\(\int \frac {(a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{c i+d i x} \, dx\) [85]

Optimal result

Integrand size = 42, antiderivative size = 536 \[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c i+d i x} \, dx=-\frac {B (b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^2 i}-\frac {4 B (b c-a d)^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i}-\frac {2 (b c-a d) g^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^2 i}+\frac {b^2 g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^3 i}-\frac {(b c-a d)^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i}+\frac {B^2 (b c-a d)^2 g^2 \log (c+d x)}{d^3 i}+\frac {B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^3 i}-\frac {4 B^2 (b c-a d)^2 g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {2 B (b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i}-\frac {B^2 (b c-a d)^2 g^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^3 i}+\frac {2 B^2 (b c-a d)^2 g^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i} \] Output:

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

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 727, normalized size of antiderivative = 1.36 \[ \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} \, dx=\frac {g^2 \left (-2 b d (b c-a d) x \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+2 A^2 (b c-a d)^2 \log (c+d x)-4 A B (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )-2 B^2 (b c-a d)^2 \log ^2\left (\frac {e (a+b x)}{c+d x}\right ) \log \left (\frac {b c-a d}{b c+b d x}\right )+2 A B (b c-a d)^2 \left (\log \left (\frac {b c-a d}{b c+b d x}\right ) \left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )+\log \left (\frac {b c-a d}{b c+b d x}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )-2 B (b c-a d) \left (2 a d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b c \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-a B d \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 B c \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 (b c-a d) \left (2 A b d x+2 B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B (b c-a d) \log (c+d x)-2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+B (b c-a d) \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 )-4 B^2 (b c-a d)^2 \left (\log \left (\frac {e (a+b x)}{c+d x}\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )-\operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )\right )\right )}{2 d^3 i} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 470, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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]

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

Input:

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

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Reduce [F]

\[ \int \frac {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{c i+d i x} \, dx=\frac {g^{2} i \left (-2 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2}}{d x +c}d x \right ) a^{2} b^{2} d^{3}-4 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )}{d x +c}d x \right ) a^{3} b \,d^{3}-2 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} x^{2}}{d x +c}d x \right ) b^{4} d^{3}-4 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right )^{2} x}{d x +c}d x \right ) a \,b^{3} d^{3}-4 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x^{2}}{d x +c}d x \right ) a \,b^{3} d^{3}-8 \left (\int \frac {\mathrm {log}\left (\frac {b e x +a e}{d x +c}\right ) x}{d x +c}d x \right ) a^{2} b^{2} d^{3}-2 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{2}+4 \,\mathrm {log}\left (d x +c \right ) a^{3} b c d -2 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2}-4 a^{3} b \,d^{2} x +2 a^{2} b^{2} c d x -a^{2} b^{2} d^{2} x^{2}\right )}{2 d^{3}} \] Input:

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

Output:

(g**2*i*( - 2*int(log((a*e + b*e*x)/(c + d*x))**2/(c + d*x),x)*a**2*b**2*d 
**3 - 4*int(log((a*e + b*e*x)/(c + d*x))/(c + d*x),x)*a**3*b*d**3 - 2*int( 
(log((a*e + b*e*x)/(c + d*x))**2*x**2)/(c + d*x),x)*b**4*d**3 - 4*int((log 
((a*e + b*e*x)/(c + d*x))**2*x)/(c + d*x),x)*a*b**3*d**3 - 4*int((log((a*e 
 + b*e*x)/(c + d*x))*x**2)/(c + d*x),x)*a*b**3*d**3 - 8*int((log((a*e + b* 
e*x)/(c + d*x))*x)/(c + d*x),x)*a**2*b**2*d**3 - 2*log(c + d*x)*a**4*d**2 
+ 4*log(c + d*x)*a**3*b*c*d - 2*log(c + d*x)*a**2*b**2*c**2 - 4*a**3*b*d** 
2*x + 2*a**2*b**2*c*d*x - a**2*b**2*d**2*x**2))/(2*d**3)