\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(f+g x)^5} \, dx\) [75]

Optimal result

Integrand size = 32, antiderivative size = 1208 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx =\text {Too large to display} \] Output:

-1/12*B^2*(-a*d+b*c)^2*g^3*n^2*(d*x+c)^2/(-a*g+b*f)^2/(-c*g+d*f)^4/(g*x+f) 
^2-1/6*B^2*(-a*d+b*c)^3*g^3*n^2*(d*x+c)/(-a*g+b*f)^3/(-c*g+d*f)^4/(g*x+f)+ 
1/4*B^2*(-a*d+b*c)^2*g^2*(-3*a*d*g-b*c*g+4*b*d*f)*n^2*(d*x+c)/(-a*g+b*f)^3 
/(-c*g+d*f)^4/(g*x+f)+1/6*B*(-a*d+b*c)*g^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/ 
(d*x+c))^n))/(-a*g+b*f)/(-c*g+d*f)^4/(g*x+f)^3-1/4*B*(-a*d+b*c)*g^2*(-3*a* 
d*g-b*c*g+4*b*d*f)*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*g+b*f)^ 
2/(-c*g+d*f)^4/(g*x+f)^2+1/2*B*(-a*d+b*c)*g*(3*a^2*d^2*g^2-2*a*b*d*g*(-c*g 
+4*d*f)+b^2*(c^2*g^2-4*c*d*f*g+6*d^2*f^2))*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d 
*x+c))^n))/(-a*g+b*f)^4/(-c*g+d*f)^3/(g*x+f)+1/4*b^4*(A+B*ln(e*((b*x+a)/(d 
*x+c))^n))^2/g/(-a*g+b*f)^4-1/4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/g/(g*x+f 
)^4-1/6*B^2*(-a*d+b*c)^4*g^3*n^2*ln((b*x+a)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d* 
f)^4+1/4*B^2*(-a*d+b*c)^3*g^2*(-3*a*d*g-b*c*g+4*b*d*f)*n^2*ln((b*x+a)/(d*x 
+c))/(-a*g+b*f)^4/(-c*g+d*f)^4+1/6*B^2*(-a*d+b*c)^4*g^3*n^2*ln((g*x+f)/(d* 
x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4-1/4*B^2*(-a*d+b*c)^3*g^2*(-3*a*d*g-b*c*g+4 
*b*d*f)*n^2*ln((g*x+f)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4+1/2*B^2*(-a*d+b* 
c)^2*g*(3*a^2*d^2*g^2-2*a*b*d*g*(-c*g+4*d*f)+b^2*(c^2*g^2-4*c*d*f*g+6*d^2* 
f^2))*n^2*ln((g*x+f)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4-1/2*B*(-a*d+b*c)*( 
-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2* 
d^2*f^2))*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-(-c*g+d*f)*(b*x+a)/(-a*g+ 
b*f)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4-1/2*B^2*(-a*d+b*c)*(-a*d*g-b*c*...
 

Mathematica [A] (verified)

Time = 7.42 (sec) , antiderivative size = 1329, normalized size of antiderivative = 1.10 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx =\text {Too large to display} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^5,x]
 

Output:

-1/12*(3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(f + g*x)*(2*(b*c 
 - a*d)*g*(b*f - a*g)^3*(d*f - c*g)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n 
]) - 3*(b*c - a*d)*g*(b*f - a*g)^2*(d*f - c*g)^2*(-2*b*d*f + b*c*g + a*d*g 
)*(f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*(b*c - a*d)*g*(b*f 
- a*g)*(d*f - c*g)*(a^2*d^2*g^2 + a*b*d*g*(-3*d*f + c*g) + b^2*(3*d^2*f^2 
- 3*c*d*f*g + c^2*g^2))*(f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
 - 6*b^4*(d*f - c*g)^4*(f + g*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c 
 + d*x))^n]) + 6*d^4*(b*f - a*g)^4*(f + g*x)^3*(A + B*Log[e*((a + b*x)/(c 
+ d*x))^n])*Log[c + d*x] + 6*(b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g)*(-2* 
a*b*d^2*f*g + a^2*d^2*g^2 + b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*(f + g* 
x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[f + g*x] - 6*B*(b*c - a*d) 
*g*(a^2*d^2*g^2 + a*b*d*g*(-3*d*f + c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^ 
2*g^2))*n*(f + g*x)^3*(b*(d*f - c*g)*Log[a + b*x] + (-(b*d*f) + a*d*g)*Log 
[c + d*x] + (b*c - a*d)*g*Log[f + g*x]) + 3*B*(b*c - a*d)*g*(2*b*d*f - b*c 
*g - a*d*g)*n*(f + g*x)^2*((b*c - a*d)*g*(b*f - a*g)*(d*f - c*g) - b^2*(d* 
f - c*g)^2*(f + g*x)*Log[a + b*x] + d^2*(b*f - a*g)^2*(f + g*x)*Log[c + d* 
x] + (b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g)*(f + g*x)*Log[f + g*x]) + B* 
(b*c - a*d)*g*n*(f + g*x)*((b*c - a*d)*g*(b*f - a*g)^2*(d*f - c*g)^2 + 2*( 
b*c - a*d)*g*(b*f - a*g)*(-(d*f) + c*g)*(-2*b*d*f + b*c*g + a*d*g)*(f + g* 
x) - 2*b^3*(d*f - c*g)^3*(f + g*x)^2*Log[a + b*x] + 2*d^3*(b*f - a*g)^3...
 

Rubi [A] (verified)

Time = 2.42 (sec) , antiderivative size = 1429, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2953, 2798, 2804, 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.

\(\displaystyle \int \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(f+g x)^5} \, dx\)

\(\Big \downarrow \) 2953

\(\displaystyle (b c-a d) \int \frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2798

\(\displaystyle (b c-a d) \left (\frac {B n \int \frac {(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 g (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g (b c-a d) \left (-\frac {(a+b x) (d f-c g)}{c+d x}-a g+b f\right )^4}\right )\)

\(\Big \downarrow \) 2804

\(\displaystyle (b c-a d) \left (\frac {B n \int \left (\frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^4}{(b f-a g)^4 (a+b x)}+\frac {(b c-a d) g (2 b d f-b c g-a d g) \left (-2 d^2 f^2 b^2-c^2 g^2 b^2+2 c d f g b^2+2 a d^2 f g b-a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^4 (d f-c g)^3 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}+\frac {(b c-a d)^2 g^2 \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^3 (d f-c g)^3 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d)^3 g^3 (-4 b d f+b c g+3 a d g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^2 (d f-c g)^3 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^3}+\frac {(b c-a d)^4 g^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g) (d f-c g)^3 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^4}\right )d\frac {a+b x}{c+d x}}{2 g (b c-a d)}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g (b c-a d) \left (-\frac {(a+b x) (d f-c g)}{c+d x}-a g+b f\right )^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle (b c-a d) \left (\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^4}{2 B (b f-a g)^4 n}+\frac {(b c-a d)^2 g^2 \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^4 (d f-c g)^3 (c+d x) \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}-\frac {(b c-a d)^3 g^3 (4 b d f-b c g-3 a d g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^2 (d f-c g)^4 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d)^4 g^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b f-a g) (d f-c g)^4 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^3}+\frac {B (b c-a d)^3 g^3 (4 b d f-b c g-3 a d g) n \log \left (\frac {a+b x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d)^4 g^4 n \log \left (\frac {a+b x}{c+d x}\right )}{3 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d)^3 g^3 (4 b d f-b c g-3 a d g) n \log \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B (b c-a d)^2 g^2 \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) n \log \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}{(b f-a g)^4 (d f-c g)^4}+\frac {B (b c-a d)^4 g^4 n \log \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}{3 (b f-a g)^4 (d f-c g)^4}-\frac {(b c-a d) g (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) g (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^4 (d f-c g)^4}+\frac {B (b c-a d)^3 g^3 (4 b d f-b c g-3 a d g) n}{2 (b f-a g)^3 (d f-c g)^4 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}-\frac {B (b c-a d)^4 g^4 n}{3 (b f-a g)^3 (d f-c g)^4 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}-\frac {B (b c-a d)^4 g^4 n}{6 (b f-a g)^2 (d f-c g)^4 \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^2}\right )}{2 (b c-a d) g}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d) g \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )^4}\right )\)

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(f + g*x)^5,x]
 

Output:

(b*c - a*d)*(-1/4*((b - (d*(a + b*x))/(c + d*x))^4*(A + B*Log[e*((a + b*x) 
/(c + d*x))^n])^2)/((b*c - a*d)*g*(b*f - a*g - ((d*f - c*g)*(a + b*x))/(c 
+ d*x))^4) + (B*n*(-1/6*(B*(b*c - a*d)^4*g^4*n)/((b*f - a*g)^2*(d*f - c*g) 
^4*(b*f - a*g - ((d*f - c*g)*(a + b*x))/(c + d*x))^2) - (B*(b*c - a*d)^4*g 
^4*n)/(3*(b*f - a*g)^3*(d*f - c*g)^4*(b*f - a*g - ((d*f - c*g)*(a + b*x))/ 
(c + d*x))) + (B*(b*c - a*d)^3*g^3*(4*b*d*f - b*c*g - 3*a*d*g)*n)/(2*(b*f 
- a*g)^3*(d*f - c*g)^4*(b*f - a*g - ((d*f - c*g)*(a + b*x))/(c + d*x))) + 
((b*c - a*d)^4*g^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*f - a*g)* 
(d*f - c*g)^4*(b*f - a*g - ((d*f - c*g)*(a + b*x))/(c + d*x))^3) - ((b*c - 
 a*d)^3*g^3*(4*b*d*f - b*c*g - 3*a*d*g)*(A + B*Log[e*((a + b*x)/(c + d*x)) 
^n]))/(2*(b*f - a*g)^2*(d*f - c*g)^4*(b*f - a*g - ((d*f - c*g)*(a + b*x))/ 
(c + d*x))^2) + ((b*c - a*d)^2*g^2*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g 
) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*(a + b*x)*(A + B*Log[e*((a + b* 
x)/(c + d*x))^n]))/((b*f - a*g)^4*(d*f - c*g)^3*(c + d*x)*(b*f - a*g - ((d 
*f - c*g)*(a + b*x))/(c + d*x))) + (b^4*(A + B*Log[e*((a + b*x)/(c + d*x)) 
^n])^2)/(2*B*(b*f - a*g)^4*n) - (B*(b*c - a*d)^4*g^4*n*Log[(a + b*x)/(c + 
d*x)])/(3*(b*f - a*g)^4*(d*f - c*g)^4) + (B*(b*c - a*d)^3*g^3*(4*b*d*f - b 
*c*g - 3*a*d*g)*n*Log[(a + b*x)/(c + d*x)])/(2*(b*f - a*g)^4*(d*f - c*g)^4 
) + (B*(b*c - a*d)^4*g^4*n*Log[b*f - a*g - ((d*f - c*g)*(a + b*x))/(c + d* 
x)])/(3*(b*f - a*g)^4*(d*f - c*g)^4) - (B*(b*c - a*d)^3*g^3*(4*b*d*f - ...
 

Defintions of rubi rules used

rule 2009

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

rule 2798

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( 
f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 
 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) 
*(e*f - d*g)))   Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] 
)^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f 
 - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
 

rule 2804

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ 
u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / 
; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
 

rule 2953

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)   Sub 
st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2 
)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n} 
, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
 

Maple [F]

\[\int \frac {{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}{\left (g x +f \right )^{5}}d x\]

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x)
 

Output:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x)
 

Fricas [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\int { \frac {{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{5}} \,d x } \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x, algorithm="frica 
s")
 

Output:

integral((B^2*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*A*B*log(e*((b*x + a)/(d 
*x + c))^n) + A^2)/(g^5*x^5 + 5*f*g^4*x^4 + 10*f^2*g^3*x^3 + 10*f^3*g^2*x^ 
2 + 5*f^4*g*x + f^5), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\text {Timed out} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(g*x+f)**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\int { \frac {{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{5}} \,d x } \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x, algorithm="maxim 
a")
 

Output:

1/12*(6*b^4*log(b*x + a)/(b^4*f^4*g - 4*a*b^3*f^3*g^2 + 6*a^2*b^2*f^2*g^3 
- 4*a^3*b*f*g^4 + a^4*g^5) - 6*d^4*log(d*x + c)/(d^4*f^4*g - 4*c*d^3*f^3*g 
^2 + 6*c^2*d^2*f^2*g^3 - 4*c^3*d*f*g^4 + c^4*g^5) + 6*(4*(b^4*c*d^3 - a*b^ 
3*d^4)*f^3 - 6*(b^4*c^2*d^2 - a^2*b^2*d^4)*f^2*g + 4*(b^4*c^3*d - a^3*b*d^ 
4)*f*g^2 - (b^4*c^4 - a^4*d^4)*g^3)*log(g*x + f)/(b^4*d^4*f^8 + a^4*c^4*g^ 
8 - 4*(b^4*c*d^3 + a*b^3*d^4)*f^7*g + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3 + 3 
*a^2*b^2*d^4)*f^6*g^2 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + 
 a^3*b*d^4)*f^5*g^3 + (b^4*c^4 + 16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16* 
a^3*b*c*d^3 + a^4*d^4)*f^4*g^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*a^3*b* 
c^2*d^2 + a^4*c*d^3)*f^3*g^5 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^ 
2*d^2)*f^2*g^6 - 4*(a^3*b*c^4 + a^4*c^3*d)*f*g^7) - (26*(b^3*c*d^2 - a*b^2 
*d^3)*f^4 - 31*(b^3*c^2*d - a^2*b*d^3)*f^3*g + (11*b^3*c^3 + 15*a*b^2*c^2* 
d - 15*a^2*b*c*d^2 - 11*a^3*d^3)*f^2*g^2 - 7*(a*b^2*c^3 - a^3*c*d^2)*f*g^3 
 + 2*(a^2*b*c^3 - a^3*c^2*d)*g^4 + 6*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2*g^2 - 
3*(b^3*c^2*d - a^2*b*d^3)*f*g^3 + (b^3*c^3 - a^3*d^3)*g^4)*x^2 + 3*(14*(b^ 
3*c*d^2 - a*b^2*d^3)*f^3*g - 15*(b^3*c^2*d - a^2*b*d^3)*f^2*g^2 + (5*b^3*c 
^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 - 5*a^3*d^3)*f*g^3 - (a*b^2*c^3 - a^3*c 
*d^2)*g^4)*x)/(b^3*d^3*f^9 + a^3*c^3*f^3*g^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*f 
^8*g + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^7*g^2 - (b^3*c^3 + 9*a* 
b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^6*g^3 + 3*(a*b^2*c^3 + 3*a^2*b*c...
 

Giac [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\int { \frac {{\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{5}} \,d x } \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x, algorithm="giac" 
)
 

Output:

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(g*x + f)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^5} \,d x \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(f + g*x)^5,x)
 

Output:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(f + g*x)^5, x)
 

Reduce [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\text {too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x)
 

Output:

( - 48*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a**5*c**2*d**3*f**5*g** 
5 + 5*a**5*c**2*d**3*f**4*g**6*x + 10*a**5*c**2*d**3*f**3*g**7*x**2 + 10*a 
**5*c**2*d**3*f**2*g**8*x**3 + 5*a**5*c**2*d**3*f*g**9*x**4 + a**5*c**2*d* 
*3*g**10*x**5 + a**5*c*d**4*f**5*g**5*x + 5*a**5*c*d**4*f**4*g**6*x**2 + 1 
0*a**5*c*d**4*f**3*g**7*x**3 + 10*a**5*c*d**4*f**2*g**8*x**4 + 5*a**5*c*d* 
*4*f*g**9*x**5 + a**5*c*d**4*g**10*x**6 + a**4*b*c**3*d**2*f**5*g**5 + 5*a 
**4*b*c**3*d**2*f**4*g**6*x + 10*a**4*b*c**3*d**2*f**3*g**7*x**2 + 10*a**4 
*b*c**3*d**2*f**2*g**8*x**3 + 5*a**4*b*c**3*d**2*f*g**9*x**4 + a**4*b*c**3 
*d**2*g**10*x**5 - 5*a**4*b*c**2*d**3*f**6*g**4 - 23*a**4*b*c**2*d**3*f**5 
*g**5*x - 40*a**4*b*c**2*d**3*f**4*g**6*x**2 - 30*a**4*b*c**2*d**3*f**3*g* 
*7*x**3 - 5*a**4*b*c**2*d**3*f**2*g**8*x**4 + 5*a**4*b*c**2*d**3*f*g**9*x* 
*5 + 2*a**4*b*c**2*d**3*g**10*x**6 - 5*a**4*b*c*d**4*f**6*g**4*x - 24*a**4 
*b*c*d**4*f**5*g**5*x**2 - 45*a**4*b*c*d**4*f**4*g**6*x**3 - 40*a**4*b*c*d 
**4*f**3*g**7*x**4 - 15*a**4*b*c*d**4*f**2*g**8*x**5 + a**4*b*c*d**4*g**10 
*x**7 + a**3*b**2*c**4*d*f**5*g**5 + 5*a**3*b**2*c**4*d*f**4*g**6*x + 10*a 
**3*b**2*c**4*d*f**3*g**7*x**2 + 10*a**3*b**2*c**4*d*f**2*g**8*x**3 + 5*a* 
*3*b**2*c**4*d*f*g**9*x**4 + a**3*b**2*c**4*d*g**10*x**5 - 5*a**3*b**2*c** 
3*d**2*f**6*g**4 - 23*a**3*b**2*c**3*d**2*f**5*g**5*x - 40*a**3*b**2*c**3* 
d**2*f**4*g**6*x**2 - 30*a**3*b**2*c**3*d**2*f**3*g**7*x**3 - 5*a**3*b**2* 
c**3*d**2*f**2*g**8*x**4 + 5*a**3*b**2*c**3*d**2*f*g**9*x**5 + 2*a**3*b...