Optimal. Leaf size=831 \[ \frac {n (a+b x) \log (a+b x)}{b h}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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 ) \log (a+b x)}{2 h^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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 ) \log (a+b x)}{2 h^2}-\frac {n (c+d x) \log (c+d x)}{d h}-\frac {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}+\frac {\left (g^2-2 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^2 \sqrt {g^2-4 f h}}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}+\frac {g \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^2}-\frac {\left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {\left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (g+\frac {g^2-2 f h}{\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^2} \]
[Out]
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Rubi [A] time = 1.07, antiderivative size = 831, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 12, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2513, 2418, 2389, 2295, 2394, 2393, 2391, 703, 634, 618, 206, 628} \[ \frac {n (a+b x) \log (a+b x)}{b h}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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 ) \log (a+b x)}{2 h^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 f h}}\right ) n \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 ) \log (a+b x)}{2 h^2}-\frac {n (c+d x) \log (c+d x)}{d h}-\frac {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}+\frac {\left (g^2-2 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^2 \sqrt {g^2-4 f h}}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}+\frac {g \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^2}-\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}-\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}+\frac {\left (g-\frac {g^2-2 f h}{\sqrt {g^2-4 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^2}+\frac {\left (g+\frac {g^2-2 f h}{\sqrt {g^2-4 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^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 618
Rule 628
Rule 634
Rule 703
Rule 2295
Rule 2389
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2513
Rubi steps
\begin {align*} \int \frac {x^2 \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^2 \log (a+b x)}{f+g x+h x^2} \, dx-n \int \frac {x^2 \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^2}{f+g x+h x^2} \, dx\\ &=-\frac {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}+n \int \left (\frac {\log (a+b x)}{h}-\frac {(f+g x) \log (a+b x)}{h \left (f+g x+h x^2\right )}\right ) \, dx-n \int \left (\frac {\log (c+d x)}{h}-\frac {(f+g x) \log (c+d x)}{h \left (f+g x+h x^2\right )}\right ) \, dx-\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 x}{f+g x+h x^2} \, dx}{h}\\ &=-\frac {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}+\frac {n \int \log (a+b x) \, dx}{h}-\frac {n \int \frac {(f+g x) \log (a+b x)}{f+g x+h x^2} \, dx}{h}-\frac {n \int \log (c+d x) \, dx}{h}+\frac {n \int \frac {(f+g x) \log (c+d x)}{f+g x+h x^2} \, dx}{h}+\frac {\left (g \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^2}-\frac {\left (\left (g^2-2 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 {1}{f+g x+h x^2} \, dx}{2 h^2}\\ &=-\frac {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}+\frac {g \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^2}-\frac {n \int \left (\frac {\left (g+\frac {-g^2+2 f h}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g-\frac {-g^2+2 f h}{\sqrt {g^2-4 f h}}\right ) \log (a+b x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{h}+\frac {n \int \left (\frac {\left (g+\frac {-g^2+2 f h}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g-\sqrt {g^2-4 f h}+2 h x}+\frac {\left (g-\frac {-g^2+2 f h}{\sqrt {g^2-4 f h}}\right ) \log (c+d x)}{g+\sqrt {g^2-4 f h}+2 h x}\right ) \, dx}{h}+\frac {n \operatorname {Subst}(\int \log (x) \, dx,x,a+b x)}{b h}-\frac {n \operatorname {Subst}(\int \log (x) \, dx,x,c+d x)}{d h}+\frac {\left (\left (g^2-2 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 ) \operatorname {Subst}\left (\int \frac {1}{g^2-4 f h-x^2} \, dx,x,g+2 h x\right )}{h^2}\\ &=\frac {n (a+b x) \log (a+b x)}{b h}-\frac {n (c+d x) \log (c+d x)}{d h}-\frac {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}+\frac {\left (g^2-2 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^2 \sqrt {g^2-4 f h}}+\frac {g \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^2}-\frac {\left (\left (g-\frac {g^2-2 f h}{\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}+\frac {\left (\left (g-\frac {g^2-2 f h}{\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}-\frac {\left (\left (g+\frac {g^2-2 f h}{\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}+\frac {\left (\left (g+\frac {g^2-2 f h}{\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}\\ &=\frac {n (a+b x) \log (a+b x)}{b h}-\frac {n (c+d x) \log (c+d x)}{d h}-\frac {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}+\frac {\left (g^2-2 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^2 \sqrt {g^2-4 f h}}-\frac {\left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {g \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^2}+\frac {\left (b \left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (d \left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (b \left (g+\frac {g^2-2 f h}{\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^2}-\frac {\left (d \left (g+\frac {g^2-2 f h}{\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^2}\\ &=\frac {n (a+b x) \log (a+b x)}{b h}-\frac {n (c+d x) \log (c+d x)}{d h}-\frac {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}+\frac {\left (g^2-2 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^2 \sqrt {g^2-4 f h}}-\frac {\left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {g \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^2}+\frac {\left (\left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (\left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (\left (g+\frac {g^2-2 f h}{\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^2}-\frac {\left (\left (g+\frac {g^2-2 f h}{\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^2}\\ &=\frac {n (a+b x) \log (a+b x)}{b h}-\frac {n (c+d x) \log (c+d x)}{d h}-\frac {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}+\frac {\left (g^2-2 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^2 \sqrt {g^2-4 f h}}-\frac {\left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {g \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^2}-\frac {\left (g-\frac {g^2-2 f h}{\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^2}-\frac {\left (g+\frac {g^2-2 f h}{\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^2}+\frac {\left (g-\frac {g^2-2 f h}{\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^2}+\frac {\left (g+\frac {g^2-2 f h}{\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^2}\\ \end {align*}
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Mathematica [A] time = 4.11, size = 1105, normalized size = 1.33 \[ \frac {2 d h \sqrt {g^2-4 f h} (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 b d f h \log \left (g+2 h x-\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b d g \left (g-\sqrt {g^2-4 f h}\right ) \log \left (g+2 h x-\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+2 b d f h \log \left (g+2 h x+\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-b d g \left (g+\sqrt {g^2-4 f h}\right ) \log \left (g+2 h x+\sqrt {g^2-4 f h}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 (b c-a d) h \sqrt {g^2-4 f h} n \log (c+d x)+2 b d f h 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 )-b d g \left (g-\sqrt {g^2-4 f h}\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 )-2 b d f h 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 )+b d g \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 )}{2 b d h^2 \sqrt {g^2-4 f h}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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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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