\(\int (a g+b g x)^2 (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [109]

Optimal result

Integrand size = 41, antiderivative size = 190 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^3 g^2 i n x}{12 b d^2}-\frac {B (b c-a d)^2 g^2 i n (a+b x)^2}{24 b^2 d}+\frac {g^2 i (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b}+\frac {(b c-a d) g^2 i (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{12 b^2}-\frac {B (b c-a d)^4 g^2 i n \log (c+d x)}{12 b^2 d^3} \] Output:

1/12*B*(-a*d+b*c)^3*g^2*i*n*x/b/d^2-1/24*B*(-a*d+b*c)^2*g^2*i*n*(b*x+a)^2/ 
b^2/d+1/4*g^2*i*(b*x+a)^3*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b+1/12*( 
-a*d+b*c)*g^2*i*(b*x+a)^3*(A-B*n+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2-1/12*B*( 
-a*d+b*c)^4*g^2*i*n*ln(d*x+c)/b^2/d^3
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.18 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^2 i \left (8 (b c-a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 d (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {4 B (b c-a d)^2 n \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )}{d^3}-\frac {B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^3}\right )}{24 b^2} \] Input:

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

Output:

(g^2*i*(8*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 
 6*d*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + (4*B*(b*c - a*d) 
^2*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x] 
))/d^3 - (B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a 
 + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]))/d^3))/(24*b 
^2)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2959, 27, 2947, 49, 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.

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

\(\Big \downarrow \) 2959

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2947

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

\(\Big \downarrow \) 49

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 49

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

rule 2009

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

rule 2947

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + 
 B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d) 
/(g*(m + 1)))   Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; Free 
Q[{a, b, c, d, e, f, g, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] 
&& NeQ[m, -2]
 

rule 2959

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_)), x_Symbol] :> Simp[(f 
 + g*x)^(m + 1)*(h + i*x)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 2 
))), x] + Simp[i*((b*c - a*d)/(b*d*(m + 2)))   Int[(f + g*x)^m*(A - B*n + B 
*Log[e*((a + b*x)/(c + d*x))^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, 
i, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c 
*i, 0] && IGtQ[m, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(875\) vs. \(2(180)=360\).

Time = 4.98 (sec) , antiderivative size = 876, normalized size of antiderivative = 4.61

method	result	size
parallelrisch	\(\frac {6 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{4} g^{2} i n +8 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d \,g^{2} i \,n^{2}+12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} c^{2} d^{2} g^{2} i n -2 B \,b^{4} c^{4} g^{2} i \,n^{2}-2 B \,a^{4} d^{4} g^{2} i \,n^{2}+24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} c \,d^{3} g^{2} i n +24 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} c \,d^{3} g^{2} i n +2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{4} g^{2} i n -2 B \ln \left (b x +a \right ) b^{4} c^{4} g^{2} i \,n^{2}-2 B \ln \left (b x +a \right ) a^{4} d^{4} g^{2} i \,n^{2}+6 A \,x^{4} b^{4} d^{4} g^{2} i n -48 A \,a^{2} b^{2} c^{2} d^{2} g^{2} i n +2 B \,x^{3} a \,b^{3} d^{4} g^{2} i \,n^{2}-2 B \,x^{3} b^{4} c \,d^{3} g^{2} i \,n^{2}-8 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} c^{3} d \,g^{2} i n +16 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} d^{4} g^{2} i n +8 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{3} g^{2} i n +12 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} d^{4} g^{2} i n -4 B \,x^{2} a \,b^{3} c \,d^{3} g^{2} i \,n^{2}+24 A \,x^{2} a \,b^{3} c \,d^{3} g^{2} i n +4 B x \,a^{2} b^{2} c \,d^{3} g^{2} i \,n^{2}-8 B x a \,b^{3} c^{2} d^{2} g^{2} i \,n^{2}+24 A x \,a^{2} b^{2} c \,d^{3} g^{2} i n +8 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g^{2} i \,n^{2}-12 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g^{2} i \,n^{2}-11 B \,a^{3} b c \,d^{3} g^{2} i \,n^{2}+8 B \,a^{2} b^{2} c^{2} d^{2} g^{2} i \,n^{2}+7 B a \,b^{3} c^{3} d \,g^{2} i \,n^{2}-36 A \,a^{3} b c \,d^{3} g^{2} i n +16 A \,x^{3} a \,b^{3} d^{4} g^{2} i n +8 A \,x^{3} b^{4} c \,d^{3} g^{2} i n +5 B \,x^{2} a^{2} b^{2} d^{4} g^{2} i \,n^{2}-B \,x^{2} b^{4} c^{2} d^{2} g^{2} i \,n^{2}+12 A \,x^{2} a^{2} b^{2} d^{4} g^{2} i n +2 B x \,a^{3} b \,d^{4} g^{2} i \,n^{2}+2 B x \,b^{4} c^{3} d \,g^{2} i \,n^{2}}{24 b^{2} d^{3} n}\)	\(876\)

Input:

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

Output:

1/24*(6*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^4*d^4*g^2*i*n+8*B*ln(b*x+a)*a*b^ 
3*c^3*d*g^2*i*n^2+12*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c^2*d^2*g^2*i*n-2 
*B*b^4*c^4*g^2*i*n^2-2*B*a^4*d^4*g^2*i*n^2+24*B*x*ln(e*((b*x+a)/(d*x+c))^n 
)*a^2*b^2*c*d^3*g^2*i*n+24*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*c*d^3*g^2 
*i*n+2*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^4*g^2*i*n-2*B*ln(b*x+a)*b^4*c^4*g 
^2*i*n^2-2*B*ln(b*x+a)*a^4*d^4*g^2*i*n^2+6*A*x^4*b^4*d^4*g^2*i*n-48*A*a^2* 
b^2*c^2*d^2*g^2*i*n+2*B*x^3*a*b^3*d^4*g^2*i*n^2-2*B*x^3*b^4*c*d^3*g^2*i*n^ 
2-8*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*c^3*d*g^2*i*n+16*B*x^3*ln(e*((b*x+a) 
/(d*x+c))^n)*a*b^3*d^4*g^2*i*n+8*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^3 
*g^2*i*n+12*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*d^4*g^2*i*n-4*B*x^2*a* 
b^3*c*d^3*g^2*i*n^2+24*A*x^2*a*b^3*c*d^3*g^2*i*n+4*B*x*a^2*b^2*c*d^3*g^2*i 
*n^2-8*B*x*a*b^3*c^2*d^2*g^2*i*n^2+24*A*x*a^2*b^2*c*d^3*g^2*i*n+8*B*ln(b*x 
+a)*a^3*b*c*d^3*g^2*i*n^2-12*B*ln(b*x+a)*a^2*b^2*c^2*d^2*g^2*i*n^2-11*B*a^ 
3*b*c*d^3*g^2*i*n^2+8*B*a^2*b^2*c^2*d^2*g^2*i*n^2+7*B*a*b^3*c^3*d*g^2*i*n^ 
2-36*A*a^3*b*c*d^3*g^2*i*n+16*A*x^3*a*b^3*d^4*g^2*i*n+8*A*x^3*b^4*c*d^3*g^ 
2*i*n+5*B*x^2*a^2*b^2*d^4*g^2*i*n^2-B*x^2*b^4*c^2*d^2*g^2*i*n^2+12*A*x^2*a 
^2*b^2*d^4*g^2*i*n+2*B*x*a^3*b*d^4*g^2*i*n^2+2*B*x*b^4*c^3*d*g^2*i*n^2)/b^ 
2/d^3/n
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (182) = 364\).

Time = 0.11 (sec) , antiderivative size = 529, normalized size of antiderivative = 2.78 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} g^{2} i x^{4} + 2 \, {\left (4 \, B a^{3} b c d^{3} - B a^{4} d^{4}\right )} g^{2} i n \log \left (b x + a\right ) - 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d + 6 \, B a^{2} b^{2} c^{2} d^{2}\right )} g^{2} i n \log \left (d x + c\right ) - 2 \, {\left ({\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g^{2} i n - 4 \, {\left (A b^{4} c d^{3} + 2 \, A a b^{3} d^{4}\right )} g^{2} i\right )} x^{3} - {\left ({\left (B b^{4} c^{2} d^{2} + 4 \, B a b^{3} c d^{3} - 5 \, B a^{2} b^{2} d^{4}\right )} g^{2} i n - 12 \, {\left (2 \, A a b^{3} c d^{3} + A a^{2} b^{2} d^{4}\right )} g^{2} i\right )} x^{2} + 2 \, {\left (12 \, A a^{2} b^{2} c d^{3} g^{2} i + {\left (B b^{4} c^{3} d - 4 \, B a b^{3} c^{2} d^{2} + 2 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g^{2} i n\right )} x + 2 \, {\left (3 \, B b^{4} d^{4} g^{2} i x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i x + 4 \, {\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i x^{3} + 6 \, {\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i x^{2}\right )} \log \left (e\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g^{2} i n x^{4} + 12 \, B a^{2} b^{2} c d^{3} g^{2} i n x + 4 \, {\left (B b^{4} c d^{3} + 2 \, B a b^{3} d^{4}\right )} g^{2} i n x^{3} + 6 \, {\left (2 \, B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} g^{2} i n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{2} d^{3}} \] Input:

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

Output:

1/24*(6*A*b^4*d^4*g^2*i*x^4 + 2*(4*B*a^3*b*c*d^3 - B*a^4*d^4)*g^2*i*n*log( 
b*x + a) - 2*(B*b^4*c^4 - 4*B*a*b^3*c^3*d + 6*B*a^2*b^2*c^2*d^2)*g^2*i*n*l 
og(d*x + c) - 2*((B*b^4*c*d^3 - B*a*b^3*d^4)*g^2*i*n - 4*(A*b^4*c*d^3 + 2* 
A*a*b^3*d^4)*g^2*i)*x^3 - ((B*b^4*c^2*d^2 + 4*B*a*b^3*c*d^3 - 5*B*a^2*b^2* 
d^4)*g^2*i*n - 12*(2*A*a*b^3*c*d^3 + A*a^2*b^2*d^4)*g^2*i)*x^2 + 2*(12*A*a 
^2*b^2*c*d^3*g^2*i + (B*b^4*c^3*d - 4*B*a*b^3*c^2*d^2 + 2*B*a^2*b^2*c*d^3 
+ B*a^3*b*d^4)*g^2*i*n)*x + 2*(3*B*b^4*d^4*g^2*i*x^4 + 12*B*a^2*b^2*c*d^3* 
g^2*i*x + 4*(B*b^4*c*d^3 + 2*B*a*b^3*d^4)*g^2*i*x^3 + 6*(2*B*a*b^3*c*d^3 + 
 B*a^2*b^2*d^4)*g^2*i*x^2)*log(e) + 2*(3*B*b^4*d^4*g^2*i*n*x^4 + 12*B*a^2* 
b^2*c*d^3*g^2*i*n*x + 4*(B*b^4*c*d^3 + 2*B*a*b^3*d^4)*g^2*i*n*x^3 + 6*(2*B 
*a*b^3*c*d^3 + B*a^2*b^2*d^4)*g^2*i*n*x^2)*log((b*x + a)/(d*x + c)))/(b^2* 
d^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 740 vs. \(2 (182) = 364\).

Time = 0.06 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.89 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*B*b^2*d*g^2*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^2*d 
*g^2*i*x^4 + 1/3*B*b^2*c*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) 
+ 2/3*B*a*b*d*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A*b^2 
*c*g^2*i*x^3 + 2/3*A*a*b*d*g^2*i*x^3 + B*a*b*c*g^2*i*x^2*log(e*(b*x/(d*x + 
 c) + a/(d*x + c))^n) + 1/2*B*a^2*d*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d* 
x + c))^n) + A*a*b*c*g^2*i*x^2 + 1/2*A*a^2*d*g^2*i*x^2 - 1/24*B*b^2*d*g^2* 
i*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b 
^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^ 
3*d^3)) + 1/6*B*b^2*c*g^2*i*n*(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)) + 1/ 
3*B*a*b*d*g^2*i*n*(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)) - B*a*b*c*g^2*i* 
n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - 1/ 
2*B*a^2*d*g^2*i*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a* 
d)*x/(b*d)) + B*a^2*c*g^2*i*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + B*a^ 
2*c*g^2*i*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a^2*c*g^2*i*x
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2571 vs. \(2 (182) = 364\).

Time = 0.85 (sec) , antiderivative size = 2571, normalized size of antiderivative = 13.53 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/24*(2*(B*b^7*c^5*g^2*i*n - 5*B*a*b^6*c^4*d*g^2*i*n - 4*(b*x + a)*B*b^6*c 
^5*d*g^2*i*n/(d*x + c) + 10*B*a^2*b^5*c^3*d^2*g^2*i*n + 20*(b*x + a)*B*a*b 
^5*c^4*d^2*g^2*i*n/(d*x + c) + 6*(b*x + a)^2*B*b^5*c^5*d^2*g^2*i*n/(d*x + 
c)^2 - 10*B*a^3*b^4*c^2*d^3*g^2*i*n - 40*(b*x + a)*B*a^2*b^4*c^3*d^3*g^2*i 
*n/(d*x + c) - 30*(b*x + a)^2*B*a*b^4*c^4*d^3*g^2*i*n/(d*x + c)^2 + 5*B*a^ 
4*b^3*c*d^4*g^2*i*n + 40*(b*x + a)*B*a^3*b^3*c^2*d^4*g^2*i*n/(d*x + c) + 6 
0*(b*x + a)^2*B*a^2*b^3*c^3*d^4*g^2*i*n/(d*x + c)^2 - B*a^5*b^2*d^5*g^2*i* 
n - 20*(b*x + a)*B*a^4*b^2*c*d^5*g^2*i*n/(d*x + c) - 60*(b*x + a)^2*B*a^3* 
b^2*c^2*d^5*g^2*i*n/(d*x + c)^2 + 4*(b*x + a)*B*a^5*b*d^6*g^2*i*n/(d*x + c 
) + 30*(b*x + a)^2*B*a^4*b*c*d^6*g^2*i*n/(d*x + c)^2 - 6*(b*x + a)^2*B*a^5 
*d^7*g^2*i*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/(b^4*d^3 - 4*(b*x + a)* 
b^3*d^4/(d*x + c) + 6*(b*x + a)^2*b^2*d^5/(d*x + c)^2 - 4*(b*x + a)^3*b*d^ 
6/(d*x + c)^3 + (b*x + a)^4*d^7/(d*x + c)^4) + (B*b^8*c^5*g^2*i*n - 5*B*a* 
b^7*c^4*d*g^2*i*n - 2*(b*x + a)*B*b^7*c^5*d*g^2*i*n/(d*x + c) + 10*B*a^2*b 
^6*c^3*d^2*g^2*i*n + 10*(b*x + a)*B*a*b^6*c^4*d^2*g^2*i*n/(d*x + c) - (b*x 
 + a)^2*B*b^6*c^5*d^2*g^2*i*n/(d*x + c)^2 - 10*B*a^3*b^5*c^2*d^3*g^2*i*n - 
 20*(b*x + a)*B*a^2*b^5*c^3*d^3*g^2*i*n/(d*x + c) + 5*(b*x + a)^2*B*a*b^5* 
c^4*d^3*g^2*i*n/(d*x + c)^2 + 2*(b*x + a)^3*B*b^5*c^5*d^3*g^2*i*n/(d*x + c 
)^3 + 5*B*a^4*b^4*c*d^4*g^2*i*n + 20*(b*x + a)*B*a^3*b^4*c^2*d^4*g^2*i*n/( 
d*x + c) - 10*(b*x + a)^2*B*a^2*b^4*c^3*d^4*g^2*i*n/(d*x + c)^2 - 10*(b...
 

Mupad [B] (verification not implemented)

Time = 26.14 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.49 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,a^2\,c\,g^2\,i\,x+\frac {B\,a\,g^2\,i\,x^2\,\left (a\,d+2\,b\,c\right )}{2}+\frac {B\,b\,g^2\,i\,x^3\,\left (2\,a\,d+b\,c\right )}{3}+\frac {B\,b^2\,d\,g^2\,i\,x^4}{4}\right )+x^3\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{12\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\,n\right )}{3\,d}+A\,a\,b\,c\,g^2\,i\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {a\,g^2\,i\,\left (2\,A\,a^2\,d^2+6\,A\,b^2\,c^2+B\,a^2\,d^2\,n-2\,B\,b^2\,c^2\,n+12\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\,n\right )}{2\,b\,d}\right )-x^2\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {b\,g^2\,i\,\left (12\,A\,a\,d+8\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,b\,g^2\,i\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{24\,b\,d}-\frac {g^2\,i\,\left (9\,A\,a^2\,d^2+3\,A\,b^2\,c^2+2\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+18\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\,n\right )}{6\,d}+\frac {A\,a\,b\,c\,g^2\,i}{2}\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^4\,d\,g^2\,i\,n-4\,B\,a^3\,b\,c\,g^2\,i\,n\right )}{12\,b^2}-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,i\,n\,a^2\,c^2\,d^2\,g^2-4\,B\,i\,n\,a\,b\,c^3\,d\,g^2+B\,i\,n\,b^2\,c^4\,g^2\right )}{12\,d^3}+\frac {A\,b^2\,d\,g^2\,i\,x^4}{4} \] Input:

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

Output:

log(e*((a + b*x)/(c + d*x))^n)*(B*a^2*c*g^2*i*x + (B*a*g^2*i*x^2*(a*d + 2* 
b*c))/2 + (B*b*g^2*i*x^3*(2*a*d + b*c))/3 + (B*b^2*d*g^2*i*x^4)/4) + x^3*( 
(b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d*n - B*b*c*n))/12 - (A*b*g^2*i*(12*a*d 
 + 12*b*c))/36) + x*(((12*a*d + 12*b*c)*(((12*a*d + 12*b*c)*((b*g^2*i*(12* 
A*a*d + 8*A*b*c + B*a*d*n - B*b*c*n))/4 - (A*b*g^2*i*(12*a*d + 12*b*c))/12 
))/(12*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 2*B*a^2*d^2*n - B*b^2*c^ 
2*n + 18*A*a*b*c*d - B*a*b*c*d*n))/(3*d) + A*a*b*c*g^2*i))/(12*b*d) - (a*c 
*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d*n - B*b*c*n))/4 - (A*b*g^2*i*(12*a* 
d + 12*b*c))/12))/(b*d) + (a*g^2*i*(2*A*a^2*d^2 + 6*A*b^2*c^2 + B*a^2*d^2* 
n - 2*B*b^2*c^2*n + 12*A*a*b*c*d + B*a*b*c*d*n))/(2*b*d)) - x^2*(((12*a*d 
+ 12*b*c)*((b*g^2*i*(12*A*a*d + 8*A*b*c + B*a*d*n - B*b*c*n))/4 - (A*b*g^2 
*i*(12*a*d + 12*b*c))/12))/(24*b*d) - (g^2*i*(9*A*a^2*d^2 + 3*A*b^2*c^2 + 
2*B*a^2*d^2*n - B*b^2*c^2*n + 18*A*a*b*c*d - B*a*b*c*d*n))/(6*d) + (A*a*b* 
c*g^2*i)/2) - (log(a + b*x)*(B*a^4*d*g^2*i*n - 4*B*a^3*b*c*g^2*i*n))/(12*b 
^2) - (log(c + d*x)*(B*b^2*c^4*g^2*i*n + 6*B*a^2*c^2*d^2*g^2*i*n - 4*B*a*b 
*c^3*d*g^2*i*n))/(12*d^3) + (A*b^2*d*g^2*i*x^4)/4
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 573, normalized size of antiderivative = 3.02 \[ \int (a g+b g x)^2 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^{2} i \left (12 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{2} c^{2} d^{2}-8 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} c^{3} d +6 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} d^{4} x^{4}+12 a^{3} b \,d^{4} x^{2}+16 a^{2} b^{2} d^{4} x^{3}+6 a \,b^{3} d^{4} x^{4}+24 a^{3} b c \,d^{3} x +24 a^{2} b^{2} c \,d^{3} x^{2}+4 a^{2} b^{2} c \,d^{3} n x -8 a \,b^{3} c^{2} d^{2} n x -4 a \,b^{3} c \,d^{3} n \,x^{2}+12 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{2} d^{4} x^{2}+16 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} d^{4} x^{3}+2 a^{3} b \,d^{4} n x +5 a^{2} b^{2} d^{4} n \,x^{2}+2 a \,b^{3} d^{4} n \,x^{3}+2 b^{4} c^{3} d n x -b^{4} c^{2} d^{2} n \,x^{2}-2 b^{4} c \,d^{3} n \,x^{3}+8 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c \,d^{3} x^{3}+8 a \,b^{3} c \,d^{3} x^{3}+2 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c^{4}-2 \,\mathrm {log}\left (b x +a \right ) a^{4} d^{4} n -2 \,\mathrm {log}\left (b x +a \right ) b^{4} c^{4} n +8 \,\mathrm {log}\left (b x +a \right ) a^{3} b c \,d^{3} n -12 \,\mathrm {log}\left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} n +8 \,\mathrm {log}\left (b x +a \right ) a \,b^{3} c^{3} d n +24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{2} c \,d^{3} x +24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} c \,d^{3} x^{2}\right )}{24 b \,d^{3}} \] Input:

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

Output:

(g**2*i*( - 2*log(a + b*x)*a**4*d**4*n + 8*log(a + b*x)*a**3*b*c*d**3*n - 
12*log(a + b*x)*a**2*b**2*c**2*d**2*n + 8*log(a + b*x)*a*b**3*c**3*d*n - 2 
*log(a + b*x)*b**4*c**4*n + 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b** 
2*c**2*d**2 + 24*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*c*d**3*x + 1 
2*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**2*d**4*x**2 - 8*log(((a + b*x 
)**n*e)/(c + d*x)**n)*a*b**3*c**3*d + 24*log(((a + b*x)**n*e)/(c + d*x)**n 
)*a*b**3*c*d**3*x**2 + 16*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**3*d**4*x 
**3 + 2*log(((a + b*x)**n*e)/(c + d*x)**n)*b**4*c**4 + 8*log(((a + b*x)**n 
*e)/(c + d*x)**n)*b**4*c*d**3*x**3 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)* 
b**4*d**4*x**4 + 24*a**3*b*c*d**3*x + 2*a**3*b*d**4*n*x + 12*a**3*b*d**4*x 
**2 + 4*a**2*b**2*c*d**3*n*x + 24*a**2*b**2*c*d**3*x**2 + 5*a**2*b**2*d**4 
*n*x**2 + 16*a**2*b**2*d**4*x**3 - 8*a*b**3*c**2*d**2*n*x - 4*a*b**3*c*d** 
3*n*x**2 + 8*a*b**3*c*d**3*x**3 + 2*a*b**3*d**4*n*x**3 + 6*a*b**3*d**4*x** 
4 + 2*b**4*c**3*d*n*x - b**4*c**2*d**2*n*x**2 - 2*b**4*c*d**3*n*x**3))/(24 
*b*d**3)