Integrand size = 31, antiderivative size = 1471 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\frac {p q r^2 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )}{h}+\frac {p^2 r^2 \log ^2(a+b x) \log (g+h x)}{h}+\frac {2 p q r^2 \log (a+b x) \log (c+d x) \log (g+h x)}{h}+\frac {q^2 r^2 \log ^2(c+d x) \log (g+h x)}{h}-\frac {2 p r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {p^2 r^2 \log ^2(a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \log (a+b x) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {p q r^2 \log ^2\left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {p q r^2 \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {2 p r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \log (a+b x) \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {q^2 r^2 \log ^2(c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \log (a+b x) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {p q r^2 \log ^2\left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}+\frac {2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h}-\frac {p q r^2 \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )}{h}-\frac {2 p r \left (q r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{h}+\frac {2 q r \left (p r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right )}{h}+\frac {2 p q r^2 \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{h}-\frac {2 p q r^2 \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )}{h}-\frac {2 p^2 r^2 \operatorname {PolyLog}\left (3,-\frac {h (a+b x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \operatorname {PolyLog}\left (3,-\frac {h (a+b x)}{b g-a h}\right )}{h}-\frac {2 p q r^2 \operatorname {PolyLog}\left (3,-\frac {h (c+d x)}{d g-c h}\right )}{h}-\frac {2 q^2 r^2 \operatorname {PolyLog}\left (3,-\frac {h (c+d x)}{d g-c h}\right )}{h}-\frac {2 p q r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{h}+\frac {2 p q r^2 \operatorname {PolyLog}\left (3,\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )}{h} \]
[Out]
Time = 1.36 (sec) , antiderivative size = 2096, normalized size of antiderivative = 1.42, number of steps used = 29, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {2583, 2586, 2441, 2440, 2438, 2481, 2422, 2354, 2421, 6724, 2490, 2487, 2485, 2352} \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=-\frac {p q \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (\frac {b g-a h}{b (g+h x)}\right )-\log \left (\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) r^2}{h}+\frac {p q \left (\log \left (\frac {b (c+d x)}{b c-a d}\right )-\log \left (-\frac {h (c+d x)}{d g-c h}\right )\right ) \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right )^2 r^2}{h}-\frac {p q \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )+\log \left (\frac {d g-c h}{d (g+h x)}\right )-\log \left (-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right )\right ) \log ^2\left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) r^2}{h}+\frac {p q \left (\log \left (-\frac {d (a+b x)}{b c-a d}\right )-\log \left (-\frac {h (a+b x)}{b g-a h}\right )\right ) \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right )^2 r^2}{h}-\frac {2 p q \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \log (g+h x) r^2}{h}-\frac {2 p q \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (g+h x) r^2}{h}-\frac {2 p q \left (\log (g+h x)-\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) r^2}{h}-\frac {2 p q \left (\log (g+h x)-\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) r^2}{h}+\frac {2 p q \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {h (a+b x)}{b (g+h x)}\right ) r^2}{h}-\frac {2 p q \log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right ) \operatorname {PolyLog}\left (2,-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right ) r^2}{h}+\frac {2 p q \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \operatorname {PolyLog}\left (2,\frac {h (c+d x)}{d (g+h x)}\right ) r^2}{h}-\frac {2 p q \log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right ) r^2}{h}-\frac {2 p q \left (\log (c+d x)+\log \left (\frac {(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}-\frac {2 p q \left (\log (a+b x)+\log \left (-\frac {(b c-a d) (g+h x)}{(d g-c h) (a+b x)}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}+\frac {2 p q \operatorname {PolyLog}\left (3,-\frac {d (a+b x)}{b c-a d}\right ) r^2}{h}-\frac {2 p^2 \operatorname {PolyLog}\left (3,-\frac {h (a+b x)}{b g-a h}\right ) r^2}{h}+\frac {2 p q \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{b c-a d}\right ) r^2}{h}-\frac {2 q^2 \operatorname {PolyLog}\left (3,-\frac {h (c+d x)}{d g-c h}\right ) r^2}{h}+\frac {2 p q \operatorname {PolyLog}\left (3,\frac {h (a+b x)}{b (g+h x)}\right ) r^2}{h}-\frac {2 p q \operatorname {PolyLog}\left (3,-\frac {(d g-c h) (a+b x)}{(b c-a d) (g+h x)}\right ) r^2}{h}+\frac {2 p q \operatorname {PolyLog}\left (3,\frac {h (c+d x)}{d (g+h x)}\right ) r^2}{h}-\frac {2 p q \operatorname {PolyLog}\left (3,\frac {(b g-a h) (c+d x)}{(b c-a d) (g+h x)}\right ) r^2}{h}+\frac {2 p q \operatorname {PolyLog}\left (3,\frac {b (g+h x)}{b g-a h}\right ) r^2}{h}+\frac {2 p q \operatorname {PolyLog}\left (3,\frac {d (g+h x)}{d g-c h}\right ) r^2}{h}+\frac {2 q \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \log (g+h x) r}{h}+\frac {2 p \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \log (g+h x) r}{h}+\frac {2 p \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h}+\frac {2 q \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h}+\frac {2 p \log \left ((a+b x)^{p r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right ) r}{h}+\frac {2 q \log \left ((c+d x)^{q r}\right ) \operatorname {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) r}{h}+\frac {2 p \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) r}{h}+\frac {2 p \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (g+h x)}{b g-a h}\right ) r}{h}+\frac {2 q \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) r}{h}+\frac {2 q \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (g+h x)}{d g-c h}\right ) r}{h}-\frac {\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac {\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\log ^2\left ((a+b x)^{p r}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h}+\frac {\log ^2\left ((c+d x)^{q r}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h} \]
[In]
[Out]
Rule 2352
Rule 2354
Rule 2421
Rule 2422
Rule 2438
Rule 2440
Rule 2441
Rule 2481
Rule 2485
Rule 2487
Rule 2490
Rule 2583
Rule 2586
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {(2 b p r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac {(2 d q r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{c+d x} \, dx}{h} \\ & = \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {(2 b p r) \int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac {(2 b p r) \int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{a+b x} \, dx}{h}-\frac {(2 d q r) \int \frac {\log \left ((a+b x)^{p r}\right ) \log (g+h x)}{c+d x} \, dx}{h}-\frac {(2 d q r) \int \frac {\log \left ((c+d x)^{q r}\right ) \log (g+h x)}{c+d x} \, dx}{h}-\frac {\left (2 b p r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {\log (g+h x)}{a+b x} \, dx}{h}-\frac {\left (2 d q r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {\log (g+h x)}{c+d x} \, dx}{h} \\ & = \frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}-\frac {(2 p r) \text {Subst}\left (\int \frac {\log \left (x^{p r}\right ) \log \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {(2 p r) \text {Subst}\left (\int \frac {\log \left (\left (\frac {b c-a d}{b}+\frac {d x}{b}\right )^{q r}\right ) \log \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {(2 q r) \text {Subst}\left (\int \frac {\log \left (x^{q r}\right ) \log \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}-\frac {(2 q r) \text {Subst}\left (\int \frac {\log \left (\left (\frac {-b c+a d}{d}+\frac {b x}{d}\right )^{p r}\right ) \log \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\left (2 p r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {\log \left (\frac {h (a+b x)}{-b g+a h}\right )}{g+h x} \, dx+\left (2 q r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {\log \left (\frac {h (c+d x)}{-d g+c h}\right )}{g+h x} \, dx \\ & = -\frac {\log ^2\left ((a+b x)^{p r}\right ) \log (g+h x)}{h}-\frac {\log ^2\left ((c+d x)^{q r}\right ) \log (g+h x)}{h}+\frac {2 p r \log \left (-\frac {h (a+b x)}{b g-a h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac {2 q r \log \left (-\frac {h (c+d x)}{d g-c h}\right ) \left (\log \left ((a+b x)^{p r}\right )+\log \left ((c+d x)^{q r}\right )-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h}+\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)}{h}+\frac {\text {Subst}\left (\int \frac {\log ^2\left (x^{p r}\right )}{\frac {b g-a h}{b}+\frac {h x}{b}} \, dx,x,a+b x\right )}{b}+\frac {\text {Subst}\left (\int \frac {\log ^2\left (x^{q r}\right )}{\frac {d g-c h}{d}+\frac {h x}{d}} \, dx,x,c+d x\right )}{d}-\frac {\left (2 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {b c-a d}{b}+\frac {d x}{b}\right ) \log \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}-\frac {\left (2 p q r^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {-b c+a d}{d}+\frac {b x}{d}\right ) \log \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\frac {\left (2 q r \left (p r \log (a+b x)-\log \left ((a+b x)^{p r}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )}{x} \, dx,x,c+d x\right )}{h}+\frac {\left (2 p r \left (q r \log (c+d x)-\log \left ((c+d x)^{q r}\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )}{x} \, dx,x,a+b x\right )}{h}+\frac {\left (2 p r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b g+a h}\right )}{x} \, dx,x,g+h x\right )}{h}+\frac {\left (2 q r \left (-\log \left ((a+b x)^{p r}\right )-\log \left ((c+d x)^{q r}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d g+c h}\right )}{x} \, dx,x,g+h x\right )}{h} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 1370, normalized size of antiderivative = 0.93 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\frac {p q r^2 \log \left (\frac {-b c+a d}{d (a+b x)}\right ) \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )+p^2 r^2 \log ^2(a+b x) \log (g+h x)+2 p q r^2 \log (a+b x) \log (c+d x) \log (g+h x)+q^2 r^2 \log ^2(c+d x) \log (g+h x)-2 p r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)-2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)+\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log (g+h x)-p^2 r^2 \log ^2(a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )-2 p q r^2 \log (a+b x) \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )+p q r^2 \log ^2\left (\frac {h (c+d x)}{-d g+c h}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )-2 p q r^2 \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )+p q r^2 \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )+2 p r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {b (g+h x)}{b g-a h}\right )-2 p q r^2 \log (a+b x) \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )-q^2 r^2 \log ^2(c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )+2 p q r^2 \log (a+b x) \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )-p q r^2 \log ^2\left (\frac {h (c+d x)}{-d g+c h}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )+2 p q r^2 \log \left (\frac {h (c+d x)}{-d g+c h}\right ) \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )+2 q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \log \left (\frac {d (g+h x)}{d g-c h}\right )-p q r^2 \log ^2\left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \log \left (\frac {(-b c+a d) (g+h x)}{(d g-c h) (a+b x)}\right )+2 p r \left (-q r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (a+b x)}{-b g+a h}\right )+2 q r \left (p r \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \operatorname {PolyLog}\left (2,\frac {h (c+d x)}{-d g+c h}\right )+2 p q r^2 \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )-2 p q r^2 \log \left (\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right ) \operatorname {PolyLog}\left (2,\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )-2 p^2 r^2 \operatorname {PolyLog}\left (3,\frac {h (a+b x)}{-b g+a h}\right )-2 p q r^2 \operatorname {PolyLog}\left (3,\frac {h (a+b x)}{-b g+a h}\right )-2 p q r^2 \operatorname {PolyLog}\left (3,\frac {h (c+d x)}{-d g+c h}\right )-2 q^2 r^2 \operatorname {PolyLog}\left (3,\frac {h (c+d x)}{-d g+c h}\right )-2 p q r^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )+2 p q r^2 \operatorname {PolyLog}\left (3,\frac {(b g-a h) (c+d x)}{(d g-c h) (a+b x)}\right )}{h} \]
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\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{h x +g}d x\]
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\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\text {Timed out} \]
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\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g} \,d x } \]
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\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h x + g} \,d x } \]
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Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{g+h x} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{g+h\,x} \,d x \]
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