\(\int (a g+b g x)^3 (c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [108]

Optimal result

Integrand size = 41, antiderivative size = 223 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^4 g^3 i n x}{20 b d^3}+\frac {B (b c-a d)^3 g^3 i n (a+b x)^2}{40 b^2 d^2}-\frac {B (b c-a d)^2 g^3 i n (a+b x)^3}{60 b^2 d}+\frac {g^3 i (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 b}+\frac {(b c-a d) g^3 i (a+b x)^4 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^2}+\frac {B (b c-a d)^5 g^3 i n \log (c+d x)}{20 b^2 d^4} \] Output:

-1/20*B*(-a*d+b*c)^4*g^3*i*n*x/b/d^3+1/40*B*(-a*d+b*c)^3*g^3*i*n*(b*x+a)^2 
/b^2/d^2-1/60*B*(-a*d+b*c)^2*g^3*i*n*(b*x+a)^3/b^2/d+1/5*g^3*i*(b*x+a)^4*( 
d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b+1/20*(-a*d+b*c)*g^3*i*(b*x+a)^4*( 
A-B*n+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2+1/20*B*(-a*d+b*c)^5*g^3*i*n*ln(d*x+ 
c)/b^2/d^4
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.21 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 i \left (30 (b c-a d) (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+24 d (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-\frac {5 B (b c-a d)^2 n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )}{d^4}+\frac {2 B (b c-a d) n \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )}{d^4}\right )}{120 b^2} \] Input:

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

Output:

(g^3*i*(30*(b*c - a*d)*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) 
+ 24*d*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - (5*B*(b*c - a* 
d)^2*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^4 + (2*B*(b*c - a*d)*n*(12*b 
*d*(b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*( 
a + b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]))/d^4))/(12 
0*b^2)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.88, 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.

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

\(\Big \downarrow \) 2959

\(\displaystyle \frac {i (b c-a d) \int g^3 (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )dx}{5 b}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {g^3 i (b c-a d) \int (a+b x)^3 \left (A-B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )dx}{5 b}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}\)

\(\Big \downarrow \) 2947

\(\displaystyle \frac {g^3 i (b c-a d) \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{4 b}-\frac {B n (b c-a d) \int \frac {(a+b x)^3}{c+d x}dx}{4 b}\right )}{5 b}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {g^3 i (b c-a d) \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{4 b}-\frac {B n (b c-a d) \int \left (\frac {(a d-b c)^3}{d^3 (c+d x)}+\frac {b (b c-a d)^2}{d^3}+\frac {b (a+b x)^2}{d}-\frac {b (b c-a d) (a+b x)}{d^2}\right )dx}{4 b}\right )}{5 b}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g^3 i (b c-a d) \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A-B n\right )}{4 b}-\frac {B n (b c-a d) \left (-\frac {(b c-a d)^3 \log (c+d x)}{d^4}+\frac {b x (b c-a d)^2}{d^3}-\frac {(a+b x)^2 (b c-a d)}{2 d^2}+\frac {(a+b x)^3}{3 d}\right )}{4 b}\right )}{5 b}+\frac {g^3 i (a+b x)^4 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b}\)

Input:

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

Output:

(g^3*i*(a + b*x)^4*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b) 
 + ((b*c - a*d)*g^3*i*(((a + b*x)^4*(A - B*n + B*Log[e*((a + b*x)/(c + d*x 
))^n]))/(4*b) - (B*(b*c - a*d)*n*((b*(b*c - a*d)^2*x)/d^3 - ((b*c - a*d)*( 
a + b*x)^2)/(2*d^2) + (a + b*x)^3/(3*d) - ((b*c - a*d)^3*Log[c + d*x])/d^4 
))/(4*b)))/(5*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. \(1194\) vs. \(2(211)=422\).

Time = 11.74 (sec) , antiderivative size = 1195, normalized size of antiderivative = 5.36

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1195\)

Input:

int((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x,method=_RE 
TURNVERBOSE)
 

Output:

1/120*(24*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^5*d^5*g^3*i*n-20*B*x^3*a*b^4*c 
*d^4*g^3*i*n^2+120*A*x^3*a*b^4*c*d^4*g^3*i*n+60*B*x^2*ln(e*((b*x+a)/(d*x+c 
))^n)*a^3*b^2*d^5*g^3*i*n+30*B*x*a^3*b^2*c*d^4*g^3*i*n^2-60*B*x*a^2*b^3*c^ 
2*d^3*g^3*i*n^2+30*B*x*a*b^4*c^3*d^2*g^3*i*n^2+30*B*ln(e*((b*x+a)/(d*x+c)) 
^n)*a*b^4*c^4*d*g^3*i*n-60*B*ln(b*x+a)*a^3*b^2*c^2*d^3*g^3*i*n^2+60*B*ln(b 
*x+a)*a^2*b^3*c^3*d^2*g^3*i*n^2-30*B*ln(b*x+a)*a*b^4*c^4*d*g^3*i*n^2+30*B* 
ln(b*x+a)*a^4*b*c*d^4*g^3*i*n^2-15*B*x^2*a^2*b^3*c*d^4*g^3*i*n^2-15*B*x^2* 
a*b^4*c^2*d^3*g^3*i*n^2+180*A*x^2*a^2*b^3*c*d^4*g^3*i*n+90*B*x^4*ln(e*((b* 
x+a)/(d*x+c))^n)*a*b^4*d^5*g^3*i*n+6*B*x^4*a*b^4*d^5*g^3*i*n^2-6*B*x^4*b^5 
*c*d^4*g^3*i*n^2+90*A*x^4*a*b^4*d^5*g^3*i*n+30*A*x^4*b^5*c*d^4*g^3*i*n+22* 
B*x^3*a^2*b^3*d^5*g^3*i*n^2-2*B*x^3*b^5*c^2*d^3*g^3*i*n^2+120*A*x^3*a^2*b^ 
3*d^5*g^3*i*n+27*B*x^2*a^3*b^2*d^5*g^3*i*n^2+3*B*x^2*b^5*c^3*d^2*g^3*i*n^2 
+60*A*x^2*a^3*b^2*d^5*g^3*i*n+6*B*x*a^4*b*d^5*g^3*i*n^2-6*B*x*b^5*c^4*d*g^ 
3*i*n^2+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*c*d^4*g^3*i*n+180*B*x^2* 
ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c*d^4*g^3*i*n+120*B*x*ln(e*((b*x+a)/(d*x 
+c))^n)*a^3*b^2*c*d^4*g^3*i*n+120*A*x*a^3*b^2*c*d^4*g^3*i*n+60*B*ln(e*((b* 
x+a)/(d*x+c))^n)*a^3*b^2*c^2*d^3*g^3*i*n-60*B*ln(e*((b*x+a)/(d*x+c))^n)*a^ 
2*b^3*c^3*d^2*g^3*i*n+30*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c*d^4*g^3*i*n 
+120*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*d^5*g^3*i*n-6*B*ln(b*x+a)*a^5 
*d^5*g^3*i*n^2+24*A*x^5*b^5*d^5*g^3*i*n-6*B*ln(e*((b*x+a)/(d*x+c))^n)*b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (213) = 426\).

Time = 0.17 (sec) , antiderivative size = 720, normalized size of antiderivative = 3.23 \[ \int (a g+b g x)^3 (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 {24 \, A b^{5} d^{5} g^{3} i x^{5} + 6 \, {\left (5 \, B a^{4} b c d^{4} - B a^{5} d^{5}\right )} g^{3} i n \log \left (b x + a\right ) + 6 \, {\left (B b^{5} c^{5} - 5 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - 10 \, B a^{3} b^{2} c^{2} d^{3}\right )} g^{3} i n \log \left (d x + c\right ) - 6 \, {\left ({\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} g^{3} i n - 5 \, {\left (A b^{5} c d^{4} + 3 \, A a b^{4} d^{5}\right )} g^{3} i\right )} x^{4} - 2 \, {\left ({\left (B b^{5} c^{2} d^{3} + 10 \, B a b^{4} c d^{4} - 11 \, B a^{2} b^{3} d^{5}\right )} g^{3} i n - 60 \, {\left (A a b^{4} c d^{4} + A a^{2} b^{3} d^{5}\right )} g^{3} i\right )} x^{3} + 3 \, {\left ({\left (B b^{5} c^{3} d^{2} - 5 \, B a b^{4} c^{2} d^{3} - 5 \, B a^{2} b^{3} c d^{4} + 9 \, B a^{3} b^{2} d^{5}\right )} g^{3} i n + 20 \, {\left (3 \, A a^{2} b^{3} c d^{4} + A a^{3} b^{2} d^{5}\right )} g^{3} i\right )} x^{2} + 6 \, {\left (20 \, A a^{3} b^{2} c d^{4} g^{3} i - {\left (B b^{5} c^{4} d - 5 \, B a b^{4} c^{3} d^{2} + 10 \, B a^{2} b^{3} c^{2} d^{3} - 5 \, B a^{3} b^{2} c d^{4} - B a^{4} b d^{5}\right )} g^{3} i n\right )} x + 6 \, {\left (4 \, B b^{5} d^{5} g^{3} i x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i x + 5 \, {\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i x^{4} + 20 \, {\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i x^{3} + 10 \, {\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i x^{2}\right )} \log \left (e\right ) + 6 \, {\left (4 \, B b^{5} d^{5} g^{3} i n x^{5} + 20 \, B a^{3} b^{2} c d^{4} g^{3} i n x + 5 \, {\left (B b^{5} c d^{4} + 3 \, B a b^{4} d^{5}\right )} g^{3} i n x^{4} + 20 \, {\left (B a b^{4} c d^{4} + B a^{2} b^{3} d^{5}\right )} g^{3} i n x^{3} + 10 \, {\left (3 \, B a^{2} b^{3} c d^{4} + B a^{3} b^{2} d^{5}\right )} g^{3} i n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{120 \, b^{2} d^{4}} \] Input:

integrate((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, al 
gorithm="fricas")
 

Output:

1/120*(24*A*b^5*d^5*g^3*i*x^5 + 6*(5*B*a^4*b*c*d^4 - B*a^5*d^5)*g^3*i*n*lo 
g(b*x + a) + 6*(B*b^5*c^5 - 5*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - 10*B* 
a^3*b^2*c^2*d^3)*g^3*i*n*log(d*x + c) - 6*((B*b^5*c*d^4 - B*a*b^4*d^5)*g^3 
*i*n - 5*(A*b^5*c*d^4 + 3*A*a*b^4*d^5)*g^3*i)*x^4 - 2*((B*b^5*c^2*d^3 + 10 
*B*a*b^4*c*d^4 - 11*B*a^2*b^3*d^5)*g^3*i*n - 60*(A*a*b^4*c*d^4 + A*a^2*b^3 
*d^5)*g^3*i)*x^3 + 3*((B*b^5*c^3*d^2 - 5*B*a*b^4*c^2*d^3 - 5*B*a^2*b^3*c*d 
^4 + 9*B*a^3*b^2*d^5)*g^3*i*n + 20*(3*A*a^2*b^3*c*d^4 + A*a^3*b^2*d^5)*g^3 
*i)*x^2 + 6*(20*A*a^3*b^2*c*d^4*g^3*i - (B*b^5*c^4*d - 5*B*a*b^4*c^3*d^2 + 
 10*B*a^2*b^3*c^2*d^3 - 5*B*a^3*b^2*c*d^4 - B*a^4*b*d^5)*g^3*i*n)*x + 6*(4 
*B*b^5*d^5*g^3*i*x^5 + 20*B*a^3*b^2*c*d^4*g^3*i*x + 5*(B*b^5*c*d^4 + 3*B*a 
*b^4*d^5)*g^3*i*x^4 + 20*(B*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g^3*i*x^3 + 10*(3 
*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*g^3*i*x^2)*log(e) + 6*(4*B*b^5*d^5*g^3*i 
*n*x^5 + 20*B*a^3*b^2*c*d^4*g^3*i*n*x + 5*(B*b^5*c*d^4 + 3*B*a*b^4*d^5)*g^ 
3*i*n*x^4 + 20*(B*a*b^4*c*d^4 + B*a^2*b^3*d^5)*g^3*i*n*x^3 + 10*(3*B*a^2*b 
^3*c*d^4 + B*a^3*b^2*d^5)*g^3*i*n*x^2)*log((b*x + a)/(d*x + c)))/(b^2*d^4)
 

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \] Input:

integrate((b*g*x+a*g)**3*(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. 1118 vs. \(2 (213) = 426\).

Time = 0.07 (sec) , antiderivative size = 1118, normalized size of antiderivative = 5.01 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, al 
gorithm="maxima")
 

Output:

1/5*B*b^3*d*g^3*i*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A*b^3*d 
*g^3*i*x^5 + 1/4*B*b^3*c*g^3*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) 
+ 3/4*B*a*b^2*d*g^3*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b 
^3*c*g^3*i*x^4 + 3/4*A*a*b^2*d*g^3*i*x^4 + B*a*b^2*c*g^3*i*x^3*log(e*(b*x/ 
(d*x + c) + a/(d*x + c))^n) + B*a^2*b*d*g^3*i*x^3*log(e*(b*x/(d*x + c) + a 
/(d*x + c))^n) + A*a*b^2*c*g^3*i*x^3 + A*a^2*b*d*g^3*i*x^3 + 3/2*B*a^2*b*c 
*g^3*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*B*a^3*d*g^3*i*x^2* 
log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A*a^2*b*c*g^3*i*x^2 + 1/2*A*a 
^3*d*g^3*i*x^2 + 1/60*B*b^3*d*g^3*i*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*lo 
g(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2 
*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4 
*d^4)) - 1/24*B*b^3*c*g^3*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)) - 1/8*B*a*b^2*d*g^3*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)) + 1/2*B*a*b^ 
2*c*g^3*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)) + 1/2*B*a^2*b*d*g^3*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)) - 3/2*B*a^2*b*c*g^3*i*n*(a^2*...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3945 vs. \(2 (213) = 426\).

Time = 0.99 (sec) , antiderivative size = 3945, normalized size of antiderivative = 17.69 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, al 
gorithm="giac")
 

Output:

-1/120*(6*(B*b^9*c^6*g^3*i*n - 6*B*a*b^8*c^5*d*g^3*i*n - 5*(b*x + a)*B*b^8 
*c^6*d*g^3*i*n/(d*x + c) + 15*B*a^2*b^7*c^4*d^2*g^3*i*n + 30*(b*x + a)*B*a 
*b^7*c^5*d^2*g^3*i*n/(d*x + c) + 10*(b*x + a)^2*B*b^7*c^6*d^2*g^3*i*n/(d*x 
 + c)^2 - 20*B*a^3*b^6*c^3*d^3*g^3*i*n - 75*(b*x + a)*B*a^2*b^6*c^4*d^3*g^ 
3*i*n/(d*x + c) - 60*(b*x + a)^2*B*a*b^6*c^5*d^3*g^3*i*n/(d*x + c)^2 - 10* 
(b*x + a)^3*B*b^6*c^6*d^3*g^3*i*n/(d*x + c)^3 + 15*B*a^4*b^5*c^2*d^4*g^3*i 
*n + 100*(b*x + a)*B*a^3*b^5*c^3*d^4*g^3*i*n/(d*x + c) + 150*(b*x + a)^2*B 
*a^2*b^5*c^4*d^4*g^3*i*n/(d*x + c)^2 + 60*(b*x + a)^3*B*a*b^5*c^5*d^4*g^3* 
i*n/(d*x + c)^3 - 6*B*a^5*b^4*c*d^5*g^3*i*n - 75*(b*x + a)*B*a^4*b^4*c^2*d 
^5*g^3*i*n/(d*x + c) - 200*(b*x + a)^2*B*a^3*b^4*c^3*d^5*g^3*i*n/(d*x + c) 
^2 - 150*(b*x + a)^3*B*a^2*b^4*c^4*d^5*g^3*i*n/(d*x + c)^3 + B*a^6*b^3*d^6 
*g^3*i*n + 30*(b*x + a)*B*a^5*b^3*c*d^6*g^3*i*n/(d*x + c) + 150*(b*x + a)^ 
2*B*a^4*b^3*c^2*d^6*g^3*i*n/(d*x + c)^2 + 200*(b*x + a)^3*B*a^3*b^3*c^3*d^ 
6*g^3*i*n/(d*x + c)^3 - 5*(b*x + a)*B*a^6*b^2*d^7*g^3*i*n/(d*x + c) - 60*( 
b*x + a)^2*B*a^5*b^2*c*d^7*g^3*i*n/(d*x + c)^2 - 150*(b*x + a)^3*B*a^4*b^2 
*c^2*d^7*g^3*i*n/(d*x + c)^3 + 10*(b*x + a)^2*B*a^6*b*d^8*g^3*i*n/(d*x + c 
)^2 + 60*(b*x + a)^3*B*a^5*b*c*d^8*g^3*i*n/(d*x + c)^3 - 10*(b*x + a)^3*B* 
a^6*d^9*g^3*i*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(b^5*d^4 - 5*(b*x + 
a)*b^4*d^5/(d*x + c) + 10*(b*x + a)^2*b^3*d^6/(d*x + c)^2 - 10*(b*x + a)^3 
*b^2*d^7/(d*x + c)^3 + 5*(b*x + a)^4*b*d^8/(d*x + c)^4 - (b*x + a)^5*d^...
 

Mupad [B] (verification not implemented)

Time = 27.08 (sec) , antiderivative size = 1237, normalized size of antiderivative = 5.55 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

int((a*g + b*g*x)^3*(c*i + d*i*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n)),x 
)
 

Output:

x*((a*c*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B 
*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(20*b*d) - (b*g^3*i*(24* 
A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d - 2*B 
*a*b*c*d*n))/(4*d) + A*a*b^2*c*g^3*i))/(b*d) - ((20*a*d + 20*b*c)*(((20*a* 
d + 20*b*c)*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n 
 - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(20*b*d) - (b*g^3*i* 
(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 32*A*a*b*c*d - 
 2*B*a*b*c*d*n))/(4*d) + A*a*b^2*c*g^3*i))/(20*b*d) - (a*c*((b^2*g^3*i*(20 
*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c) 
)/20))/(b*d) + (a*g^3*i*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2*d^2*n - B*b^2*c 
^2*n + 12*A*a*b*c*d))/d))/(20*b*d) + (a^2*g^3*i*(2*A*a^2*d^2 + 12*A*b^2*c^ 
2 + B*a^2*d^2*n - 3*B*b^2*c^2*n + 16*A*a*b*c*d + 2*B*a*b*c*d*n))/(2*b*d)) 
+ x^2*(((20*a*d + 20*b*c)*(((20*a*d + 20*b*c)*((b^2*g^3*i*(20*A*a*d + 10*A 
*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20*a*d + 20*b*c))/20))/(20*b* 
d) - (b*g^3*i*(24*A*a^2*d^2 + 4*A*b^2*c^2 + 3*B*a^2*d^2*n - B*b^2*c^2*n + 
32*A*a*b*c*d - 2*B*a*b*c*d*n))/(4*d) + A*a*b^2*c*g^3*i))/(40*b*d) - (a*c*( 
(b^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(20 
*a*d + 20*b*c))/20))/(2*b*d) + (a*g^3*i*(4*A*a^2*d^2 + 4*A*b^2*c^2 + B*a^2 
*d^2*n - B*b^2*c^2*n + 12*A*a*b*c*d))/(2*d)) - x^3*(((20*a*d + 20*b*c)*((b 
^2*g^3*i*(20*A*a*d + 10*A*b*c + B*a*d*n - B*b*c*n))/5 - (A*b^2*g^3*i*(2...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 801, normalized size of antiderivative = 3.59 \[ \int (a g+b g x)^3 (c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

int((b*g*x+a*g)^3*(d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x)
 

Output:

(g**3*i*( - 6*log(a + b*x)*a**5*d**5*n + 30*log(a + b*x)*a**4*b*c*d**4*n - 
 60*log(a + b*x)*a**3*b**2*c**2*d**3*n + 60*log(a + b*x)*a**2*b**3*c**3*d* 
*2*n - 30*log(a + b*x)*a*b**4*c**4*d*n + 6*log(a + b*x)*b**5*c**5*n + 60*l 
og(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**2*c**2*d**3 + 120*log(((a + b*x) 
**n*e)/(c + d*x)**n)*a**3*b**2*c*d**4*x + 60*log(((a + b*x)**n*e)/(c + d*x 
)**n)*a**3*b**2*d**5*x**2 - 60*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b** 
3*c**3*d**2 + 180*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3*c*d**4*x**2 
 + 120*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3*d**5*x**3 + 30*log(((a 
 + b*x)**n*e)/(c + d*x)**n)*a*b**4*c**4*d + 120*log(((a + b*x)**n*e)/(c + 
d*x)**n)*a*b**4*c*d**4*x**3 + 90*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**4 
*d**5*x**4 - 6*log(((a + b*x)**n*e)/(c + d*x)**n)*b**5*c**5 + 30*log(((a + 
 b*x)**n*e)/(c + d*x)**n)*b**5*c*d**4*x**4 + 24*log(((a + b*x)**n*e)/(c + 
d*x)**n)*b**5*d**5*x**5 + 120*a**4*b*c*d**4*x + 6*a**4*b*d**5*n*x + 60*a** 
4*b*d**5*x**2 + 30*a**3*b**2*c*d**4*n*x + 180*a**3*b**2*c*d**4*x**2 + 27*a 
**3*b**2*d**5*n*x**2 + 120*a**3*b**2*d**5*x**3 - 60*a**2*b**3*c**2*d**3*n* 
x - 15*a**2*b**3*c*d**4*n*x**2 + 120*a**2*b**3*c*d**4*x**3 + 22*a**2*b**3* 
d**5*n*x**3 + 90*a**2*b**3*d**5*x**4 + 30*a*b**4*c**3*d**2*n*x - 15*a*b**4 
*c**2*d**3*n*x**2 - 20*a*b**4*c*d**4*n*x**3 + 30*a*b**4*c*d**4*x**4 + 6*a* 
b**4*d**5*n*x**4 + 24*a*b**4*d**5*x**5 - 6*b**5*c**4*d*n*x + 3*b**5*c**3*d 
**2*n*x**2 - 2*b**5*c**2*d**3*n*x**3 - 6*b**5*c*d**4*n*x**4))/(120*b*d*...