Optimal. Leaf size=1046 \[ -\frac {n \log (a+b x) a^2}{2 b^2 h}+\frac {n x a}{2 b h}-\frac {c n x}{2 d h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {c^2 n \log (c+d x)}{2 d^2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (h x^2+g x+f\right )}{2 h^3}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.72, antiderivative size = 1046, normalized size of antiderivative = 1.00, number of steps used = 37, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {2513, 2418, 2389, 2295, 2395, 43, 2394, 2393, 2391, 701, 634, 618, 206, 628} \[ -\frac {n \log (a+b x) a^2}{2 b^2 h}+\frac {n x a}{2 b h}-\frac {c n x}{2 d h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {c^2 n \log (c+d x)}{2 d^2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x-\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (a+b x) \log \left (-\frac {b \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \log (c+d x) \log \left (-\frac {d \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (h x^2+g x+f\right )}{2 h^3}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {PolyLog}\left (2,\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2+\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) n \text {PolyLog}\left (2,\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 206
Rule 618
Rule 628
Rule 634
Rule 701
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2418
Rule 2513
Rubi steps
\begin {align*} \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx &=n \int \frac {x^3 \log (a+b x)}{f+g x+h x^2} \, dx-n \int \frac {x^3 \log (c+d x)}{f+g x+h x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {x^3}{f+g x+h x^2} \, dx\\ &=n \int \left (-\frac {g \log (a+b x)}{h^2}+\frac {x \log (a+b x)}{h}+\frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (a+b x)}{h^2 \left (f+g x+h x^2\right )}\right ) \, dx-n \int \left (-\frac {g \log (c+d x)}{h^2}+\frac {x \log (c+d x)}{h}+\frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (c+d x)}{h^2 \left (f+g x+h x^2\right )}\right ) \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \left (-\frac {g}{h^2}+\frac {x}{h}+\frac {f g+\left (g^2-f h\right ) x}{h^2 \left (f+g x+h x^2\right )}\right ) \, dx\\ &=\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {n \int \frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (a+b x)}{f+g x+h x^2} \, dx}{h^2}-\frac {n \int \frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (c+d x)}{f+g x+h x^2} \, dx}{h^2}-\frac {(g n) \int \log (a+b x) \, dx}{h^2}+\frac {(g n) \int \log (c+d x) \, dx}{h^2}+\frac {n \int x \log (a+b x) \, dx}{h}-\frac {n \int x \log (c+d x) \, dx}{h}-\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {f g+\left (g^2-f h\right ) x}{f+g x+h x^2} \, dx}{h^2}\\ &=\frac {n x^2 \log (a+b x)}{2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}+\frac {n \int \left (\frac {\left (g^2-f h+\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g^2-f h-\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{h^2}-\frac {n \int \left (\frac {\left (g^2-f h+\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g^2-f h-\frac {g \left (-g^2+3 f h\right )}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{h^2}-\frac {(g n) \operatorname {Subst}(\int \log (x) \, dx,x,a+b x)}{b h^2}+\frac {(g n) \operatorname {Subst}(\int \log (x) \, dx,x,c+d x)}{d h^2}-\frac {(b n) \int \frac {x^2}{a+b x} \, dx}{2 h}+\frac {(d n) \int \frac {x^2}{c+d x} \, dx}{2 h}+\frac {\left (g \left (g^2-3 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac {1}{f+g x+h x^2} \, dx}{2 h^3}-\frac {\left (\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac {g+2 h x}{f+g x+h x^2} \, dx}{2 h^3}\\ &=\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {(b n) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{2 h}+\frac {(d n) \int \left (-\frac {c}{d^2}+\frac {x}{d}+\frac {c^2}{d^2 (c+d x)}\right ) \, dx}{2 h}+\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}-\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}+\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}-\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x} \, dx}{h^2}-\frac {\left (g \left (g^2-3 f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{g^2-4 f h-x^2} \, dx,x,g+2 h x\right )}{h^3}\\ &=\frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {\left (b \left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{a+b x} \, dx}{2 h^3}+\frac {\left (d \left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{c+d x} \, dx}{2 h^3}-\frac {\left (b \left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{a+b x} \, dx}{2 h^3}+\frac {\left (d \left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \int \frac {\log \left (\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{c+d x} \, dx}{2 h^3}\\ &=\frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 a h+b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 h^3}+\frac {\left (\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 c h+d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 h^3}-\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 a h+b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,a+b x\right )}{2 h^3}+\frac {\left (\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 h x}{-2 c h+d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{x} \, dx,x,c+d x\right )}{2 h^3}\\ &=\frac {a n x}{2 b h}-\frac {c n x}{2 d h}-\frac {a^2 n \log (a+b x)}{2 b^2 h}+\frac {n x^2 \log (a+b x)}{2 h}-\frac {g n (a+b x) \log (a+b x)}{b h^2}+\frac {c^2 n \log (c+d x)}{2 d^2 h}-\frac {n x^2 \log (c+d x)}{2 h}+\frac {g n (c+d x) \log (c+d x)}{d h^2}+\frac {g x \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^2}-\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 h}-\frac {g \left (g^2-3 f h\right ) \tanh ^{-1}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (a+b x) \log \left (-\frac {b \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \log (c+d x) \log \left (-\frac {d \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h\right ) \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f+g x+h x^2\right )}{2 h^3}+\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}+\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g-\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}-\frac {\left (g^2-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \text {Li}_2\left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )}{2 h^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.48, size = 1240, normalized size = 1.19 \[ \frac {x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) h^2+\frac {n \left (b \left (b \log (c+d x) c^2+d (a d-b c) x\right )-a^2 d^2 \log (a+b x)\right ) h^2}{b^2 d^2}-\frac {2 g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) h}{b}+\frac {2 (b c-a d) g n \log (c+d x) h}{b d}+\frac {2 f g \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x-\sqrt {g^2-4 f h}\right ) h}{\sqrt {g^2-4 f h}}-\frac {2 f g \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x+\sqrt {g^2-4 f h}\right ) h}{\sqrt {g^2-4 f h}}-\frac {2 f g n \left (\left (\log \left (\frac {2 h (a+b x)}{-g b+\sqrt {g^2-4 f h} b+2 a h}\right )-\log \left (\frac {2 h (c+d x)}{-g d+\sqrt {g^2-4 f h} d+2 c h}\right )\right ) \log \left (g+2 h x-\sqrt {g^2-4 f h}\right )+\text {Li}_2\left (\frac {b \left (-g-2 h x+\sqrt {g^2-4 f h}\right )}{-g b+\sqrt {g^2-4 f h} b+2 a h}\right )-\text {Li}_2\left (\frac {d \left (-g-2 h x+\sqrt {g^2-4 f h}\right )}{2 c h+d \left (\sqrt {g^2-4 f h}-g\right )}\right )\right ) h}{\sqrt {g^2-4 f h}}+\frac {2 f g n \left (\left (\log \left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\log \left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right ) \log \left (g+2 h x+\sqrt {g^2-4 f h}\right )+\text {Li}_2\left (\frac {b \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{b \left (g+\sqrt {g^2-4 f h}\right )-2 a h}\right )-\text {Li}_2\left (\frac {d \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{d \left (g+\sqrt {g^2-4 f h}\right )-2 c h}\right )\right ) h}{\sqrt {g^2-4 f h}}+\left (g^2-f h\right ) \left (1-\frac {g}{\sqrt {g^2-4 f h}}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x-\sqrt {g^2-4 f h}\right )+\left (g^2-f h\right ) \left (\frac {g}{\sqrt {g^2-4 f h}}+1\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+2 h x+\sqrt {g^2-4 f h}\right )-\frac {\left (g^2-f h\right ) \left (\sqrt {g^2-4 f h}-g\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{-g b+\sqrt {g^2-4 f h} b+2 a h}\right )-\log \left (\frac {2 h (c+d x)}{-g d+\sqrt {g^2-4 f h} d+2 c h}\right )\right ) \log \left (g+2 h x-\sqrt {g^2-4 f h}\right )+\text {Li}_2\left (\frac {b \left (-g-2 h x+\sqrt {g^2-4 f h}\right )}{-g b+\sqrt {g^2-4 f h} b+2 a h}\right )-\text {Li}_2\left (\frac {d \left (-g-2 h x+\sqrt {g^2-4 f h}\right )}{2 c h+d \left (\sqrt {g^2-4 f h}-g\right )}\right )\right )}{\sqrt {g^2-4 f h}}-\frac {\left (g^2-f h\right ) \left (g+\sqrt {g^2-4 f h}\right ) n \left (\left (\log \left (\frac {2 h (a+b x)}{2 a h-b \left (g+\sqrt {g^2-4 f h}\right )}\right )-\log \left (\frac {2 h (c+d x)}{2 c h-d \left (g+\sqrt {g^2-4 f h}\right )}\right )\right ) \log \left (g+2 h x+\sqrt {g^2-4 f h}\right )+\text {Li}_2\left (\frac {b \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{b \left (g+\sqrt {g^2-4 f h}\right )-2 a h}\right )-\text {Li}_2\left (\frac {d \left (g+2 h x+\sqrt {g^2-4 f h}\right )}{d \left (g+\sqrt {g^2-4 f h}\right )-2 c h}\right )\right )}{\sqrt {g^2-4 f h}}}{2 h^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.95, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________