\(\int (f+g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [68]

Optimal result

Integrand size = 32, antiderivative size = 565 \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 g^2 n^2 x}{3 b^2 d^2}-\frac {2 B (b c-a d) g (3 b d f-2 b c g-a d g) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b d^3}-\frac {(b f-a g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 g}+\frac {2 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^2 n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d)^2 g (3 b d f-2 b c g-a d g) n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3} \] Output:

1/3*B^2*(-a*d+b*c)^2*g^2*n^2*x/b^2/d^2-2/3*B*(-a*d+b*c)*g*(-a*d*g-2*b*c*g+ 
3*b*d*f)*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d^2-1/3*B*(-a*d+b*c 
)*g^2*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^3-1/3*(-a*g+b*f)^3*( 
A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3/g+1/3*(g*x+f)^3*(A+B*ln(e*((b*x+a)/(d 
*x+c))^n))^2/g+2/3*B*(-a*d+b*c)*(a^2*d^2*g^2-a*b*d*g*(-c*g+3*d*f)+b^2*(c^2 
*g^2-3*c*d*f*g+3*d^2*f^2))*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c) 
/b/(d*x+c))/b^3/d^3+1/3*B^2*(-a*d+b*c)^3*g^2*n^2*ln((b*x+a)/(d*x+c))/b^3/d 
^3+1/3*B^2*(-a*d+b*c)^3*g^2*n^2*ln(d*x+c)/b^3/d^3+2/3*B^2*(-a*d+b*c)^2*g*( 
-a*d*g-2*b*c*g+3*b*d*f)*n^2*ln(d*x+c)/b^3/d^3+2/3*B^2*(-a*d+b*c)*(a^2*d^2* 
g^2-a*b*d*g*(-c*g+3*d*f)+b^2*(c^2*g^2-3*c*d*f*g+3*d^2*f^2))*n^2*polylog(2, 
d*(b*x+a)/b/(d*x+c))/b^3/d^3
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.90 \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (2 A b d (b c-a d) g^2 (3 b d f-b c g-a d g) x+2 B d (b c-a d) g^2 (3 b d f-b c g-a d g) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 d^2 (b c-a d) g^3 x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^3 (b f-a g)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 B (b c-a d)^2 g^2 (-3 b d f+b c g+a d g) n \log (c+d x)-2 b^3 (d f-c g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-B (b c-a d) g^3 n \left (a^2 d^2 \log (a+b x)-b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-B d^3 (b f-a g)^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b^3 B (d f-c g)^3 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{b^3 d^3}}{3 g} \] Input:

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

Output:

((f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (B*n*(2*A*b*d*(b*c 
 - a*d)*g^2*(3*b*d*f - b*c*g - a*d*g)*x + 2*B*d*(b*c - a*d)*g^2*(3*b*d*f - 
 b*c*g - a*d*g)*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + b^2*d^2*(b*c - 
a*d)*g^3*x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d^3*(b*f - a*g)^3* 
Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*B*(b*c - a*d)^2*g^ 
2*(-3*b*d*f + b*c*g + a*d*g)*n*Log[c + d*x] - 2*b^3*(d*f - c*g)^3*(A + B*L 
og[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - B*(b*c - a*d)*g^3*n*(a^2*d^2 
*Log[a + b*x] - b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c + d*x])) - B*d^3*(b*f 
- a*g)^3*n*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) 
 - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + b^3*B*(d*f - c*g)^3*n*((2 
*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLo 
g[2, (b*(c + d*x))/(b*c - a*d)])))/(b^3*d^3))/(3*g)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.25, 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.

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

\(\Big \downarrow \) 2953

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

\(\Big \downarrow \) 2798

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

\(\Big \downarrow \) 2804

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1659 vs. \(2 (544) = 1088\).

Time = 0.59 (sec) , antiderivative size = 1659, normalized size of antiderivative = 2.94 \[ \int (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x 
**2),x)*a**3*b**2*d**4*g**2*n - 6*int((log(((a + b*x)**n*e)/(c + d*x)**n)* 
x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**3*d**4*f*g*n + 6*int((log(( 
(a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b** 
4*d**4*f**2*n - 2*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x 
+ b*c*x + b*d*x**2),x)*b**5*c**3*d*g**2*n + 6*int((log(((a + b*x)**n*e)/(c 
 + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**5*c**2*d**2*f*g*n - 
6*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x* 
*2),x)*b**5*c*d**3*f**2*n + 2*log(c + d*x)*a**4*d**3*g**2*n - 6*log(c + d* 
x)*a**3*b*d**3*f*g*n - 3*log(c + d*x)*a**3*b*d**3*g**2*n**2 + 3*log(c + d* 
x)*a**2*b**2*c*d**2*g**2*n**2 + 6*log(c + d*x)*a**2*b**2*d**3*f**2*n + 6*l 
og(c + d*x)*a**2*b**2*d**3*f*g*n**2 - 2*log(c + d*x)*a*b**3*c**3*g**2*n + 
6*log(c + d*x)*a*b**3*c**2*d*f*g*n + 3*log(c + d*x)*a*b**3*c**2*d*g**2*n** 
2 - 6*log(c + d*x)*a*b**3*c*d**2*f**2*n - 12*log(c + d*x)*a*b**3*c*d**2*f* 
g*n**2 - 3*log(c + d*x)*b**4*c**3*g**2*n**2 + 6*log(c + d*x)*b**4*c**2*d*f 
*g*n**2 - log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**2*c*d**2*g**2 - lo 
g(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**3*c**2*d*g**2 + 3*log(((a + b*x)* 
*n*e)/(c + d*x)**n)**2*a*b**3*c*d**2*f*g + 3*log(((a + b*x)**n*e)/(c + d*x 
)**n)**2*b**4*d**3*f**2*x + 3*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**4*d 
**3*f*g*x**2 + log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**4*d**3*g**2*x**...