Optimal. Leaf size=832 \[ \frac {2 b p q \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) r^2}{h (b g-a h)}+\frac {2 d p q \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac {2 d p q \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h (d g-c h)}-\frac {2 b p q \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h (b g-a h)}-\frac {2 b p^2 \log (a+b x) \log \left (\frac {b g-a h}{h (a+b x)}+1\right ) r^2}{h (b g-a h)}-\frac {2 d q^2 \log (c+d x) \log \left (\frac {d g-c h}{h (c+d x)}+1\right ) r^2}{h (d g-c h)}+\frac {2 b p^2 \text {Li}_2\left (-\frac {b g-a h}{h (a+b x)}\right ) r^2}{h (b g-a h)}+\frac {2 d p q \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac {2 d p q \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right ) r^2}{h (d g-c h)}+\frac {2 d q^2 \text {Li}_2\left (-\frac {d g-c h}{h (c+d x)}\right ) r^2}{h (d g-c h)}+\frac {2 b p q \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right ) r^2}{h (b g-a h)}-\frac {2 b p q \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right ) r^2}{h (b g-a h)}-\frac {2 b p \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (b g-a h)}-\frac {2 d q \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (d g-c h)}+\frac {2 b p \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (b g-a h)}+\frac {2 d q \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)} \]
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Rubi [A] time = 0.93, antiderivative size = 878, normalized size of antiderivative = 1.06, number of steps used = 35, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2498, 2513, 2411, 2344, 2301, 2317, 2391, 2418, 2394, 2393, 36, 31} \[ \frac {b p^2 \log ^2(a+b x) r^2}{h (b g-a h)}+\frac {d q^2 \log ^2(c+d x) r^2}{h (d g-c h)}+\frac {2 b p q \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) r^2}{h (b g-a h)}+\frac {2 d p q \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac {2 b p^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h (b g-a h)}-\frac {2 d p q \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right ) r^2}{h (d g-c h)}-\frac {2 d q^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h (d g-c h)}-\frac {2 b p q \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) r^2}{h (b g-a h)}+\frac {2 d p q \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) r^2}{h (d g-c h)}-\frac {2 b p^2 \text {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right ) r^2}{h (b g-a h)}-\frac {2 d p q \text {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right ) r^2}{h (d g-c h)}+\frac {2 b p q \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) r^2}{h (b g-a h)}-\frac {2 d q^2 \text {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) r^2}{h (d g-c h)}-\frac {2 b p q \text {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) r^2}{h (b g-a h)}-\frac {2 b p \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (b g-a h)}-\frac {2 d q \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) r}{h (d g-c h)}+\frac {2 b p \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (b g-a h)}+\frac {2 d q \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x) r}{h (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 36
Rule 2301
Rule 2317
Rule 2344
Rule 2391
Rule 2393
Rule 2394
Rule 2411
Rule 2418
Rule 2498
Rule 2513
Rubi steps
\begin {align*} \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^2} \, dx &=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {(2 b p r) \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (g+h 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 )}{(c+d x) (g+h x)} \, dx}{h}\\ &=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {\left (2 b p^2 r^2\right ) \int \frac {\log (a+b x)}{(a+b x) (g+h x)} \, dx}{h}+\frac {\left (2 b p q r^2\right ) \int \frac {\log (c+d x)}{(a+b x) (g+h x)} \, dx}{h}+\frac {\left (2 d p q r^2\right ) \int \frac {\log (a+b x)}{(c+d x) (g+h x)} \, dx}{h}+\frac {\left (2 d q^2 r^2\right ) \int \frac {\log (c+d x)}{(c+d x) (g+h x)} \, dx}{h}-\frac {\left (2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {1}{(a+b x) (g+h x)} \, dx}{h}-\frac {\left (2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {1}{(c+d x) (g+h x)} \, dx}{h}\\ &=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {\left (2 p^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )} \, dx,x,a+b x\right )}{h}+\frac {\left (2 b p q r^2\right ) \int \left (\frac {b \log (c+d x)}{(b g-a h) (a+b x)}-\frac {h \log (c+d x)}{(b g-a h) (g+h x)}\right ) \, dx}{h}+\frac {\left (2 d p q r^2\right ) \int \left (\frac {d \log (a+b x)}{(d g-c h) (c+d x)}-\frac {h \log (a+b x)}{(d g-c h) (g+h x)}\right ) \, dx}{h}+\frac {\left (2 q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )} \, dx,x,c+d x\right )}{h}+\frac {\left (2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {1}{g+h x} \, dx}{b g-a h}-\frac {\left (2 b^2 p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {1}{a+b x} \, dx}{h (b g-a h)}+\frac {\left (2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {1}{g+h x} \, dx}{d g-c h}-\frac {\left (2 d^2 q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )\right ) \int \frac {1}{c+d x} \, dx}{h (d g-c h)}\\ &=-\frac {2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac {2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac {2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac {\left (2 p^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {b g-a h}{b}+\frac {h x}{b}} \, dx,x,a+b x\right )}{b g-a h}+\frac {\left (2 b p^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{h (b g-a h)}-\frac {\left (2 b p q r^2\right ) \int \frac {\log (c+d x)}{g+h x} \, dx}{b g-a h}+\frac {\left (2 b^2 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{h (b g-a h)}-\frac {\left (2 d p q r^2\right ) \int \frac {\log (a+b x)}{g+h x} \, dx}{d g-c h}+\frac {\left (2 d^2 p q r^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{h (d g-c h)}-\frac {\left (2 q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{\frac {d g-c h}{d}+\frac {h x}{d}} \, dx,x,c+d x\right )}{d g-c h}+\frac {\left (2 d q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{h (d g-c h)}\\ &=\frac {b p^2 r^2 \log ^2(a+b x)}{h (b g-a h)}+\frac {2 b p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{h (b g-a h)}+\frac {d q^2 r^2 \log ^2(c+d x)}{h (d g-c h)}+\frac {2 d p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{h (d g-c h)}-\frac {2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac {2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac {2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac {2 b p^2 r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (b g-a h)}-\frac {2 d p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (d g-c h)}-\frac {2 b p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (b g-a h)}-\frac {2 d q^2 r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (d g-c h)}+\frac {\left (2 b p^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h (b g-a h)}-\frac {\left (2 b d p q r^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{h (b g-a h)}+\frac {\left (2 b d p q r^2\right ) \int \frac {\log \left (\frac {d (g+h x)}{d g-c h}\right )}{c+d x} \, dx}{h (b g-a h)}-\frac {\left (2 b d p q r^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{h (d g-c h)}+\frac {\left (2 b d p q r^2\right ) \int \frac {\log \left (\frac {b (g+h x)}{b g-a h}\right )}{a+b x} \, dx}{h (d g-c h)}+\frac {\left (2 d q^2 r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h (d g-c h)}\\ &=\frac {b p^2 r^2 \log ^2(a+b x)}{h (b g-a h)}+\frac {2 b p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{h (b g-a h)}+\frac {d q^2 r^2 \log ^2(c+d x)}{h (d g-c h)}+\frac {2 d p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{h (d g-c h)}-\frac {2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac {2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac {2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac {2 b p^2 r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (b g-a h)}-\frac {2 d p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (d g-c h)}-\frac {2 b p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (b g-a h)}-\frac {2 d q^2 r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (d g-c h)}-\frac {2 b p^2 r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (b g-a h)}-\frac {2 d q^2 r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h (d g-c h)}-\frac {\left (2 b p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{h (b g-a h)}+\frac {\left (2 b p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {h x}{d g-c h}\right )}{x} \, dx,x,c+d x\right )}{h (b g-a h)}-\frac {\left (2 d p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{h (d g-c h)}+\frac {\left (2 d p q r^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {h x}{b g-a h}\right )}{x} \, dx,x,a+b x\right )}{h (d g-c h)}\\ &=\frac {b p^2 r^2 \log ^2(a+b x)}{h (b g-a h)}+\frac {2 b p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{h (b g-a h)}+\frac {d q^2 r^2 \log ^2(c+d x)}{h (d g-c h)}+\frac {2 d p q r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{h (d g-c h)}-\frac {2 b p r \log (a+b x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (b g-a h)}-\frac {2 d q r \log (c+d x) \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )}{h (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{h (g+h x)}+\frac {2 b p r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (b g-a h)}+\frac {2 d q r \left (p r \log (a+b x)+q r \log (c+d x)-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right ) \log (g+h x)}{h (d g-c h)}-\frac {2 b p^2 r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (b g-a h)}-\frac {2 d p q r^2 \log (a+b x) \log \left (\frac {b (g+h x)}{b g-a h}\right )}{h (d g-c h)}-\frac {2 b p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (b g-a h)}-\frac {2 d q^2 r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right )}{h (d g-c h)}+\frac {2 d p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h (d g-c h)}-\frac {2 b p^2 r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (b g-a h)}-\frac {2 d p q r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (d g-c h)}+\frac {2 b p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{h (b g-a h)}-\frac {2 b p q r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h (b g-a h)}-\frac {2 d q^2 r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h (d g-c h)}\\ \end {align*}
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Mathematica [B] time = 1.34, size = 2930, normalized size = 3.52 \[ \text {Result too large to show} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{h^{2} x^{2} + 2 \, g h x + g^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}}{\left (h x +g \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.80, size = 745, normalized size = 0.90 \[ \frac {2 \, {\left (\frac {b f p \log \left (b x + a\right )}{b g - a h} + \frac {d f q \log \left (d x + c\right )}{d g - c h} - \frac {{\left (a d f h q - {\left (d f g {\left (p + q\right )} - c f h p\right )} b\right )} \log \left (h x + g\right )}{{\left (d g h - c h^{2}\right )} a - {\left (d g^{2} - c g h\right )} b}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{f h} - \frac {{\left (\frac {2 \, {\left (b c f^{2} h p q - a d f^{2} h p q\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )}}{{\left (d g h - c h^{2}\right )} a - {\left (d g^{2} - c g h\right )} b} + \frac {2 \, {\left (a d f^{2} h p q + {\left (c f^{2} h p^{2} - {\left (p^{2} + p q\right )} d f^{2} g\right )} b\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b h x + a h}{b g - a h} + 1\right ) + {\rm Li}_2\left (-\frac {b h x + a h}{b g - a h}\right )\right )}}{{\left (d g h - c h^{2}\right )} a - {\left (d g^{2} - c g h\right )} b} + \frac {2 \, {\left (a d f^{2} h q^{2} + {\left (c f^{2} h p q - {\left (p q + q^{2}\right )} d f^{2} g\right )} b\right )} {\left (\log \left (d x + c\right ) \log \left (\frac {d h x + c h}{d g - c h} + 1\right ) + {\rm Li}_2\left (-\frac {d h x + c h}{d g - c h}\right )\right )}}{{\left (d g h - c h^{2}\right )} a - {\left (d g^{2} - c g h\right )} b} - \frac {{\left (d f^{2} g p^{2} - c f^{2} h p^{2}\right )} b \log \left (b x + a\right )^{2} + 2 \, {\left (b d f^{2} g p q - a d f^{2} h p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) + {\left (b d f^{2} g q^{2} - a d f^{2} h q^{2}\right )} \log \left (d x + c\right )^{2} + 2 \, {\left ({\left (a d f^{2} h p q + {\left (c f^{2} h p^{2} - {\left (p^{2} + p q\right )} d f^{2} g\right )} b\right )} \log \left (b x + a\right ) + {\left (a d f^{2} h q^{2} + {\left (c f^{2} h p q - {\left (p q + q^{2}\right )} d f^{2} g\right )} b\right )} \log \left (d x + c\right )\right )} \log \left (h x + g\right )}{{\left (d g h - c h^{2}\right )} a - {\left (d g^{2} - c g h\right )} b}\right )} r^{2}}{f^{2} h} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (h x + g\right )} h} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (g+h\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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