\(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [117]

Optimal result

Integrand size = 43, antiderivative size = 442 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^5 g^3 i^2 n x}{60 b^2 d^3}+\frac {B (b c-a d)^4 g^3 i^2 n (c+d x)^2}{120 b d^4}-\frac {19 B (b c-a d)^3 g^3 i^2 n (c+d x)^3}{180 d^4}+\frac {13 b B (b c-a d)^2 g^3 i^2 n (c+d x)^4}{120 d^4}-\frac {b^2 B (b c-a d) g^3 i^2 n (c+d x)^5}{30 d^4}-\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^4}-\frac {3 b^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^4}+\frac {B (b c-a d)^6 g^3 i^2 n \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}+\frac {B (b c-a d)^6 g^3 i^2 n \log (c+d x)}{60 b^3 d^4} \] Output:

1/60*B*(-a*d+b*c)^5*g^3*i^2*n*x/b^2/d^3+1/120*B*(-a*d+b*c)^4*g^3*i^2*n*(d* 
x+c)^2/b/d^4-19/180*B*(-a*d+b*c)^3*g^3*i^2*n*(d*x+c)^3/d^4+13/120*b*B*(-a* 
d+b*c)^2*g^3*i^2*n*(d*x+c)^4/d^4-1/30*b^2*B*(-a*d+b*c)*g^3*i^2*n*(d*x+c)^5 
/d^4-1/3*(-a*d+b*c)^3*g^3*i^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^ 
4+3/4*b*(-a*d+b*c)^2*g^3*i^2*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4 
-3/5*b^2*(-a*d+b*c)*g^3*i^2*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+ 
1/6*b^3*g^3*i^2*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+1/60*B*(-a*d 
+b*c)^6*g^3*i^2*n*ln((b*x+a)/(d*x+c))/b^3/d^4+1/60*B*(-a*d+b*c)^6*g^3*i^2* 
n*ln(d*x+c)/b^3/d^4
 

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.00 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^3 i^2 \left (90 d^4 (b c-a d)^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+144 d^5 (b c-a d) (a+b x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+60 d^6 (a+b x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-15 B (b c-a d)^3 n \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^2 n \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )-B (b c-a d) n \left (60 b d (b c-a d)^4 x+30 d^2 (-b c+a d)^3 (a+b x)^2+20 d^3 (b c-a d)^2 (a+b x)^3+15 d^4 (-b c+a d) (a+b x)^4+12 d^5 (a+b x)^5-60 (b c-a d)^5 \log (c+d x)\right )\right )}{360 b^3 d^4} \] Input:

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

Output:

(g^3*i^2*(90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d* 
x))^n]) + 144*d^5*(b*c - a*d)*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x 
))^n]) + 60*d^6*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 15*B* 
(b*c - a*d)^3*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]) + 12*B*(b*c - a*d)^2*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]) - B* 
(b*c - a*d)*n*(60*b*d*(b*c - a*d)^4*x + 30*d^2*(-(b*c) + a*d)^3*(a + b*x)^ 
2 + 20*d^3*(b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b*x)^4 + 
 12*d^5*(a + b*x)^5 - 60*(b*c - a*d)^5*Log[c + d*x])))/(360*b^3*d^4)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.88, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2782, 27, 2123, 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.

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

\(\Big \downarrow \) 2961

\(\displaystyle g^3 i^2 (b c-a d)^6 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2782

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (-B n \int -\frac {(c+d x) \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {15 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {20 d^3 (a+b x)^3}{(c+d x)^3}\right )}{60 d^4 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {B n \int \frac {(c+d x) \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {15 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {20 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{60 d^4}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\)

\(\Big \downarrow \) 2123

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {B n \int \left (-\frac {10 d b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {26 d b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {19 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3 b}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2 b^2}+\frac {c+d x}{(a+b x) b^3}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^3}\right )d\frac {a+b x}{c+d x}}{60 d^4}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{60 d^4}\right )\)

Input:

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

Output:

(b*c - a*d)^6*g^3*i^2*((b^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^4 
*(b - (d*(a + b*x))/(c + d*x))^6) - (3*b^2*(A + B*Log[e*((a + b*x)/(c + d* 
x))^n]))/(5*d^4*(b - (d*(a + b*x))/(c + d*x))^5) + (3*b*(A + B*Log[e*((a + 
 b*x)/(c + d*x))^n]))/(4*d^4*(b - (d*(a + b*x))/(c + d*x))^4) - (A + B*Log 
[e*((a + b*x)/(c + d*x))^n])/(3*d^4*(b - (d*(a + b*x))/(c + d*x))^3) + (B* 
n*((-2*b^2)/(b - (d*(a + b*x))/(c + d*x))^5 + (13*b)/(2*(b - (d*(a + b*x)) 
/(c + d*x))^4) - 19/(3*(b - (d*(a + b*x))/(c + d*x))^3) + 1/(2*b*(b - (d*( 
a + b*x))/(c + d*x))^2) + 1/(b^2*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + 
 b*x)/(c + d*x)]/b^3 - Log[b - (d*(a + b*x))/(c + d*x)]/b^3))/(60*d^4))
 

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 2009

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

rule 2123

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

rule 2782

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q 
_), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* 
x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
 

rule 2961

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1760\) vs. \(2(420)=840\).

Time = 26.82 (sec) , antiderivative size = 1761, normalized size of antiderivative = 3.98

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1761\)

Input:

int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n)),x,method=_ 
RETURNVERBOSE)
 

Output:

1/360*(540*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^5*g^3*i^2*n+720*B*x^3 
*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c*d^5*g^3*i^2*n+360*B*x^3*ln(e*((b*x+a) 
/(d*x+c))^n)*a*b^5*c^2*d^4*g^3*i^2*n+360*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a 
^3*b^3*c*d^5*g^3*i^2*n+540*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^2*d^4 
*g^3*i^2*n+360*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^3*c^2*d^4*g^3*i^2*n+60* 
B*x^6*ln(e*((b*x+a)/(d*x+c))^n)*b^6*d^6*g^3*i^2*n+216*B*x^5*ln(e*((b*x+a)/ 
(d*x+c))^n)*a*b^5*d^6*g^3*i^2*n+144*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c* 
d^5*g^3*i^2*n+270*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*d^6*g^3*i^2*n+36 
*B*x*a^4*b^2*c*d^5*g^3*i^2*n^2+30*B*x*a^3*b^3*c^2*d^4*g^3*i^2*n^2-90*B*x*a 
^2*b^4*c^3*d^3*g^3*i^2*n^2+36*B*x*a*b^5*c^4*d^2*g^3*i^2*n^2+360*A*x*a^3*b^ 
3*c^2*d^4*g^3*i^2*n+120*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^3*c^3*d^3*g^3*i^ 
2*n-90*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^4*d^2*g^3*i^2*n+36*B*ln(e*((b 
*x+a)/(d*x+c))^n)*a*b^5*c^5*d*g^3*i^2*n-36*B*ln(b*x+a)*a^5*b*c*d^5*g^3*i^2 
*n^2+90*B*ln(b*x+a)*a^4*b^2*c^2*d^4*g^3*i^2*n^2-33*B*a^5*b*c*d^5*g^3*i^2*n 
^2-168*B*a^4*b^2*c^2*d^4*g^3*i^2*n^2+150*B*a^3*b^3*c^3*d^3*g^3*i^2*n^2+72* 
B*a^2*b^4*c^4*d^2*g^3*i^2*n^2-33*B*a*b^5*c^5*d*g^3*i^2*n^2-720*A*a^4*b^2*c 
^2*d^4*g^3*i^2*n-900*A*a^3*b^3*c^3*d^3*g^3*i^2*n-36*B*ln(b*x+a)*a*b^5*c^5* 
d*g^3*i^2*n^2+360*A*x^2*a^3*b^3*c*d^5*g^3*i^2*n+540*A*x^2*a^2*b^4*c^2*d^4* 
g^3*i^2*n-120*B*ln(b*x+a)*a^3*b^3*c^3*d^3*g^3*i^2*n^2+90*B*ln(b*x+a)*a^2*b 
^4*c^4*d^2*g^3*i^2*n^2+90*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^4*g...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1074 vs. \(2 (420) = 840\).

Time = 0.29 (sec) , antiderivative size = 1074, normalized size of antiderivative = 2.43 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, 
algorithm="fricas")
 

Output:

1/360*(60*A*b^6*d^6*g^3*i^2*x^6 + 6*(15*B*a^4*b^2*c^2*d^4 - 6*B*a^5*b*c*d^ 
5 + B*a^6*d^6)*g^3*i^2*n*log(b*x + a) + 6*(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 1 
5*B*a^2*b^4*c^4*d^2 - 20*B*a^3*b^3*c^3*d^3)*g^3*i^2*n*log(d*x + c) - 12*(( 
B*b^6*c*d^5 - B*a*b^5*d^6)*g^3*i^2*n - 6*(2*A*b^6*c*d^5 + 3*A*a*b^5*d^6)*g 
^3*i^2)*x^5 - 3*((7*B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 - 13*B*a^2*b^4*d^6)*g^ 
3*i^2*n - 30*(A*b^6*c^2*d^4 + 6*A*a*b^5*c*d^5 + 3*A*a^2*b^4*d^6)*g^3*i^2)* 
x^4 - 2*((B*b^6*c^3*d^3 + 39*B*a*b^5*c^2*d^4 - 21*B*a^2*b^4*c*d^5 - 19*B*a 
^3*b^3*d^6)*g^3*i^2*n - 60*(3*A*a*b^5*c^2*d^4 + 6*A*a^2*b^4*c*d^5 + A*a^3* 
b^3*d^6)*g^3*i^2)*x^3 + 3*((B*b^6*c^4*d^2 - 6*B*a*b^5*c^3*d^3 - 30*B*a^2*b 
^4*c^2*d^4 + 34*B*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^3*i^2*n + 60*(3*A*a^2*b 
^4*c^2*d^4 + 2*A*a^3*b^3*c*d^5)*g^3*i^2)*x^2 + 6*(60*A*a^3*b^3*c^2*d^4*g^3 
*i^2 - (B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 15*B*a^2*b^4*c^3*d^3 - 5*B*a^3*b 
^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^3*i^2*n)*x + 6*(10*B*b^6*d 
^6*g^3*i^2*x^6 + 60*B*a^3*b^3*c^2*d^4*g^3*i^2*x + 12*(2*B*b^6*c*d^5 + 3*B* 
a*b^5*d^6)*g^3*i^2*x^5 + 15*(B*b^6*c^2*d^4 + 6*B*a*b^5*c*d^5 + 3*B*a^2*b^4 
*d^6)*g^3*i^2*x^4 + 20*(3*B*a*b^5*c^2*d^4 + 6*B*a^2*b^4*c*d^5 + B*a^3*b^3* 
d^6)*g^3*i^2*x^3 + 30*(3*B*a^2*b^4*c^2*d^4 + 2*B*a^3*b^3*c*d^5)*g^3*i^2*x^ 
2)*log(e) + 6*(10*B*b^6*d^6*g^3*i^2*n*x^6 + 60*B*a^3*b^3*c^2*d^4*g^3*i^2*n 
*x + 12*(2*B*b^6*c*d^5 + 3*B*a*b^5*d^6)*g^3*i^2*n*x^5 + 15*(B*b^6*c^2*d^4 
+ 6*B*a*b^5*c*d^5 + 3*B*a^2*b^4*d^6)*g^3*i^2*n*x^4 + 20*(3*B*a*b^5*c^2*...
 

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Timed out} \] Input:

integrate((b*g*x+a*g)**3*(d*i*x+c*i)**2*(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. 1978 vs. \(2 (420) = 840\).

Time = 0.10 (sec) , antiderivative size = 1978, normalized size of antiderivative = 4.48 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, 
algorithm="maxima")
 

Output:

1/6*B*b^3*d^2*g^3*i^2*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A*b 
^3*d^2*g^3*i^2*x^6 + 2/5*B*b^3*c*d*g^3*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d 
*x + c))^n) + 3/5*B*a*b^2*d^2*g^3*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x + 
c))^n) + 2/5*A*b^3*c*d*g^3*i^2*x^5 + 3/5*A*a*b^2*d^2*g^3*i^2*x^5 + 1/4*B*b 
^3*c^2*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*B*a*b^2*c* 
d*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*B*a^2*b*d^2*g^3 
*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A*b^3*c^2*g^3*i^2*x^ 
4 + 3/2*A*a*b^2*c*d*g^3*i^2*x^4 + 3/4*A*a^2*b*d^2*g^3*i^2*x^4 + B*a*b^2*c^ 
2*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*B*a^2*b*c*d*g^3*i 
^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*B*a^3*d^2*g^3*i^2*x^3* 
log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*a*b^2*c^2*g^3*i^2*x^3 + 2*A*a^2 
*b*c*d*g^3*i^2*x^3 + 1/3*A*a^3*d^2*g^3*i^2*x^3 + 3/2*B*a^2*b*c^2*g^3*i^2*x 
^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + B*a^3*c*d*g^3*i^2*x^2*log(e*(b 
*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A*a^2*b*c^2*g^3*i^2*x^2 + A*a^3*c*d*g 
^3*i^2*x^2 - 1/360*B*b^3*d^2*g^3*i^2*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*l 
og(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2* 
b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b* 
d^5)*x^2 + 60*(b^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/30*B*b^3*c*d*g^3*i^2*n 
*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(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...
                                                                                    
                                                                                    
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5133 vs. \(2 (420) = 840\).

Time = 1.63 (sec) , antiderivative size = 5133, normalized size of antiderivative = 11.61 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x, 
algorithm="giac")
 

Output:

-1/360*(6*(B*b^10*c^7*g^3*i^2*n - 7*B*a*b^9*c^6*d*g^3*i^2*n - 6*(b*x + a)* 
B*b^9*c^7*d*g^3*i^2*n/(d*x + c) + 21*B*a^2*b^8*c^5*d^2*g^3*i^2*n + 42*(b*x 
 + a)*B*a*b^8*c^6*d^2*g^3*i^2*n/(d*x + c) + 15*(b*x + a)^2*B*b^8*c^7*d^2*g 
^3*i^2*n/(d*x + c)^2 - 35*B*a^3*b^7*c^4*d^3*g^3*i^2*n - 126*(b*x + a)*B*a^ 
2*b^7*c^5*d^3*g^3*i^2*n/(d*x + c) - 105*(b*x + a)^2*B*a*b^7*c^6*d^3*g^3*i^ 
2*n/(d*x + c)^2 - 20*(b*x + a)^3*B*b^7*c^7*d^3*g^3*i^2*n/(d*x + c)^3 + 35* 
B*a^4*b^6*c^3*d^4*g^3*i^2*n + 210*(b*x + a)*B*a^3*b^6*c^4*d^4*g^3*i^2*n/(d 
*x + c) + 315*(b*x + a)^2*B*a^2*b^6*c^5*d^4*g^3*i^2*n/(d*x + c)^2 + 140*(b 
*x + a)^3*B*a*b^6*c^6*d^4*g^3*i^2*n/(d*x + c)^3 - 21*B*a^5*b^5*c^2*d^5*g^3 
*i^2*n - 210*(b*x + a)*B*a^4*b^5*c^3*d^5*g^3*i^2*n/(d*x + c) - 525*(b*x + 
a)^2*B*a^3*b^5*c^4*d^5*g^3*i^2*n/(d*x + c)^2 - 420*(b*x + a)^3*B*a^2*b^5*c 
^5*d^5*g^3*i^2*n/(d*x + c)^3 + 7*B*a^6*b^4*c*d^6*g^3*i^2*n + 126*(b*x + a) 
*B*a^5*b^4*c^2*d^6*g^3*i^2*n/(d*x + c) + 525*(b*x + a)^2*B*a^4*b^4*c^3*d^6 
*g^3*i^2*n/(d*x + c)^2 + 700*(b*x + a)^3*B*a^3*b^4*c^4*d^6*g^3*i^2*n/(d*x 
+ c)^3 - B*a^7*b^3*d^7*g^3*i^2*n - 42*(b*x + a)*B*a^6*b^3*c*d^7*g^3*i^2*n/ 
(d*x + c) - 315*(b*x + a)^2*B*a^5*b^3*c^2*d^7*g^3*i^2*n/(d*x + c)^2 - 700* 
(b*x + a)^3*B*a^4*b^3*c^3*d^7*g^3*i^2*n/(d*x + c)^3 + 6*(b*x + a)*B*a^7*b^ 
2*d^8*g^3*i^2*n/(d*x + c) + 105*(b*x + a)^2*B*a^6*b^2*c*d^8*g^3*i^2*n/(d*x 
 + c)^2 + 420*(b*x + a)^3*B*a^5*b^2*c^2*d^8*g^3*i^2*n/(d*x + c)^3 - 15*(b* 
x + a)^2*B*a^7*b*d^9*g^3*i^2*n/(d*x + c)^2 - 140*(b*x + a)^3*B*a^6*b*c*...
 

Mupad [B] (verification not implemented)

Time = 27.56 (sec) , antiderivative size = 2555, normalized size of antiderivative = 5.78 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

int((a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)) 
,x)
 

Output:

x^2*((a*c*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d*n - B*b*c*n))/6 - 
 (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b* 
g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2*n - 2*B*b^2*c^2*n + 60* 
A*a*b*c*d - B*a*b*c*d*n))/5 + A*a*b^2*c*d*g^3*i^2))/(2*b*d) - ((60*a*d + 6 
0*b*c)*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^3*n 
 + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d*n + 3*B*a^2*b*c*d 
^2*n))/(4*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + 
 B*a*d*n - B*b*c*n))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 
 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 
*n - 2*B*b^2*c^2*n + 60*A*a*b*c*d - B*a*b*c*d*n))/5 + A*a*b^2*c*d*g^3*i^2) 
)/(60*b*d) - (a*c*((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d*n - B*b*c*n 
))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60))/(b*d)))/(120*b*d) + (a*g^3 
*i^2*(3*A*a^3*d^3 + 12*A*b^3*c^3 + B*a^3*d^3*n - 3*B*b^3*c^3*n + 54*A*a*b^ 
2*c^2*d + 36*A*a^2*b*c*d^2 - 3*B*a*b^2*c^2*d*n + 5*B*a^2*b*c*d^2*n))/(6*b* 
d)) + x^3*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3*n - B*b^3*c^ 
3*n + 48*A*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d*n + 3*B*a^2*b* 
c*d^2*n))/(12*d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b 
*c + B*a*d*n - B*b*c*n))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a 
*d + 60*b*c))/(60*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2 
*d^2*n - 2*B*b^2*c^2*n + 60*A*a*b*c*d - B*a*b*c*d*n))/5 + A*a*b^2*c*d*g...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1093, normalized size of antiderivative = 2.47 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n)),x)
 

Output:

(g**3*( - 6*log(c + d*x)*a**6*d**6*n + 36*log(c + d*x)*a**5*b*c*d**5*n - 9 
0*log(c + d*x)*a**4*b**2*c**2*d**4*n + 120*log(c + d*x)*a**3*b**3*c**3*d** 
3*n - 90*log(c + d*x)*a**2*b**4*c**4*d**2*n + 36*log(c + d*x)*a*b**5*c**5* 
d*n - 6*log(c + d*x)*b**6*c**6*n - 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a* 
*6*d**6 + 36*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*b*c*d**5 - 90*log(((a 
 + b*x)**n*e)/(c + d*x)**n)*a**4*b**2*c**2*d**4 - 360*log(((a + b*x)**n*e) 
/(c + d*x)**n)*a**3*b**3*c**2*d**4*x - 360*log(((a + b*x)**n*e)/(c + d*x)* 
*n)*a**3*b**3*c*d**5*x**2 - 120*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b* 
*3*d**6*x**3 - 540*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c**2*d**4* 
x**2 - 720*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c*d**5*x**3 - 270* 
log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*d**6*x**4 - 360*log(((a + b*x 
)**n*e)/(c + d*x)**n)*a*b**5*c**2*d**4*x**3 - 540*log(((a + b*x)**n*e)/(c 
+ d*x)**n)*a*b**5*c*d**5*x**4 - 216*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b 
**5*d**6*x**5 - 90*log(((a + b*x)**n*e)/(c + d*x)**n)*b**6*c**2*d**4*x**4 
- 144*log(((a + b*x)**n*e)/(c + d*x)**n)*b**6*c*d**5*x**5 - 60*log(((a + b 
*x)**n*e)/(c + d*x)**n)*b**6*d**6*x**6 + 6*a**5*b*d**6*n*x - 360*a**4*b**2 
*c**2*d**4*x - 36*a**4*b**2*c*d**5*n*x - 360*a**4*b**2*c*d**5*x**2 - 3*a** 
4*b**2*d**6*n*x**2 - 120*a**4*b**2*d**6*x**3 - 30*a**3*b**3*c**2*d**4*n*x 
- 540*a**3*b**3*c**2*d**4*x**2 - 102*a**3*b**3*c*d**5*n*x**2 - 720*a**3*b* 
*3*c*d**5*x**3 - 38*a**3*b**3*d**6*n*x**3 - 270*a**3*b**3*d**6*x**4 + 9...