\(\int (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2 \, dx\) [67]

Optimal result

Integrand size = 32, antiderivative size = 334 \[ \int (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 i^2 x}{3 b^2}+\frac {B^2 (b c-a d)^3 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d}-\frac {2 B (b c-a d)^2 i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^3}-\frac {B (b c-a d) i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b d}+\frac {i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 d}+\frac {B^2 (b c-a d)^3 i^2 \log (c+d x)}{b^3 d}+\frac {2 B (b c-a d)^3 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^3 d}-\frac {2 B^2 (b c-a d)^3 i^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^3 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.86 \[ \int (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\frac {i^2 \left ((c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-\frac {B (b c-a d) \left (2 A b d (b c-a d) x-B (b c-a d) (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 B (b c-a d)^2 \log (c+d x)-B (b c-a d)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}\right )}{3 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {2952, 2756, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

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

rule 2751

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x 
_Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* 
(n/d)   Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, 
x] && EqQ[r*(q + 1) + 1, 0]
 

rule 2756

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

rule 2779

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r 
_.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) 
, x] + Simp[b*n*(p/(d*r))   Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 
 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
 

rule 2789

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ 
(x_), x_Symbol] :> Simp[1/d   Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x 
), x], x] - Simp[e/d   Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free 
Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 2952

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/d)^m   Subst[Int[(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] && EqQ[ 
n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f 
 - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1202 vs. \(2 (319) = 638\).

Time = 0.14 (sec) , antiderivative size = 1202, normalized size of antiderivative = 3.60 \[ \int (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*A^2*d^2*i^2*x^3 + A^2*c*d*i^2*x^2 + 2*(x*log(b*e*x/(d*x + c) + a*e/(d* 
x + c)) + a*log(b*x + a)/b - c*log(d*x + c)/d)*A*B*c^2*i^2 + 2*(x^2*log(b* 
e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d 
^2 - (b*c - a*d)*x/(b*d))*A*B*c*d*i^2 + 1/3*(2*x^3*log(b*e*x/(d*x + c) + a 
*e/(d*x + c)) + 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))*A*B*d^2*i^2 + A^2*c 
^2*i^2*x - 1/3*(5*a*b*c^2*d*i^2 - 2*a^2*c*d^2*i^2 + (2*i^2*log(e) - 3*i^2) 
*b^2*c^3)*B^2*log(d*x + c)/(b^2*d) - 2/3*(b^3*c^3*i^2 - 3*a*b^2*c^2*d*i^2 
+ 3*a^2*b*c*d^2*i^2 - a^3*d^3*i^2)*(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) + 1/3*(B^2*b^3* 
d^3*i^2*x^3*log(e)^2 + (a*b^2*d^3*i^2*log(e) + (3*i^2*log(e)^2 - i^2*log(e 
))*b^3*c*d^2)*B^2*x^2 + ((3*i^2*log(e)^2 - 4*i^2*log(e) + i^2)*b^3*c^2*d + 
 2*(3*i^2*log(e) - i^2)*a*b^2*c*d^2 - (2*i^2*log(e) - i^2)*a^2*b*d^3)*B^2* 
x + (B^2*b^3*d^3*i^2*x^3 + 3*B^2*b^3*c*d^2*i^2*x^2 + 3*B^2*b^3*c^2*d*i^2*x 
 + (3*a*b^2*c^2*d*i^2 - 3*a^2*b*c*d^2*i^2 + a^3*d^3*i^2)*B^2)*log(b*x + a) 
^2 + (B^2*b^3*d^3*i^2*x^3 + 3*B^2*b^3*c*d^2*i^2*x^2 + 3*B^2*b^3*c^2*d*i^2* 
x + B^2*b^3*c^3*i^2)*log(d*x + c)^2 + (2*B^2*b^3*d^3*i^2*x^3*log(e) + (a*b 
^2*d^3*i^2 + (6*i^2*log(e) - i^2)*b^3*c*d^2)*B^2*x^2 + 2*(3*a*b^2*c*d^2*i^ 
2 - a^2*b*d^3*i^2 + (3*i^2*log(e) - 2*i^2)*b^3*c^2*d)*B^2*x + (2*(3*i^2*lo 
g(e) - 2*i^2)*a*b^2*c^2*d - (6*i^2*log(e) - 7*i^2)*a^2*b*c*d^2 + (2*i^2...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2 
),x)*a**3*b**2*d**4 + 6*int((log((a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x 
+ b*c*x + b*d*x**2),x)*a**2*b**3*c*d**3 - 6*int((log((a*e + b*e*x)/(c + d* 
x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a*b**4*c**2*d**2 + 2*int((log(( 
a*e + b*e*x)/(c + d*x))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**5*c**3*d 
 - 2*log(a + b*x)*a**4*d**3 + 6*log(a + b*x)*a**3*b*c*d**2 + 3*log(a + b*x 
)*a**3*b*d**3 - 6*log(a + b*x)*a**2*b**2*c**2*d - 9*log(a + b*x)*a**2*b**2 
*c*d**2 + 2*log(a + b*x)*a*b**3*c**3 + 9*log(a + b*x)*a*b**3*c**2*d - 3*lo 
g(a + b*x)*b**4*c**3 + log((a*e + b*e*x)/(c + d*x))**2*a**2*b**2*c*d**2 - 
2*log((a*e + b*e*x)/(c + d*x))**2*a*b**3*c**2*d - 3*log((a*e + b*e*x)/(c + 
 d*x))**2*b**4*c**2*d*x - 3*log((a*e + b*e*x)/(c + d*x))**2*b**4*c*d**2*x* 
*2 - log((a*e + b*e*x)/(c + d*x))**2*b**4*d**3*x**3 + 2*log((a*e + b*e*x)/ 
(c + d*x))*a**2*b**2*c*d**2 + 2*log((a*e + b*e*x)/(c + d*x))*a**2*b**2*d** 
3*x - 2*log((a*e + b*e*x)/(c + d*x))*a*b**3*c**3 - 6*log((a*e + b*e*x)/(c 
+ d*x))*a*b**3*c**2*d*x - 5*log((a*e + b*e*x)/(c + d*x))*a*b**3*c**2*d - 6 
*log((a*e + b*e*x)/(c + d*x))*a*b**3*c*d**2*x**2 - 6*log((a*e + b*e*x)/(c 
+ d*x))*a*b**3*c*d**2*x - 2*log((a*e + b*e*x)/(c + d*x))*a*b**3*d**3*x**3 
- log((a*e + b*e*x)/(c + d*x))*a*b**3*d**3*x**2 + 3*log((a*e + b*e*x)/(c + 
 d*x))*b**4*c**3 + 4*log((a*e + b*e*x)/(c + d*x))*b**4*c**2*d*x + log((a*e 
 + b*e*x)/(c + d*x))*b**4*c*d**2*x**2 + 2*a**3*b*d**3*x - 3*a**2*b**2*c...