLog}\left (3,\frac {2 f (a+b x)}{-b e+2 a f+b \sqrt {e^2-4 d f}}\right )+2 \sqrt {-e^2+4 d f} n^2 \operatorname {PolyLog}\left (3,\frac {2 f (a+b x)}{2 a f-b \left (e+\sqrt {e^2-4 d f}\right )}\right )}{\sqrt {-\left (e^2-4 d f\right )^2}} \]
[In]
[Out]
\[\int \frac {\ln \left (c \left (b x +a \right )^{n}\right )^{2}}{f \,x^{2}+e x +d}d x\]
[In]
[Out]
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{f x^{2} + e x + d} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\text {Timed out} \]
[In]
[Out]
Exception generated. \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{2}}{f x^{2} + e x + d} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\log ^2\left (c (a+b x)^n\right )}{d+e x+f x^2} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^2}{f\,x^2+e\,x+d} \,d x \]
[In]
[Out]