3.1.82 \(\int \frac {x^3 \log (e (\frac {a+b x}{c+d x})^n)}{f+g x+h x^2} \, dx\) [82]

3.1.82.1 Optimal result

Integrand size = 34, antiderivative size = 1046 \[ \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=\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 ) \text {arctanh}\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 \operatorname {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-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {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-f h-\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {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-f h+\frac {g \left (g^2-3 f h\right )}{\sqrt {g^2-4 f h}}\right ) n \operatorname {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} \]

output

1/2*a*n*x/b/h-1/2*c*n*x/d/h-1/2*a^2*n*ln(b*x+a)/b^2/h+1/2*n*x^2*ln(b*x+a)/ 
h-g*n*(b*x+a)*ln(b*x+a)/b/h^2+1/2*c^2*n*ln(d*x+c)/d^2/h-1/2*n*x^2*ln(d*x+c 
)/h+g*n*(d*x+c)*ln(d*x+c)/d/h^2+g*x*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n) 
-n*ln(d*x+c))/h^2-1/2*x^2*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+ 
c))/h-1/2*(-f*h+g^2)*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))*l 
n(h*x^2+g*x+f)/h^3+1/2*n*ln(b*x+a)*ln(-b*(g+2*h*x-(-4*f*h+g^2)^(1/2))/(2*a 
*h-b*(g-(-4*f*h+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/ 
h^3-1/2*n*ln(d*x+c)*ln(-d*(g+2*h*x-(-4*f*h+g^2)^(1/2))/(2*c*h-d*(g-(-4*f*h 
+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3+1/2*n*polyl 
og(2,2*h*(b*x+a)/(2*a*h-b*(g-(-4*f*h+g^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2) 
/(-4*f*h+g^2)^(1/2))/h^3-1/2*n*polylog(2,2*h*(d*x+c)/(2*c*h-d*(g-(-4*f*h+g 
^2)^(1/2))))*(g^2-f*h-g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3+1/2*n*ln(b*x+ 
a)*ln(-b*(g+2*h*x+(-4*f*h+g^2)^(1/2))/(2*a*h-b*(g+(-4*f*h+g^2)^(1/2))))*(g 
^2-f*h+g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3-1/2*n*ln(d*x+c)*ln(-d*(g+2*h 
*x+(-4*f*h+g^2)^(1/2))/(2*c*h-d*(g+(-4*f*h+g^2)^(1/2))))*(g^2-f*h+g*(-3*f* 
h+g^2)/(-4*f*h+g^2)^(1/2))/h^3+1/2*n*polylog(2,2*h*(b*x+a)/(2*a*h-b*(g+(-4 
*f*h+g^2)^(1/2))))*(g^2-f*h+g*(-3*f*h+g^2)/(-4*f*h+g^2)^(1/2))/h^3-1/2*n*p 
olylog(2,2*h*(d*x+c)/(2*c*h-d*(g+(-4*f*h+g^2)^(1/2))))*(g^2-f*h+g*(-3*f*h+ 
g^2)/(-4*f*h+g^2)^(1/2))/h^3-g*(-3*f*h+g^2)*arctanh((2*h*x+g)/(-4*f*h+g^2) 
^(1/2))*(n*ln(b*x+a)-ln(e*((b*x+a)/(d*x+c))^n)-n*ln(d*x+c))/h^3/(-4*f*h...
 

3.1.82.2 Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 1240, normalized size of antiderivative = 1.19 \[ \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=\frac {h^2 x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-\frac {2 g h (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+\frac {2 (b c-a d) g h n \log (c+d x)}{b d}+\frac {h^2 n \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )}{b^2 d^2}+\frac {2 f g h \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )}{\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-\sqrt {g^2-4 f h}+2 h x\right )-\frac {2 f g h \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (g+\sqrt {g^2-4 f h}+2 h x\right )}{\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+\sqrt {g^2-4 f h}+2 h x\right )-\frac {2 f g h n \left (\left (\log \left (\frac {2 h (a+b x)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\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 )\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)}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\log \left (\frac {2 h (c+d x)}{-d g+2 c h+d \sqrt {g^2-4 f h}}\right )\right ) \log \left (g-\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (-g+\sqrt {g^2-4 f h}-2 h x\right )}{-b g+2 a h+b \sqrt {g^2-4 f h}}\right )-\operatorname {PolyLog}\left (2,\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 )\right )}{\sqrt {g^2-4 f h}}+\frac {2 f g 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+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\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 )-\operatorname {PolyLog}\left (2,\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 )\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+\sqrt {g^2-4 f h}+2 h x\right )+\operatorname {PolyLog}\left (2,\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 )-\operatorname {PolyLog}\left (2,\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 )\right )}{\sqrt {g^2-4 f h}}}{2 h^3} \]

input

Integrate[(x^3*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x + h*x^2),x]
 

output

(h^2*x^2*Log[e*((a + b*x)/(c + d*x))^n] - (2*g*h*(a + b*x)*Log[e*((a + b*x 
)/(c + d*x))^n])/b + (2*(b*c - a*d)*g*h*n*Log[c + d*x])/(b*d) + (h^2*n*(-( 
a^2*d^2*Log[a + b*x]) + b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c + d*x])))/(b^2 
*d^2) + (2*f*g*h*Log[e*((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] 
+ 2*h*x])/Sqrt[g^2 - 4*f*h] + (g^2 - f*h)*(1 - g/Sqrt[g^2 - 4*f*h])*Log[e* 
((a + b*x)/(c + d*x))^n]*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] - (2*f*g*h*Log 
[e*((a + b*x)/(c + d*x))^n]*Log[g + Sqrt[g^2 - 4*f*h] + 2*h*x])/Sqrt[g^2 - 
 4*f*h] + (g^2 - f*h)*(1 + g/Sqrt[g^2 - 4*f*h])*Log[e*((a + b*x)/(c + d*x) 
)^n]*Log[g + Sqrt[g^2 - 4*f*h] + 2*h*x] - (2*f*g*h*n*((Log[(2*h*(a + b*x)) 
/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - Log[(2*h*(c + d*x))/(-(d*g) + 2 
*c*h + d*Sqrt[g^2 - 4*f*h])])*Log[g - Sqrt[g^2 - 4*f*h] + 2*h*x] + PolyLog 
[2, (b*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4* 
f*h])] - PolyLog[2, (d*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(2*c*h + d*(-g + 
Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f*h] - ((g^2 - f*h)*(-g + Sqrt[g^2 - 4 
*f*h])*n*((Log[(2*h*(a + b*x))/(-(b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - L 
og[(2*h*(c + d*x))/(-(d*g) + 2*c*h + d*Sqrt[g^2 - 4*f*h])])*Log[g - Sqrt[g 
^2 - 4*f*h] + 2*h*x] + PolyLog[2, (b*(-g + Sqrt[g^2 - 4*f*h] - 2*h*x))/(-( 
b*g) + 2*a*h + b*Sqrt[g^2 - 4*f*h])] - PolyLog[2, (d*(-g + Sqrt[g^2 - 4*f* 
h] - 2*h*x))/(2*c*h + d*(-g + Sqrt[g^2 - 4*f*h]))]))/Sqrt[g^2 - 4*f*h] + ( 
2*f*g*h*n*((Log[(2*h*(a + b*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h]))] - ...
 

3.1.82.3 Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 959, normalized size of antiderivative = 0.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2993, 1143, 2009, 2865, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx\)

\(\Big \downarrow \) 2993

\(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \frac {x^3}{h x^2+g x+f}dx\right )+n \int \frac {x^3 \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x^3 \log (c+d x)}{h x^2+g x+f}dx\)

\(\Big \downarrow \) 1143

\(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \left (-\frac {g}{h^2}+\frac {x}{h}+\frac {f g+\left (g^2-f h\right ) x}{h^2 \left (h x^2+g x+f\right )}\right )dx\right )+n \int \frac {x^3 \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x^3 \log (c+d x)}{h x^2+g x+f}dx\)

\(\Big \downarrow \) 2009

\(\displaystyle n \int \frac {x^3 \log (a+b x)}{h x^2+g x+f}dx-n \int \frac {x^3 \log (c+d x)}{h x^2+g x+f}dx-\left (\left (\frac {g \left (g^2-3 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {g x}{h^2}+\frac {x^2}{2 h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\)

\(\Big \downarrow \) 2865

\(\displaystyle n \int \left (\frac {x \log (a+b x)}{h}+\frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (a+b x)}{h^2 \left (h x^2+g x+f\right )}-\frac {g \log (a+b x)}{h^2}\right )dx-n \int \left (\frac {x \log (c+d x)}{h}+\frac {\left (f g+\left (g^2-f h\right ) x\right ) \log (c+d x)}{h^2 \left (h x^2+g x+f\right )}-\frac {g \log (c+d x)}{h^2}\right )dx-\left (\left (\frac {g \left (g^2-3 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h\right ) \log \left (f+g x+h x^2\right )}{2 h^3}-\frac {g x}{h^2}+\frac {x^2}{2 h}\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle -\left (\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 ) \left (\frac {x^2}{2 h}-\frac {g x}{h^2}+\frac {g \left (g^2-3 f h\right ) \text {arctanh}\left (\frac {g+2 h x}{\sqrt {g^2-4 f h}}\right )}{h^3 \sqrt {g^2-4 f h}}+\frac {\left (g^2-f h\right ) \log \left (h x^2+g x+f\right )}{2 h^3}\right )\right )+n \left (-\frac {\log (a+b x) a^2}{2 b^2 h}+\frac {x a}{2 b h}-\frac {x^2}{4 h}+\frac {g x}{h^2}+\frac {x^2 \log (a+b x)}{2 h}-\frac {g (a+b x) \log (a+b x)}{b h^2}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) \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 ) \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 ) \operatorname {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 ) \operatorname {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}\right )-n \left (-\frac {\log (c+d x) c^2}{2 d^2 h}+\frac {x c}{2 d h}-\frac {x^2}{4 h}+\frac {g x}{h^2}+\frac {x^2 \log (c+d x)}{2 h}-\frac {g (c+d x) \log (c+d x)}{d h^2}+\frac {\left (g^2-\frac {\left (g^2-3 f h\right ) g}{\sqrt {g^2-4 f h}}-f h\right ) \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 ) \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 ) \operatorname {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 ) \operatorname {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}\right )\)

input

Int[(x^3*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x + h*x^2),x]
 

output

-((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*(-((g 
*x)/h^2) + x^2/(2*h) + (g*(g^2 - 3*f*h)*ArcTanh[(g + 2*h*x)/Sqrt[g^2 - 4*f 
*h]])/(h^3*Sqrt[g^2 - 4*f*h]) + ((g^2 - f*h)*Log[f + g*x + h*x^2])/(2*h^3) 
)) + n*((g*x)/h^2 + (a*x)/(2*b*h) - x^2/(4*h) - (a^2*Log[a + b*x])/(2*b^2* 
h) + (x^2*Log[a + b*x])/(2*h) - (g*(a + b*x)*Log[a + b*x])/(b*h^2) + ((g^2 
 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*Log[a + b*x]*Log[-((b*(g - S 
qrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h])))])/(2*h^3) 
+ ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*Log[a + b*x]*Log[-((b 
*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*a*h - b*(g + Sqrt[g^2 - 4*f*h])))])/( 
2*h^3) + ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2* 
h*(a + b*x))/(2*a*h - b*(g - Sqrt[g^2 - 4*f*h]))])/(2*h^3) + ((g^2 - f*h + 
 (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2*h*(a + b*x))/(2*a*h - 
b*(g + Sqrt[g^2 - 4*f*h]))])/(2*h^3)) - n*((g*x)/h^2 + (c*x)/(2*d*h) - x^2 
/(4*h) - (c^2*Log[c + d*x])/(2*d^2*h) + (x^2*Log[c + d*x])/(2*h) - (g*(c + 
 d*x)*Log[c + d*x])/(d*h^2) + ((g^2 - f*h - (g*(g^2 - 3*f*h))/Sqrt[g^2 - 4 
*f*h])*Log[c + d*x]*Log[-((d*(g - Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c*h - d*( 
g - Sqrt[g^2 - 4*f*h])))])/(2*h^3) + ((g^2 - f*h + (g*(g^2 - 3*f*h))/Sqrt[ 
g^2 - 4*f*h])*Log[c + d*x]*Log[-((d*(g + Sqrt[g^2 - 4*f*h] + 2*h*x))/(2*c* 
h - d*(g + Sqrt[g^2 - 4*f*h])))])/(2*h^3) + ((g^2 - f*h - (g*(g^2 - 3*f*h) 
)/Sqrt[g^2 - 4*f*h])*PolyLog[2, (2*h*(c + d*x))/(2*c*h - d*(g - Sqrt[g^...
 

3.1.82.3.1 Defintions of rubi rules used

rule 1143

Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] 
 :> Int[ExpandIntegrand[(d + e*x)^m/(a + b*x + c*x^2), x], x] /; FreeQ[{a, 
b, c, d, e}, x] && IGtQ[m, 1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2865

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Sy 
mbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, 
Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunctionQ[ 
RFx, x] && IntegerQ[p]
 

rule 2993

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r   Int[RFx*Log[a + b*x], x], x] + (Si 
mp[q*r   Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c 
+ d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])   Int[RFx, x], x]) /; FreeQ[ 
{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a 
*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ 
m, n]]
 

3.1.82.4 Maple [F]

\[\int \frac {x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )}{h \,x^{2}+g x +f}d x\]

input

int(x^3*ln(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x)
 

output

int(x^3*ln(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x)
 

3.1.82.5 Fricas [F]

\[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]

input

integrate(x^3*log(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x, algorithm="frica 
s")
 

output

integral(x^3*log(e*((b*x + a)/(d*x + c))^n)/(h*x^2 + g*x + f), x)
 

3.1.82.6 Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Timed out} \]

input

integrate(x**3*ln(e*((b*x+a)/(d*x+c))**n)/(h*x**2+g*x+f),x)
 

output

Timed out
 

3.1.82.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\text {Exception raised: ValueError} \]

input

integrate(x^3*log(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x, algorithm="maxim 
a")
 

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*f*h-g^2>0)', see `assume?` for 
 more deta
 

3.1.82.8 Giac [F]

\[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int { \frac {x^{3} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{h x^{2} + g x + f} \,d x } \]

input

integrate(x^3*log(e*((b*x+a)/(d*x+c))^n)/(h*x^2+g*x+f),x, algorithm="giac" 
)
 

output

integrate(x^3*log(e*((b*x + a)/(d*x + c))^n)/(h*x^2 + g*x + f), x)
 

3.1.82.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x+h x^2} \, dx=\int \frac {x^3\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{h\,x^2+g\,x+f} \,d x \]

input

int((x^3*log(e*((a + b*x)/(c + d*x))^n))/(f + g*x + h*x^2),x)
 

output

int((x^3*log(e*((a + b*x)/(c + d*x))^n))/(f + g*x + h*x^2), x)