\(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^3} \, dx\) [101]

Optimal result

Integrand size = 42, antiderivative size = 410 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=-\frac {B^2 g^2 (a+b x)^2}{4 d i^3 (c+d x)^2}+\frac {2 A b B g^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac {2 b B^2 g^2 (a+b x)}{d^2 i^3 (c+d x)}+\frac {2 b B^2 g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^3 (c+d x)}+\frac {B g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 d i^3 (c+d x)^2}-\frac {g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d i^3 (c+d x)^2}-\frac {b g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i^3 (c+d x)}-\frac {b^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^3}-\frac {2 b^2 B g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}+\frac {2 b^2 B^2 g^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3} \] Output:

-1/4*B^2*g^2*(b*x+a)^2/d/i^3/(d*x+c)^2+2*A*b*B*g^2*(b*x+a)/d^2/i^3/(d*x+c) 
-2*b*B^2*g^2*(b*x+a)/d^2/i^3/(d*x+c)+2*b*B^2*g^2*(b*x+a)*ln(e*(b*x+a)/(d*x 
+c))/d^2/i^3/(d*x+c)+1/2*B*g^2*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/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)))^2/d/i^3/(d*x+c)^2 
-b*g^2*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^2/i^3/(d*x+c)-b^2*g^2*ln((- 
a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/d^3/i^3-2*b^2*B*g^2*(A+B 
*ln(e*(b*x+a)/(d*x+c)))*polylog(2,d*(b*x+a)/b/(d*x+c))/d^3/i^3+2*b^2*B^2*g 
^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. \(3840\) vs. \(2(410)=820\).

Time = 7.26 (sec) , antiderivative size = 3840, normalized size of antiderivative = 9.37 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^3} \, dx=\text {Result too large to show} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.87, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(g^2*(-1/4*(B^2*(a + b*x)^2)/(d*(c + d*x)^2) + (2*A*b*B*(a + b*x))/(d^2*(c 
 + d*x)) - (2*b*B^2*(a + b*x))/(d^2*(c + d*x)) + (2*b*B^2*(a + b*x)*Log[(e 
*(a + b*x))/(c + d*x)])/(d^2*(c + d*x)) + (B*(a + b*x)^2*(A + B*Log[(e*(a 
+ b*x))/(c + d*x)]))/(2*d*(c + d*x)^2) - ((a + b*x)^2*(A + B*Log[(e*(a + b 
*x))/(c + d*x)])^2)/(2*d*(c + d*x)^2) - (b*(a + b*x)*(A + B*Log[(e*(a + b* 
x))/(c + d*x)])^2)/(d^2*(c + d*x)) - (b^2*(A + B*Log[(e*(a + b*x))/(c + d* 
x)])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^3 - (2*b^2*B*(A + B*Log[(e* 
(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^3 + (2*b 
^2*B^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 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [F]

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

Input:

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

Fricas [F]

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

integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^3,x, al 
gorithm="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((b*e*x + a*e)/(d*x + c))^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((b*e*x + a*e)/(d*x + c) 
))/(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]

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

integrate((b*g*x+a*g)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**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(a*e/(c + d*x) + b*e*x/(c + d*x))**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(a* 
e/(c + d*x) + b*e*x/(c + d*x))/(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(a*e/(c + d*x) + b*e*x/(c + d*x))* 
*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(a*e/(c + d*x) + b*e*x/(c + d*x))/(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(a*e/(c + d*x) + b*e*x/( 
c + d*x))**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integra 
l(4*A*B*a*b*x*log(a*e/(c + d*x) + b*e*x/(c + d*x))/(c**3 + 3*c**2*d*x + 3* 
c*d**2*x**2 + d**3*x**3), x))/i**3
 

Maxima [F]

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

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

Output:

-A*B*a*b*g^2*(2*(2*d*x + c)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^4*i^3* 
x^2 + 2*c*d^3*i^3*x + c^2*d^2*i^3) - (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*((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*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^3*i^3*x^2 + 2*c 
*d^2*i^3*x + c^2*d*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*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) - 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) + 1/6*(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)^3 + 3*(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)*log(d*x + c)^2)/(d^5*i 
^3*x^2 + 2*c*d^4*i^3*x + c^2*d^3*i^3) - integrate(-(2*B^2*a*b*d^2*g^2*x*lo 
g(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)^2 + 2*(2*B^2*a*b*d^2*g^2*x*log(e) + B^2*a^2*d^2*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Reduce [F]

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

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

Output:

(g**2*i*(4*int((log((a*e + b*e*x)/(c + d*x))**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*e + b*e 
*x)/(c + d*x))**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*e + b*e*x)/(c + d*x))**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*e + b*e*x)/(c + d*x))**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((log((a*e + b*e*x)/(c + d* 
x))**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*e + b*e*x)/(c + d*x))**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*e + 
 b*e*x)/(c + d*x))**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*e + b*e*x)/(c + d*x))**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*e + b*e*x)/(c + d*x))**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*e + b*e*x)/(c + d*x))*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*e + b*e*x)/(c + d*x))*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*e + b*e*x)/(c + 
d*x))*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 
**2*d**7*x**2 - 16*int((log((a*e + b*e*x)/(c + d*x))*x**2)/(c**3 + 3*c*...