\(\int (a g+b g x)^2 (A+B \log (\frac {e (c+d x)}{a+b x}))^2 \, dx\) [184]

Optimal result

Integrand size = 32, antiderivative size = 335 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {B^2 (b c-a d)^3 g^2 \log (a+b x)}{b d^3}-\frac {B^2 (b c-a d)^3 g^2 \log \left (\frac {c+d x}{a+b x}\right )}{3 b d^3}+\frac {B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b d}-\frac {2 B (b c-a d)^2 g^2 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b}-\frac {2 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3}+\frac {2 B^2 (b c-a d)^3 g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.87 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2-\frac {B (b c-a d) \left (2 A b d (b c-a d) x+2 B (b c-a d)^2 \log (a+b x)-B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+2 b B (b c-a d) (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-2 (b c-a d)^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{3 b} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.01 (sec) , antiderivative size = 361, 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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

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((d*e*x + c*e)/(b*x + a))^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((d*e*x + c*e)/(b*x + a)) 
, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1172 vs. \(2 (320) = 640\).

Time = 0.13 (sec) , antiderivative size = 1172, normalized size of antiderivative = 3.50 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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