\(\int (a g+b g x)^2 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2 \, dx\) [212]

Optimal result

Integrand size = 34, antiderivative size = 343 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {4 B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^3 g^2 \log (a+b x)}{b d^3}-\frac {4 B^2 (b c-a d)^3 g^2 \log \left (\frac {c+d x}{a+b x}\right )}{3 b d^3}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac {4 B (b c-a d)^2 g^2 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {4 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3}+\frac {8 B^2 (b c-a d)^3 g^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3} \] Output:

4/3*B^2*(-a*d+b*c)^2*g^2*x/d^2-4*B^2*(-a*d+b*c)^3*g^2*ln(b*x+a)/b/d^3-4/3* 
B^2*(-a*d+b*c)^3*g^2*ln((d*x+c)/(b*x+a))/b/d^3+2/3*B*(-a*d+b*c)*g^2*(b*x+a 
)^2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b/d-4/3*B*(-a*d+b*c)^2*g^2*(d*x+c)*(A+ 
B*ln(e*(d*x+c)^2/(b*x+a)^2))/d^3+1/3*g^2*(b*x+a)^3*(A+B*ln(e*(d*x+c)^2/(b* 
x+a)^2))^2/b-4/3*B*(-a*d+b*c)^3*g^2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))*ln(1-d 
*(b*x+a)/b/(d*x+c))/b/d^3+8/3*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 = 298, normalized size of antiderivative = 0.87 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2-\frac {2 B (b c-a d) \left (2 A b d (b c-a d) x+4 B (b c-a d)^2 \log (c+d x)-2 B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+2 B d (b c-a d) (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 (b c-a d)^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{3 b} \] Input:

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

Output:

(g^2*((a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 - (2*B*(b*c - 
 a*d)*(2*A*b*d*(b*c - a*d)*x + 4*B*(b*c - a*d)^2*Log[c + d*x] - 2*B*(b*c - 
 a*d)*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 2*B*d*(b*c - a*d)*(a + b*x)* 
Log[(e*(c + d*x)^2)/(a + b*x)^2] - d^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x) 
^2)/(a + b*x)^2]) - 2*(b*c - a*d)^2*Log[c + d*x]*(A + B*Log[(e*(c + d*x)^2 
)/(a + b*x)^2]) - 2*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 = 1.00 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, 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.

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

\(\Big \downarrow \) 2952

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 54

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 2789

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

\(\Big \downarrow \) 2751

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 2779

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

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^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^ 
2*x^2 + 2*a*b*x + a^2))^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^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1333 vs. \(2 (328) = 656\).

Time = 0.17 (sec) , antiderivative size = 1333, normalized size of antiderivative = 3.89 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/3*A^2*b^2*g^2*x^3 + A^2*a*b*g^2*x^2 + 2*(x*log(d^2*e*x^2/(b^2*x^2 + 2*a* 
b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b* 
x + a^2)) - 2*a*log(b*x + a)/b + 2*c*log(d*x + c)/d)*A*B*a^2*g^2 + 2*(x^2* 
log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a 
^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) + 2*a^2*log(b*x + a)/b^2 - 2*c^2*lo 
g(d*x + c)/d^2 + 2*(b*c - a*d)*x/(b*d))*A*B*a*b*g^2 + 2/3*(x^3*log(d^2*e*x 
^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e 
/(b^2*x^2 + 2*a*b*x + a^2)) - 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 + 4/3*((g^2*log(e) - 3*g^2)*b^2*c^3 - (3*g^2*log(e) 
 - 7*g^2)*a*b*c^2*d + (3*g^2*log(e) - 4*g^2)*a^2*c*d^2)*B^2*log(d*x + c)/d 
^3 - 8/3*(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 + (2*b^3*c* 
d^2*g^2*log(e) + (3*g^2*log(e)^2 - 2*g^2*log(e))*a*b^2*d^3)*B^2*x^2 - (4*( 
g^2*log(e) - g^2)*b^3*c^2*d - 4*(3*g^2*log(e) - 2*g^2)*a*b^2*c*d^2 - (3*g^ 
2*log(e)^2 - 8*g^2*log(e) + 4*g^2)*a^2*b*d^3)*B^2*x + 4*(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)*lo 
g(b*x + a)^2 + 4*(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...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(g**2*( - 4*int((log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + 
b**2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**2*d**4 + 12*int 
((log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/ 
(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b**3*c*d**3 - 12*int((log((c**2*e 
 + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d*x 
+ b*c*x + b*d*x**2),x)*a*b**4*c**2*d**2 + 4*int((log((c**2*e + 2*c*d*e*x + 
 d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*x)/(a*c + a*d*x + b*c*x + b*d* 
x**2),x)*b**5*c**3*d - 4*log(c + d*x)*a**4*d**3 + 12*log(c + d*x)*a**3*b*c 
*d**2 + 12*log(c + d*x)*a**3*b*d**3 - 12*log(c + d*x)*a**2*b**2*c**2*d - 3 
6*log(c + d*x)*a**2*b**2*c*d**2 + 4*log(c + d*x)*a*b**3*c**3 + 36*log(c + 
d*x)*a*b**3*c**2*d - 12*log(c + d*x)*b**4*c**3 + 2*log((c**2*e + 2*c*d*e*x 
 + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a**2*b**2*c*d**2 + 3*log( 
(c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a**2*b 
**2*d**3*x - log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2 
*x**2))**2*a*b**3*c**2*d + 3*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 
+ 2*a*b*x + b**2*x**2))**2*a*b**3*d**3*x**2 + log((c**2*e + 2*c*d*e*x + d* 
*2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*b**4*d**3*x**3 + 2*log((c**2*e 
 + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**4*d**3 + 6*lo 
g((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**3*b* 
d**3*x - 6*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**...