\(\int \frac {(a g+b g x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c i+d i x)^3} \, dx\) [203]

Optimal result

Integrand size = 45, antiderivative size = 441 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {B^2 g^2 n^2 (a+b x)^2}{4 d i^3 (c+d x)^2}+\frac {2 A b B g^2 n (a+b x)}{d^2 i^3 (c+d x)}-\frac {2 b B^2 g^2 n^2 (a+b x)}{d^2 i^3 (c+d x)}+\frac {2 b B^2 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d^2 i^3 (c+d x)}+\frac {B g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d i^3 (c+d x)^2}-\frac {g^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d i^3 (c+d x)^2}-\frac {b g^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2 i^3 (c+d x)}-\frac {b^2 g^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (\frac {b c-a d}{b (c+d x)}\right )}{d^3 i^3}-\frac {2 b^2 B g^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}+\frac {2 b^2 B^2 g^2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3} \] Output:

-1/4*B^2*g^2*n^2*(b*x+a)^2/d/i^3/(d*x+c)^2+2*A*b*B*g^2*n*(b*x+a)/d^2/i^3/( 
d*x+c)-2*b*B^2*g^2*n^2*(b*x+a)/d^2/i^3/(d*x+c)+2*b*B^2*g^2*n*(b*x+a)*ln(e* 
((b*x+a)/(d*x+c))^n)/d^2/i^3/(d*x+c)+1/2*B*g^2*n*(b*x+a)^2*(A+B*ln(e*((b*x 
+a)/(d*x+c))^n))/d/i^3/(d*x+c)^2-1/2*g^2*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x 
+c))^n))^2/d/i^3/(d*x+c)^2-b*g^2*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2 
/d^2/i^3/(d*x+c)-b^2*g^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2*ln((-a*d+b*c)/b 
/(d*x+c))/d^3/i^3-2*b^2*B*g^2*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*polylog(2, 
d*(b*x+a)/b/(d*x+c))/d^3/i^3+2*b^2*B^2*g^2*n^2*polylog(3,d*(b*x+a)/b/(d*x+ 
c))/d^3/i^3
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4493\) vs. \(2(441)=882\).

Time = 7.21 (sec) , antiderivative size = 4493, normalized size of antiderivative = 10.19 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

(-2*(-(A^2*b^2*c*g^2) + a*A^2*b*d*g^2 - 2*A*b^2*B*c*g^2*(Log[e*((a + b*x)/ 
(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) + 2*a*A*b*B*d*g^2*(Log[e*((a + 
 b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]) - b^2*B^2*c*g^2*(Log[e*( 
(a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2 + a*b*B^2*d*g^2*(L 
og[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2))/(d^3*i^3*( 
c + d*x)) + (-(A^2*b^2*c^2*g^2) + 2*a*A^2*b*c*d*g^2 - a^2*A^2*d^2*g^2 - 2* 
A*b^2*B*c^2*g^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x 
)]) + 4*a*A*b*B*c*d*g^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/ 
(c + d*x)]) - 2*a^2*A*B*d^2*g^2*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a 
 + b*x)/(c + d*x)]) - b^2*B^2*c^2*g^2*(Log[e*((a + b*x)/(c + d*x))^n] - n* 
Log[(a + b*x)/(c + d*x)])^2 + 2*a*b*B^2*c*d*g^2*(Log[e*((a + b*x)/(c + d*x 
))^n] - n*Log[(a + b*x)/(c + d*x)])^2 - a^2*B^2*d^2*g^2*(Log[e*((a + b*x)/ 
(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x)])^2)/(2*d^3*i^3*(c + d*x)^2) + ( 
b^2*g^2*(A + B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c + d*x) 
]))^2*Log[c + d*x])/(d^3*i^3) + (2*a^2*B*g^2*n*(A + B*(Log[e*((a + b*x)/(c 
 + d*x))^n] - n*Log[(a + b*x)/(c + d*x)]))*(((c/d + x)*(2*Log[c/d + x] + 4 
*Log[c/d + x]^2))/(8*(c + d*x)^3*Log[c/d + x]) + ((d*(a/b + x))/((-c + (a* 
d)/b)^3*(1 - (d*(a/b + x))/(-c + (a*d)/b))) - ((d^2*(a/b + x)^2)/((-c + (a 
*d)/b)^4*(1 - (d*(a/b + x))/(-c + (a*d)/b))^2) + (2*d*(a/b + x))/((-c + (a 
*d)/b)^3*(1 - (d*(a/b + x))/(-c + (a*d)/b))))*Log[a/b + x] - Log[1 - (d...
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2795, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

rule 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [F]

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

Input:

int((b*g*x+a*g)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,x)
 

Output:

int((b*g*x+a*g)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,x)
 

Fricas [F]

\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{3}} \,d x } \] Input:

integrate((b*g*x+a*g)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,x 
, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\frac {g^{2} \left (\int \frac {A^{2} a^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {A^{2} b^{2} x^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B^{2} a^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 A B a^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 A^{2} a b x}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B^{2} b^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 A B b^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 B^{2} a b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {4 A B a b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx\right )}{i^{3}} \] Input:

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

Output:

g**2*(Integral(A**2*a**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), 
x) + Integral(A**2*b**2*x**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x** 
3), x) + Integral(B**2*a**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c* 
*3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(2*A*B*a**2*log 
(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + 
d**3*x**3), x) + Integral(2*A**2*a*b*x/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 
+ d**3*x**3), x) + Integral(B**2*b**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d 
*x))**n)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral 
(2*A*B*b**2*x**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**3 + 3*c**2*d* 
x + 3*c*d**2*x**2 + d**3*x**3), x) + Integral(2*B**2*a*b*x*log(e*(a/(c + d 
*x) + b*x/(c + d*x))**n)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3 
), x) + Integral(4*A*B*a*b*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**3 
 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/i**3
 

Maxima [F]

\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{3}} \,d x } \] Input:

integrate((b*g*x+a*g)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,x 
, algorithm="maxima")
 

Output:

A*B*a*b*g^2*n*((b*c^2 - 3*a*c*d + 2*(b*c*d - 2*a*d^2)*x)/((b*c*d^4 - a*d^5 
)*i^3*x^2 + 2*(b*c^2*d^3 - a*c*d^4)*i^3*x + (b*c^3*d^2 - a*c^2*d^3)*i^3) + 
 2*(b^2*c - 2*a*b*d)*log(b*x + a)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*i 
^3) - 2*(b^2*c - 2*a*b*d)*log(d*x + c)/((b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d 
^4)*i^3)) + 1/2*A*B*a^2*g^2*n*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)* 
i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3) + 2*b 
^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x 
+ c)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3)) + 1/2*A^2*b^2*g^2*((4*c*d* 
x + 3*c^2)/(d^5*i^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) + 2*log(d*x + c)/(d 
^3*i^3)) - 2*(2*d*x + c)*A*B*a*b*g^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n 
)/(d^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - (2*d*x + c)*A^2*a*b*g^2/(d 
^4*i^3*x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - A*B*a^2*g^2*log(e*(b*x/(d*x + 
c) + a/(d*x + c))^n)/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3) - 1/2*A^2*a 
^2*g^2/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3) + 1/2*(4*(b^2*c*d*g^2 - a 
*b*d^2*g^2)*B^2*x + (3*b^2*c^2*g^2 - 2*a*b*c*d*g^2 - a^2*d^2*g^2)*B^2 + 2* 
(B^2*b^2*d^2*g^2*x^2 + 2*B^2*b^2*c*d*g^2*x + B^2*b^2*c^2*g^2)*log(d*x + c) 
)*log((d*x + c)^n)^2/(d^5*i^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) - integra 
te(-(2*B^2*a*b*d^2*g^2*x*log(e)^2 + B^2*a^2*d^2*g^2*log(e)^2 + (B^2*b^2*d^ 
2*g^2*log(e)^2 + 2*A*B*b^2*d^2*g^2*log(e))*x^2 + (B^2*b^2*d^2*g^2*x^2 + 2* 
B^2*a*b*d^2*g^2*x + B^2*a^2*d^2*g^2)*log((b*x + a)^n)^2 + 2*(2*B^2*a*b*...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

int(((a*g + b*g*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(c*i + d*i* 
x)^3,x)
 

Output:

int(((a*g + b*g*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(c*i + d*i* 
x)^3, x)
 

Reduce [F]

\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c i+d i x)^3} \, dx=\text {too large to display} \] Input:

int((b*g*x+a*g)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*i*x+c*i)^3,x)
 

Output:

(g**2*i*(4*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 + 3*c**2 
*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*b**4*c**4*d**5 + 8*int((log(((a 
+ b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d 
**3*x**3),x)*a**2*b**4*c**3*d**6*x + 4*int((log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**2*b**4 
*c**2*d**7*x**2 - 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 
 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b**5*c**5*d**4 - 16*int((l 
og(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x 
**2 + d**3*x**3),x)*a*b**5*c**4*d**5*x - 8*int((log(((a + b*x)**n*e)/(c + 
d*x)**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a*b** 
5*c**3*d**6*x**2 + 4*int((log(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c** 
3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**6*c**6*d**3 + 8*int((log 
(((a + b*x)**n*e)/(c + d*x)**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x** 
2 + d**3*x**3),x)*b**6*c**5*d**4*x + 4*int((log(((a + b*x)**n*e)/(c + d*x) 
**n)**2*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*b**6*c**4 
*d**5*x**2 + 8*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c** 
2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**3*b**3*c**4*d**5 + 16*int((log((( 
a + b*x)**n*e)/(c + d*x)**n)*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d* 
*3*x**3),x)*a**3*b**3*c**3*d**6*x + 8*int((log(((a + b*x)**n*e)/(c + d*x)* 
*n)*x**2)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*a**3*b**3*...