\(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [128]

Optimal result

Integrand size = 43, antiderivative size = 387 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=-\frac {B (b c-a d)^5 g^2 i^3 n x}{60 b^3 d^2}-\frac {B (b c-a d)^4 g^2 i^3 n (c+d x)^2}{120 b^2 d^3}-\frac {B (b c-a d)^3 g^2 i^3 n (c+d x)^3}{180 b d^3}+\frac {7 B (b c-a d)^2 g^2 i^3 n (c+d x)^4}{120 d^3}-\frac {b B (b c-a d) g^2 i^3 n (c+d x)^5}{30 d^3}+\frac {(b c-a d)^2 g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^3}-\frac {2 b (b c-a d) g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log \left (\frac {a+b x}{c+d x}\right )}{60 b^4 d^3}-\frac {B (b c-a d)^6 g^2 i^3 n \log (c+d x)}{60 b^4 d^3} \] Output:

-1/60*B*(-a*d+b*c)^5*g^2*i^3*n*x/b^3/d^2-1/120*B*(-a*d+b*c)^4*g^2*i^3*n*(d 
*x+c)^2/b^2/d^3-1/180*B*(-a*d+b*c)^3*g^2*i^3*n*(d*x+c)^3/b/d^3+7/120*B*(-a 
*d+b*c)^2*g^2*i^3*n*(d*x+c)^4/d^3-1/30*b*B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^5/ 
d^3+1/4*(-a*d+b*c)^2*g^2*i^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3 
-2/5*b*(-a*d+b*c)*g^2*i^3*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3+1/ 
6*b^2*g^2*i^3*(d*x+c)^6*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/60*B*(-a*d+b 
*c)^6*g^2*i^3*n*ln((b*x+a)/(d*x+c))/b^4/d^3-1/60*B*(-a*d+b*c)^6*g^2*i^3*n* 
ln(d*x+c)/b^4/d^3
 

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.14 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g^2 i^3 \left (-15 B (b c-a d)^3 n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )+12 B (b c-a d)^2 n \left (12 b d (b c-a d)^3 x+6 b^2 (b c-a d)^2 (c+d x)^2+4 b^3 (b c-a d) (c+d x)^3+3 b^4 (c+d x)^4+12 (b c-a d)^4 \log (a+b x)\right )-B (b c-a d) n \left (60 b d (b c-a d)^4 x+30 b^2 (b c-a d)^3 (c+d x)^2+20 b^3 (b c-a d)^2 (c+d x)^3+15 b^4 (b c-a d) (c+d x)^4+12 b^5 (c+d x)^5+60 (b c-a d)^5 \log (a+b x)\right )+90 b^4 (b c-a d)^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-144 b^5 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+60 b^6 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{360 b^4 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.87, 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, 1195, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2782

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1195

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (-\frac {B n \int \left (\frac {d}{b^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {14 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {10 b d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {c+d x}{b^4 (a+b x)}\right )d\frac {a+b x}{c+d x}}{60 d^3}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^2 i^3 (b c-a d)^6 \left (\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {2 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{4 d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^4}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^4}+\frac {1}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {2 b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {7}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {1}{3 b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{60 d^3}\right )\)

Input:

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

Output:

(b*c - a*d)^6*g^2*i^3*((b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^3 
*(b - (d*(a + b*x))/(c + d*x))^6) - (2*b*(A + B*Log[e*((a + b*x)/(c + d*x) 
)^n]))/(5*d^3*(b - (d*(a + b*x))/(c + d*x))^5) + (A + B*Log[e*((a + b*x)/( 
c + d*x))^n])/(4*d^3*(b - (d*(a + b*x))/(c + d*x))^4) - (B*n*((2*b)/(b - ( 
d*(a + b*x))/(c + d*x))^5 - 7/(2*(b - (d*(a + b*x))/(c + d*x))^4) + 1/(3*b 
*(b - (d*(a + b*x))/(c + d*x))^3) + 1/(2*b^2*(b - (d*(a + b*x))/(c + d*x)) 
^2) + 1/(b^3*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^4 
 - Log[b - (d*(a + b*x))/(c + d*x)]/b^4))/(60*d^3))
 

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 1195

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

rule 2009

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

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. \(1720\) vs. \(2(367)=734\).

Time = 26.81 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.45

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1721\)

Input:

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

Output:

1/360*(540*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^2*d^4*g^2*i^3*n+360*B 
*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^3*d^3*g^2*i^3*n+360*B*x*ln(e*((b*x+ 
a)/(d*x+c))^n)*a^2*b^4*c^3*d^3*g^2*i^3*n+540*B*x^4*ln(e*((b*x+a)/(d*x+c))^ 
n)*a*b^5*c*d^5*g^2*i^3*n+360*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c*d^5 
*g^2*i^3*n+720*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^2*d^4*g^2*i^3*n-90* 
B*ln(b*x+a)*a^2*b^4*c^4*d^2*g^2*i^3*n^2+360*A*x^2*a*b^5*c^3*d^3*g^2*i^3*n+ 
78*B*x^3*a^2*b^4*c*d^5*g^2*i^3*n^2-42*B*x^3*a*b^5*c^2*d^4*g^2*i^3*n^2+360* 
A*x^3*a^2*b^4*c*d^5*g^2*i^3*n+720*A*x^3*a*b^5*c^2*d^4*g^2*i^3*n+270*A*x^4* 
b^6*c^2*d^4*g^2*i^3*n+60*A*x^6*b^6*d^6*g^2*i^3*n+6*B*ln(e*((b*x+a)/(d*x+c) 
)^n)*b^6*c^6*g^2*i^3*n-6*B*ln(b*x+a)*b^6*c^6*g^2*i^3*n^2-6*B*ln(b*x+a)*a^6 
*d^6*g^2*i^3*n^2-36*B*x*a*b^5*c^4*d^2*g^2*i^3*n^2+360*A*x*a^2*b^4*c^3*d^3* 
g^2*i^3*n+90*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^4*c^4*d^2*g^2*i^3*n-36*B*ln 
(e*((b*x+a)/(d*x+c))^n)*a*b^5*c^5*d*g^2*i^3*n+36*B*ln(b*x+a)*a*b^5*c^5*d*g 
^2*i^3*n^2+36*B*ln(b*x+a)*a^5*b*c*d^5*g^2*i^3*n^2-90*B*ln(b*x+a)*a^4*b^2*c 
^2*d^4*g^2*i^3*n^2+120*B*ln(b*x+a)*a^3*b^3*c^3*d^3*g^2*i^3*n^2+18*B*x^2*a^ 
3*b^3*c*d^5*g^2*i^3*n^2+90*B*x^2*a^2*b^4*c^2*d^4*g^2*i^3*n^2-102*B*x^2*a*b 
^5*c^3*d^3*g^2*i^3*n^2+540*A*x^2*a^2*b^4*c^2*d^4*g^2*i^3*n+216*B*x^5*ln(e* 
((b*x+a)/(d*x+c))^n)*b^6*c*d^5*g^2*i^3*n+90*B*x^4*ln(e*((b*x+a)/(d*x+c))^n 
)*a^2*b^4*d^6*g^2*i^3*n+270*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^2*d^4*g^ 
2*i^3*n+18*B*x^4*a*b^5*c*d^5*g^2*i^3*n^2+540*A*x^4*a*b^5*c*d^5*g^2*i^3*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1075 vs. \(2 (367) = 734\).

Time = 0.37 (sec) , antiderivative size = 1075, normalized size of antiderivative = 2.78 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

integrate((b*g*x+a*g)**2*(d*i*x+c*i)**3*(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. \(2 (367) = 734\).

Time = 0.09 (sec) , antiderivative size = 1978, normalized size of antiderivative = 5.11 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4300 vs. \(2 (367) = 734\).

Time = 1.64 (sec) , antiderivative size = 4300, normalized size of antiderivative = 11.11 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/360*(6*(B*b^9*c^7*g^2*i^3*n - 7*B*a*b^8*c^6*d*g^2*i^3*n - 6*(b*x + a)*B* 
b^8*c^7*d*g^2*i^3*n/(d*x + c) + 21*B*a^2*b^7*c^5*d^2*g^2*i^3*n + 42*(b*x + 
 a)*B*a*b^7*c^6*d^2*g^2*i^3*n/(d*x + c) + 15*(b*x + a)^2*B*b^7*c^7*d^2*g^2 
*i^3*n/(d*x + c)^2 - 35*B*a^3*b^6*c^4*d^3*g^2*i^3*n - 126*(b*x + a)*B*a^2* 
b^6*c^5*d^3*g^2*i^3*n/(d*x + c) - 105*(b*x + a)^2*B*a*b^6*c^6*d^3*g^2*i^3* 
n/(d*x + c)^2 + 35*B*a^4*b^5*c^3*d^4*g^2*i^3*n + 210*(b*x + a)*B*a^3*b^5*c 
^4*d^4*g^2*i^3*n/(d*x + c) + 315*(b*x + a)^2*B*a^2*b^5*c^5*d^4*g^2*i^3*n/( 
d*x + c)^2 - 21*B*a^5*b^4*c^2*d^5*g^2*i^3*n - 210*(b*x + a)*B*a^4*b^4*c^3* 
d^5*g^2*i^3*n/(d*x + c) - 525*(b*x + a)^2*B*a^3*b^4*c^4*d^5*g^2*i^3*n/(d*x 
 + c)^2 + 7*B*a^6*b^3*c*d^6*g^2*i^3*n + 126*(b*x + a)*B*a^5*b^3*c^2*d^6*g^ 
2*i^3*n/(d*x + c) + 525*(b*x + a)^2*B*a^4*b^3*c^3*d^6*g^2*i^3*n/(d*x + c)^ 
2 - B*a^7*b^2*d^7*g^2*i^3*n - 42*(b*x + a)*B*a^6*b^2*c*d^7*g^2*i^3*n/(d*x 
+ c) - 315*(b*x + a)^2*B*a^5*b^2*c^2*d^7*g^2*i^3*n/(d*x + c)^2 + 6*(b*x + 
a)*B*a^7*b*d^8*g^2*i^3*n/(d*x + c) + 105*(b*x + a)^2*B*a^6*b*c*d^8*g^2*i^3 
*n/(d*x + c)^2 - 15*(b*x + a)^2*B*a^7*d^9*g^2*i^3*n/(d*x + c)^2)*log((b*x 
+ a)/(d*x + c))/(b^6*d^3 - 6*(b*x + a)*b^5*d^4/(d*x + c) + 15*(b*x + a)^2* 
b^4*d^5/(d*x + c)^2 - 20*(b*x + a)^3*b^3*d^6/(d*x + c)^3 + 15*(b*x + a)^4* 
b^2*d^7/(d*x + c)^4 - 6*(b*x + a)^5*b*d^8/(d*x + c)^5 + (b*x + a)^6*d^9/(d 
*x + c)^6) - (2*B*b^12*c^7*g^2*i^3*n - 14*B*a*b^11*c^6*d*g^2*i^3*n - 18*(b 
*x + a)*B*b^11*c^7*d*g^2*i^3*n/(d*x + c) + 42*B*a^2*b^10*c^5*d^2*g^2*i^...
 

Mupad [B] (verification not implemented)

Time = 27.72 (sec) , antiderivative size = 2547, normalized size of antiderivative = 6.58 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1097, normalized size of antiderivative = 2.83 \[ \int (a g+b g x)^2 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(g**2*i*(6*log(a + b*x)*a**6*d**6*n - 36*log(a + b*x)*a**5*b*c*d**5*n + 90 
*log(a + b*x)*a**4*b**2*c**2*d**4*n - 120*log(a + b*x)*a**3*b**3*c**3*d**3 
*n + 90*log(a + b*x)*a**2*b**4*c**4*d**2*n - 36*log(a + b*x)*a*b**5*c**5*d 
*n + 6*log(a + b*x)*b**6*c**6*n - 90*log(((a + b*x)**n*e)/(c + d*x)**n)*a* 
*2*b**4*c**4*d**2 - 360*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c**3* 
d**3*x - 540*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c**2*d**4*x**2 - 
 360*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*c*d**5*x**3 - 90*log(((a 
 + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*d**6*x**4 + 36*log(((a + b*x)**n*e)/ 
(c + d*x)**n)*a*b**5*c**5*d - 360*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b** 
5*c**3*d**3*x**2 - 720*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 - 144*lo 
g(((a + b*x)**n*e)/(c + d*x)**n)*a*b**5*d**6*x**5 - 6*log(((a + b*x)**n*e) 
/(c + d*x)**n)*b**6*c**6 - 120*log(((a + b*x)**n*e)/(c + d*x)**n)*b**6*c** 
3*d**3*x**3 - 270*log(((a + b*x)**n*e)/(c + d*x)**n)*b**6*c**2*d**4*x**4 - 
 216*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 + 36*a**4*b**2*c 
*d**5*n*x + 3*a**4*b**2*d**6*n*x**2 - 360*a**3*b**3*c**3*d**3*x - 90*a**3* 
b**3*c**2*d**4*n*x - 540*a**3*b**3*c**2*d**4*x**2 - 18*a**3*b**3*c*d**5*n* 
x**2 - 360*a**3*b**3*c*d**5*x**3 - 2*a**3*b**3*d**6*n*x**3 - 90*a**3*b**3* 
d**6*x**4 + 30*a**2*b**4*c**3*d**3*n*x - 360*a**2*b**4*c**3*d**3*x**2 -...