Optimal. Leaf size=1304 \[ -\frac {b d p q r^2 \log (a+b x)}{h (b g-a h) (d g-c h)}+\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {b p^2 r^2 (a+b x) \log (a+b x)}{(b g-a h)^2 (g+h x)}-\frac {b d p q r^2 \log (c+d x)}{h (b g-a h) (d g-c h)}+\frac {b p q r^2 \log (c+d x)}{h (b g-a h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {b^2 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)^2}+\frac {d^2 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)^2}-\frac {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 )}{h (b g-a h) (g+h x)}-\frac {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 )}{h (d g-c h) (g+h x)}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {b^2 p^2 r^2 \log (g+h x)}{h (b g-a h)^2}+\frac {2 b d p q r^2 \log (g+h x)}{h (b g-a h) (d g-c h)}+\frac {d^2 q^2 r^2 \log (g+h x)}{h (d g-c h)^2}+\frac {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 ) \log (g+h x)}{h (b g-a h)^2}+\frac {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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {d^2 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)^2}-\frac {b^2 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)^2}-\frac {b^2 p^2 r^2 \log (a+b x) \log \left (1+\frac {b g-a h}{h (a+b x)}\right )}{h (b g-a h)^2}-\frac {d^2 q^2 r^2 \log (c+d x) \log \left (1+\frac {d g-c h}{h (c+d x)}\right )}{h (d g-c h)^2}+\frac {b^2 p^2 r^2 \text {Li}_2\left (-\frac {b g-a h}{h (a+b x)}\right )}{h (b g-a h)^2}+\frac {d^2 p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {d^2 p q r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (d g-c h)^2}+\frac {d^2 q^2 r^2 \text {Li}_2\left (-\frac {d g-c h}{h (c+d x)}\right )}{h (d g-c h)^2}+\frac {b^2 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{h (b g-a h)^2}-\frac {b^2 p q r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h (b g-a h)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.92, antiderivative size = 1304, normalized size of antiderivative = 1.00, number
of steps used = 43, number of rules used = 14, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules
used = {2584, 2593, 2458, 2389, 2379, 2438, 2351, 31, 2465, 2441, 2440, 2442, 36, 46}
\begin {gather*} \frac {p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) b^2}{h (b g-a h)^2}-\frac {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 ) b^2}{h (b g-a h)^2}+\frac {p^2 r^2 \log (g+h x) b^2}{h (b g-a h)^2}+\frac {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) b^2}{h (b g-a h)^2}-\frac {p q r^2 \log (c+d x) \log \left (\frac {d (g+h x)}{d g-c h}\right ) b^2}{h (b g-a h)^2}-\frac {p^2 r^2 \log (a+b x) \log \left (\frac {b g-a h}{h (a+b x)}+1\right ) b^2}{h (b g-a h)^2}+\frac {p^2 r^2 \text {PolyLog}\left (2,-\frac {b g-a h}{h (a+b x)}\right ) b^2}{h (b g-a h)^2}+\frac {p q r^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) b^2}{h (b g-a h)^2}-\frac {p q r^2 \text {PolyLog}\left (2,-\frac {h (c+d x)}{d g-c h}\right ) b^2}{h (b g-a h)^2}-\frac {d p q r^2 \log (a+b x) b}{h (b g-a h) (d g-c h)}-\frac {p^2 r^2 (a+b x) \log (a+b x) b}{(b g-a h)^2 (g+h x)}-\frac {d p q r^2 \log (c+d x) b}{h (b g-a h) (d g-c h)}+\frac {p q r^2 \log (c+d x) b}{h (b g-a h) (g+h x)}-\frac {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 ) b}{h (b g-a h) (g+h x)}+\frac {2 d p q r^2 \log (g+h x) b}{h (b g-a h) (d g-c h)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {d^2 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)^2}-\frac {d^2 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)^2}-\frac {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 )}{h (d g-c h) (g+h x)}+\frac {d^2 q^2 r^2 \log (g+h x)}{h (d g-c h)^2}+\frac {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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {d^2 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)^2}-\frac {d^2 q^2 r^2 \log (c+d x) \log \left (\frac {d g-c h}{h (c+d x)}+1\right )}{h (d g-c h)^2}+\frac {d^2 p q r^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {d^2 p q r^2 \text {PolyLog}\left (2,-\frac {h (a+b x)}{b g-a h}\right )}{h (d g-c h)^2}+\frac {d^2 q^2 r^2 \text {PolyLog}\left (2,-\frac {d g-c h}{h (c+d x)}\right )}{h (d g-c h)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 36
Rule 46
Rule 2351
Rule 2379
Rule 2389
Rule 2438
Rule 2440
Rule 2441
Rule 2442
Rule 2458
Rule 2465
Rule 2584
Rule 2593
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)^3} \, dx &=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {(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)^2} \, dx}{h}+\frac {(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)^2} \, dx}{h}\\ &=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {\left (b p^2 r^2\right ) \int \frac {\log (a+b x)}{(a+b x) (g+h x)^2} \, dx}{h}+\frac {\left (b p q r^2\right ) \int \frac {\log (c+d x)}{(a+b x) (g+h x)^2} \, dx}{h}+\frac {\left (d p q r^2\right ) \int \frac {\log (a+b x)}{(c+d x) (g+h x)^2} \, dx}{h}+\frac {\left (d q^2 r^2\right ) \int \frac {\log (c+d x)}{(c+d x) (g+h x)^2} \, dx}{h}-\frac {\left (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)^2} \, dx}{h}-\frac {\left (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)^2} \, dx}{h}\\ &=-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {\left (p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {b g-a h}{b}+\frac {h x}{b}\right )^2} \, dx,x,a+b x\right )}{h}+\frac {\left (b p q r^2\right ) \int \left (\frac {b^2 \log (c+d x)}{(b g-a h)^2 (a+b x)}-\frac {h \log (c+d x)}{(b g-a h) (g+h x)^2}-\frac {b h \log (c+d x)}{(b g-a h)^2 (g+h x)}\right ) \, dx}{h}+\frac {\left (d p q r^2\right ) \int \left (\frac {d^2 \log (a+b x)}{(d g-c h)^2 (c+d x)}-\frac {h \log (a+b x)}{(d g-c h) (g+h x)^2}-\frac {d h \log (a+b x)}{(d g-c h)^2 (g+h x)}\right ) \, dx}{h}+\frac {\left (q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x \left (\frac {d g-c h}{d}+\frac {h x}{d}\right )^2} \, dx,x,c+d x\right )}{h}-\frac {\left (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 \left (\frac {b^2}{(b g-a h)^2 (a+b x)}-\frac {h}{(b g-a h) (g+h x)^2}-\frac {b h}{(b g-a h)^2 (g+h x)}\right ) \, dx}{h}-\frac {\left (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 \left (\frac {d^2}{(d g-c h)^2 (c+d x)}-\frac {h}{(d g-c h) (g+h x)^2}-\frac {d h}{(d g-c h)^2 (g+h x)}\right ) \, dx}{h}\\ &=-\frac {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 )}{h (b g-a h) (g+h x)}-\frac {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 )}{h (d g-c h) (g+h x)}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {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 ) \log (g+h x)}{h (b g-a h)^2}+\frac {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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {\left (p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\left (\frac {b g-a h}{b}+\frac {h x}{b}\right )^2} \, dx,x,a+b x\right )}{b g-a h}+\frac {\left (b p^2 r^2\right ) \text {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 (b g-a h)}-\frac {\left (b^2 p q r^2\right ) \int \frac {\log (c+d x)}{g+h x} \, dx}{(b g-a h)^2}+\frac {\left (b^3 p q r^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{h (b g-a h)^2}-\frac {\left (b p q r^2\right ) \int \frac {\log (c+d x)}{(g+h x)^2} \, dx}{b g-a h}-\frac {\left (d^2 p q r^2\right ) \int \frac {\log (a+b x)}{g+h x} \, dx}{(d g-c h)^2}+\frac {\left (d^3 p q r^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{h (d g-c h)^2}-\frac {\left (d p q r^2\right ) \int \frac {\log (a+b x)}{(g+h x)^2} \, dx}{d g-c h}-\frac {\left (q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{\left (\frac {d g-c h}{d}+\frac {h x}{d}\right )^2} \, dx,x,c+d x\right )}{d g-c h}+\frac {\left (d q^2 r^2\right ) \text {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 (d g-c h)}\\ &=\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {b p^2 r^2 (a+b x) \log (a+b x)}{(b g-a h)^2 (g+h x)}+\frac {b p q r^2 \log (c+d x)}{h (b g-a h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {b^2 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)^2}+\frac {d^2 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)^2}-\frac {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 )}{h (b g-a h) (g+h x)}-\frac {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 )}{h (d g-c h) (g+h x)}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {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 ) \log (g+h x)}{h (b g-a h)^2}+\frac {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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {d^2 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)^2}-\frac {b^2 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)^2}+\frac {\left (b p^2 r^2\right ) \text {Subst}\left (\int \frac {1}{\frac {b g-a h}{b}+\frac {h x}{b}} \, dx,x,a+b x\right )}{(b g-a h)^2}-\frac {\left (b p^2 r^2\right ) \text {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)^2}+\frac {\left (b^2 p^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{h (b g-a h)^2}-\frac {\left (b^2 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)^2}+\frac {\left (b^2 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)^2}-\frac {\left (b d p q r^2\right ) \int \frac {1}{(c+d x) (g+h x)} \, dx}{h (b g-a h)}-\frac {\left (b d^2 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)^2}+\frac {\left (b d^2 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)^2}-\frac {\left (b d p q r^2\right ) \int \frac {1}{(a+b x) (g+h x)} \, dx}{h (d g-c h)}+\frac {\left (d q^2 r^2\right ) \text {Subst}\left (\int \frac {1}{\frac {d g-c h}{d}+\frac {h x}{d}} \, dx,x,c+d x\right )}{(d g-c h)^2}-\frac {\left (d q^2 r^2\right ) \text {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)^2}+\frac {\left (d^2 q^2 r^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{h (d g-c h)^2}\\ &=\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {b p^2 r^2 (a+b x) \log (a+b x)}{(b g-a h)^2 (g+h x)}+\frac {b^2 p^2 r^2 \log ^2(a+b x)}{2 h (b g-a h)^2}+\frac {b p q r^2 \log (c+d x)}{h (b g-a h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {b^2 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)^2}+\frac {d^2 q^2 r^2 \log ^2(c+d x)}{2 h (d g-c h)^2}+\frac {d^2 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)^2}-\frac {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 )}{h (b g-a h) (g+h x)}-\frac {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 )}{h (d g-c h) (g+h x)}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {b^2 p^2 r^2 \log (g+h x)}{h (b g-a h)^2}+\frac {d^2 q^2 r^2 \log (g+h x)}{h (d g-c h)^2}+\frac {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 ) \log (g+h x)}{h (b g-a h)^2}+\frac {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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {b^2 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)^2}-\frac {d^2 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)^2}+\frac {\left (b^2 p^2 r^2\right ) \text {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)^2}-\frac {\left (b^2 p q r^2\right ) \text {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)^2}+\frac {\left (b^2 p q r^2\right ) \text {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)^2}-\frac {\left (d^2 p q r^2\right ) \text {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)^2}+\frac {\left (d^2 p q r^2\right ) \text {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)^2}+2 \frac {\left (b d p q r^2\right ) \int \frac {1}{g+h x} \, dx}{(b g-a h) (d g-c h)}-\frac {\left (b^2 d p q r^2\right ) \int \frac {1}{a+b x} \, dx}{h (b g-a h) (d g-c h)}-\frac {\left (b d^2 p q r^2\right ) \int \frac {1}{c+d x} \, dx}{h (b g-a h) (d g-c h)}+\frac {\left (d^2 q^2 r^2\right ) \text {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)^2}\\ &=-\frac {b d p q r^2 \log (a+b x)}{h (b g-a h) (d g-c h)}+\frac {d p q r^2 \log (a+b x)}{h (d g-c h) (g+h x)}-\frac {b p^2 r^2 (a+b x) \log (a+b x)}{(b g-a h)^2 (g+h x)}+\frac {b^2 p^2 r^2 \log ^2(a+b x)}{2 h (b g-a h)^2}-\frac {b d p q r^2 \log (c+d x)}{h (b g-a h) (d g-c h)}+\frac {b p q r^2 \log (c+d x)}{h (b g-a h) (g+h x)}-\frac {d q^2 r^2 (c+d x) \log (c+d x)}{(d g-c h)^2 (g+h x)}+\frac {b^2 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)^2}+\frac {d^2 q^2 r^2 \log ^2(c+d x)}{2 h (d g-c h)^2}+\frac {d^2 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)^2}-\frac {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 )}{h (b g-a h) (g+h x)}-\frac {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 )}{h (d g-c h) (g+h x)}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac {b^2 p^2 r^2 \log (g+h x)}{h (b g-a h)^2}+\frac {2 b d p q r^2 \log (g+h x)}{h (b g-a h) (d g-c h)}+\frac {d^2 q^2 r^2 \log (g+h x)}{h (d g-c h)^2}+\frac {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 ) \log (g+h x)}{h (b g-a h)^2}+\frac {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 ) \log (g+h x)}{h (d g-c h)^2}-\frac {b^2 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)^2}-\frac {d^2 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)^2}-\frac {b^2 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)^2}-\frac {d^2 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)^2}+\frac {d^2 p q r^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{h (d g-c h)^2}-\frac {b^2 p^2 r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (b g-a h)^2}-\frac {d^2 p q r^2 \text {Li}_2\left (-\frac {h (a+b x)}{b g-a h}\right )}{h (d g-c h)^2}+\frac {b^2 p q r^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{h (b g-a h)^2}-\frac {b^2 p q r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h (b g-a h)^2}-\frac {d^2 q^2 r^2 \text {Li}_2\left (-\frac {h (c+d x)}{d g-c h}\right )}{h (d g-c h)^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(12086\) vs. \(2(1304)=2608\).
time = 3.98, size = 12086, normalized size = 9.27 \begin {gather*} \text {Result too large to show} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )^{2}}{\left (h x +g \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.58, size = 1859, normalized size = 1.43 \begin {gather*} \frac {{\left (\frac {b^{2} f p \log \left (b x + a\right )}{b^{2} g^{2} - 2 \, a b g h + a^{2} h^{2}} + \frac {d^{2} f q \log \left (d x + c\right )}{d^{2} g^{2} - 2 \, c d g h + c^{2} h^{2}} + \frac {{\left (2 \, a b d^{2} f g h q - a^{2} d^{2} f h^{2} q - {\left (d^{2} f g^{2} {\left (p + q\right )} - 2 \, c d f g h p + c^{2} f h^{2} p\right )} b^{2}\right )} \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} + \frac {a d f h q - {\left (d f g {\left (p + q\right )} - c f h p\right )} b}{{\left (d g^{2} h - c g h^{2}\right )} a - {\left (d g^{3} - c g^{2} h\right )} b + {\left ({\left (d g h^{2} - c h^{3}\right )} a - {\left (d g^{2} h - c g h^{2}\right )} b\right )} x}\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 (2 \, a b d^{2} f^{2} g h p q - a^{2} d^{2} f^{2} h^{2} p q - {\left (2 \, c d f^{2} g h p q - c^{2} f^{2} h^{2} p q\right )} b^{2}\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^{2} - c h^{3}\right )} a^{2} - 2 \, {\left (d g^{2} h - c g h^{2}\right )} a b + {\left (d g^{3} - c g^{2} h\right )} b^{2}} - \frac {2 \, {\left (2 \, a b d^{2} f^{2} g h p q - a^{2} d^{2} f^{2} h^{2} p q + {\left (2 \, c d f^{2} g h p^{2} - c^{2} f^{2} h^{2} p^{2} - {\left (p^{2} + p q\right )} d^{2} f^{2} g^{2}\right )} b^{2}\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^{2} - c h^{3}\right )} a^{2} - 2 \, {\left (d g^{2} h - c g h^{2}\right )} a b + {\left (d g^{3} - c g^{2} h\right )} b^{2}} + \frac {2 \, {\left (2 \, a b d^{2} f^{2} g h q^{2} - a^{2} d^{2} f^{2} h^{2} q^{2} + {\left (2 \, c d f^{2} g h p q - c^{2} f^{2} h^{2} p q - {\left (p q + q^{2}\right )} d^{2} f^{2} g^{2}\right )} b^{2}\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^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} - \frac {2 \, {\left (a d^{2} f^{2} h q^{2} + {\left (c d f^{2} h p q - {\left (p q + q^{2}\right )} d^{2} f^{2} g\right )} b\right )} \log \left (d x + c\right )}{{\left (d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}\right )} a - {\left (d^{2} g^{3} - 2 \, c d g^{2} h + c^{2} g h^{2}\right )} b} + \frac {2 \, {\left (a^{2} d^{2} f^{2} h^{2} q^{2} + 2 \, {\left (c d f^{2} h^{2} p q - {\left (p q + q^{2}\right )} d^{2} f^{2} g h\right )} a b + {\left (c^{2} f^{2} h^{2} p^{2} + {\left (p^{2} + 2 \, p q + q^{2}\right )} d^{2} f^{2} g^{2} - 2 \, {\left (p^{2} + p q\right )} c d f^{2} g h\right )} b^{2}\right )} \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} + \frac {{\left (d^{3} f^{2} g^{3} p^{2} - 3 \, c d^{2} f^{2} g^{2} h p^{2} + 3 \, c^{2} d f^{2} g h^{2} p^{2} - c^{3} f^{2} h^{3} p^{2}\right )} b^{2} \log \left (b x + a\right )^{2} - 2 \, {\left (b^{2} d^{2} f^{2} g^{2} p q - 2 \, a b d^{2} f^{2} g h p q + a^{2} d^{2} f^{2} h^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b^{2} d^{2} f^{2} g^{2} q^{2} - 2 \, a b d^{2} f^{2} g h q^{2} + a^{2} d^{2} f^{2} h^{2} q^{2}\right )} \log \left (d x + c\right )^{2} + 2 \, {\left ({\left (d^{2} f^{2} g h p q - c d f^{2} h^{2} p q\right )} a b - {\left (c^{2} f^{2} h^{2} p^{2} + {\left (p^{2} + p q\right )} d^{2} f^{2} g^{2} - {\left (2 \, p^{2} + p q\right )} c d f^{2} g h\right )} b^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left ({\left (2 \, a b d^{2} f^{2} g h p q - a^{2} d^{2} f^{2} h^{2} p q + {\left (2 \, c d f^{2} g h p^{2} - c^{2} f^{2} h^{2} p^{2} - {\left (p^{2} + p q\right )} d^{2} f^{2} g^{2}\right )} b^{2}\right )} \log \left (b x + a\right ) + {\left (2 \, a b d^{2} f^{2} g h q^{2} - a^{2} d^{2} f^{2} h^{2} q^{2} + {\left (2 \, c d f^{2} g h p q - c^{2} f^{2} h^{2} p q - {\left (p q + q^{2}\right )} d^{2} f^{2} g^{2}\right )} b^{2}\right )} \log \left (d x + c\right )\right )} \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \, {\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b + {\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}}\right )} r^{2}}{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}}{2 \, {\left (h x + g\right )}^{2} h} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________