\(\int (a g+b g x) (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n)) \, dx\) [119]

Optimal result

Integrand size = 41, antiderivative size = 250 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {B (b c-a d)^3 g i^2 n x}{12 b^2 d}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2}{24 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3}{12 d^2}-\frac {(b c-a d) g i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d^2}+\frac {b g i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 d^2}+\frac {B (b c-a d)^4 g i^2 n \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}+\frac {B (b c-a d)^4 g i^2 n \log (c+d x)}{12 b^3 d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.90 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g i^2 \left (\frac {4 B (b c-a d)^2 n \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )}{b^3}-\frac {B (b c-a d) n \left (6 b d (b c-a d)^2 x+3 b^2 (b c-a d) (c+d x)^2+2 b^3 (c+d x)^3+6 (b c-a d)^3 \log (a+b x)\right )}{b^3}-8 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 b (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )\right )}{24 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 234, normalized size of antiderivative = 0.94, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2961, 2782, 27, 86, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2782

\(\displaystyle g i^2 (b c-a d)^4 \left (-B n \int -\frac {(c+d x) \left (b-\frac {4 d (a+b x)}{c+d x}\right )}{12 d^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g i^2 (b c-a d)^4 \left (\frac {B n \int \frac {(c+d x) \left (b-\frac {4 d (a+b x)}{c+d x}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{12 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 86

\(\displaystyle g i^2 (b c-a d)^4 \left (\frac {B n \int \left (\frac {d}{b^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {c+d x}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{12 d^2}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g i^2 (b c-a d)^4 \left (-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{12 d^2}\right )\)

Input:

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

Output:

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

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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

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. \(836\) vs. \(2(236)=472\).

Time = 4.97 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.35

method	result	size
parallelrisch	\(\frac {8 B x \,a^{2} b^{2} c \,d^{3} g \,i^{2} n^{2}-4 B x a \,b^{3} c^{2} d^{2} g \,i^{2} n^{2}+24 A x a \,b^{3} c^{2} d^{2} g \,i^{2} n -8 B \ln \left (b x +a \right ) a^{3} b c \,d^{3} g \,i^{2} n^{2}-2 B x \,a^{3} b \,d^{4} g \,i^{2} n^{2}+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 \,i^{2} n +24 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} c^{2} d^{2} g \,i^{2} n -2 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{4} g \,i^{2} n +6 A \,x^{4} b^{4} d^{4} g \,i^{2} n +2 B \ln \left (b x +a \right ) b^{4} c^{4} g \,i^{2} n^{2}+2 B \ln \left (b x +a \right ) a^{4} d^{4} g \,i^{2} n^{2}+12 B \ln \left (b x +a \right ) a^{2} b^{2} c^{2} d^{2} g \,i^{2} n^{2}-8 B \ln \left (b x +a \right ) a \,b^{3} c^{3} d g \,i^{2} 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 \,i^{2} n +8 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} d^{4} g \,i^{2} n +16 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{3} g \,i^{2} n +12 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d^{2} g \,i^{2} n +4 B \,x^{2} a \,b^{3} c \,d^{3} g \,i^{2} n^{2}+24 A \,x^{2} a \,b^{3} c \,d^{3} g \,i^{2} n +2 B \,b^{4} c^{4} g \,i^{2} n^{2}+2 B \,a^{4} d^{4} g \,i^{2} n^{2}-7 B \,a^{3} b c \,d^{3} g \,i^{2} n^{2}-8 B \,a^{2} b^{2} c^{2} d^{2} g \,i^{2} n^{2}+11 B a \,b^{3} c^{3} d g \,i^{2} n^{2}-48 A \,a^{2} b^{2} c^{2} d^{2} g \,i^{2} n -36 A a \,b^{3} c^{3} d g \,i^{2} n -2 B x \,b^{4} c^{3} d g \,i^{2} n^{2}+6 B \,x^{4} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{4} g \,i^{2} n +2 B \,x^{3} a \,b^{3} d^{4} g \,i^{2} n^{2}-2 B \,x^{3} b^{4} c \,d^{3} g \,i^{2} n^{2}+8 A \,x^{3} a \,b^{3} d^{4} g \,i^{2} n +16 A \,x^{3} b^{4} c \,d^{3} g \,i^{2} n +B \,x^{2} a^{2} b^{2} d^{4} g \,i^{2} n^{2}-5 B \,x^{2} b^{4} c^{2} d^{2} g \,i^{2} n^{2}+12 A \,x^{2} b^{4} c^{2} d^{2} g \,i^{2} n}{24 b^{3} d^{2} n}\)	\(837\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (236) = 472\).

Time = 0.12 (sec) , antiderivative size = 530, normalized size of antiderivative = 2.12 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {6 \, A b^{4} d^{4} g i^{2} x^{4} + 2 \, {\left (6 \, B a^{2} b^{2} c^{2} d^{2} - 4 \, B a^{3} b c d^{3} + B a^{4} d^{4}\right )} g i^{2} n \log \left (b x + a\right ) + 2 \, {\left (B b^{4} c^{4} - 4 \, B a b^{3} c^{3} d\right )} g i^{2} n \log \left (d x + c\right ) - 2 \, {\left ({\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} g i^{2} n - 4 \, {\left (2 \, A b^{4} c d^{3} + A a b^{3} d^{4}\right )} g i^{2}\right )} x^{3} - {\left ({\left (5 \, B b^{4} c^{2} d^{2} - 4 \, B a b^{3} c d^{3} - B a^{2} b^{2} d^{4}\right )} g i^{2} n - 12 \, {\left (A b^{4} c^{2} d^{2} + 2 \, A a b^{3} c d^{3}\right )} g i^{2}\right )} x^{2} + 2 \, {\left (12 \, A a b^{3} c^{2} d^{2} g i^{2} - {\left (B b^{4} c^{3} d + 2 \, B a b^{3} c^{2} d^{2} - 4 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} g i^{2} n\right )} x + 2 \, {\left (3 \, B b^{4} d^{4} g i^{2} x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} x + 4 \, {\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} x^{2}\right )} \log \left (e\right ) + 2 \, {\left (3 \, B b^{4} d^{4} g i^{2} n x^{4} + 12 \, B a b^{3} c^{2} d^{2} g i^{2} n x + 4 \, {\left (2 \, B b^{4} c d^{3} + B a b^{3} d^{4}\right )} g i^{2} n x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} + 2 \, B a b^{3} c d^{3}\right )} g i^{2} n x^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{24 \, b^{3} d^{2}} \] Input:

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

Sympy [F(-1)]

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

integrate((b*g*x+a*g)*(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. 740 vs. \(2 (236) = 472\).

Time = 0.07 (sec) , antiderivative size = 740, normalized size of antiderivative = 2.96 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1997 vs. \(2 (236) = 472\).

Time = 0.85 (sec) , antiderivative size = 1997, normalized size of antiderivative = 7.99 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/24*(2*(B*b^6*c^5*g*i^2*n - 5*B*a*b^5*c^4*d*g*i^2*n - 4*(b*x + a)*B*b^5* 
c^5*d*g*i^2*n/(d*x + c) + 10*B*a^2*b^4*c^3*d^2*g*i^2*n + 20*(b*x + a)*B*a* 
b^4*c^4*d^2*g*i^2*n/(d*x + c) - 10*B*a^3*b^3*c^2*d^3*g*i^2*n - 40*(b*x + a 
)*B*a^2*b^3*c^3*d^3*g*i^2*n/(d*x + c) + 5*B*a^4*b^2*c*d^4*g*i^2*n + 40*(b* 
x + a)*B*a^3*b^2*c^2*d^4*g*i^2*n/(d*x + c) - B*a^5*b*d^5*g*i^2*n - 20*(b*x 
 + a)*B*a^4*b*c*d^5*g*i^2*n/(d*x + c) + 4*(b*x + a)*B*a^5*d^6*g*i^2*n/(d*x 
 + c))*log((b*x + a)/(d*x + c))/(b^4*d^2 - 4*(b*x + a)*b^3*d^3/(d*x + c) + 
 6*(b*x + a)^2*b^2*d^4/(d*x + c)^2 - 4*(b*x + a)^3*b*d^5/(d*x + c)^3 + (b* 
x + a)^4*d^6/(d*x + c)^4) - (B*b^8*c^5*g*i^2*n - 5*B*a*b^7*c^4*d*g*i^2*n - 
 6*(b*x + a)*B*b^7*c^5*d*g*i^2*n/(d*x + c) + 10*B*a^2*b^6*c^3*d^2*g*i^2*n 
+ 30*(b*x + a)*B*a*b^6*c^4*d^2*g*i^2*n/(d*x + c) + 7*(b*x + a)^2*B*b^6*c^5 
*d^2*g*i^2*n/(d*x + c)^2 - 10*B*a^3*b^5*c^2*d^3*g*i^2*n - 60*(b*x + a)*B*a 
^2*b^5*c^3*d^3*g*i^2*n/(d*x + c) - 35*(b*x + a)^2*B*a*b^5*c^4*d^3*g*i^2*n/ 
(d*x + c)^2 - 2*(b*x + a)^3*B*b^5*c^5*d^3*g*i^2*n/(d*x + c)^3 + 5*B*a^4*b^ 
4*c*d^4*g*i^2*n + 60*(b*x + a)*B*a^3*b^4*c^2*d^4*g*i^2*n/(d*x + c) + 70*(b 
*x + a)^2*B*a^2*b^4*c^3*d^4*g*i^2*n/(d*x + c)^2 + 10*(b*x + a)^3*B*a*b^4*c 
^4*d^4*g*i^2*n/(d*x + c)^3 - B*a^5*b^3*d^5*g*i^2*n - 30*(b*x + a)*B*a^4*b^ 
3*c*d^5*g*i^2*n/(d*x + c) - 70*(b*x + a)^2*B*a^3*b^3*c^2*d^5*g*i^2*n/(d*x 
+ c)^2 - 20*(b*x + a)^3*B*a^2*b^3*c^3*d^5*g*i^2*n/(d*x + c)^3 + 6*(b*x + a 
)*B*a^5*b^2*d^6*g*i^2*n/(d*x + c) + 35*(b*x + a)^2*B*a^4*b^2*c*d^6*g*i^...
 

Mupad [B] (verification not implemented)

Time = 26.09 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.64 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (B\,a\,c^2\,g\,i^2\,x+\frac {B\,c\,g\,i^2\,x^2\,\left (2\,a\,d+b\,c\right )}{2}+\frac {B\,d\,g\,i^2\,x^3\,\left (a\,d+2\,b\,c\right )}{3}+\frac {B\,b\,d^2\,g\,i^2\,x^4}{4}\right )+x^3\,\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{12}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{36}\right )+x\,\left (\frac {\left (12\,a\,d+12\,b\,c\right )\,\left (\frac {\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{12\,b\,d}-\frac {g\,i^2\,\left (3\,A\,a^2\,d^2+9\,A\,b^2\,c^2+B\,a^2\,d^2\,n-2\,B\,b^2\,c^2\,n+18\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\,n\right )}{3\,b}+A\,a\,c\,d\,g\,i^2\right )}{12\,b\,d}-\frac {a\,c\,\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )}{b\,d}+\frac {c\,g\,i^2\,\left (6\,A\,a^2\,d^2+2\,A\,b^2\,c^2+2\,B\,a^2\,d^2\,n-B\,b^2\,c^2\,n+12\,A\,a\,b\,c\,d-B\,a\,b\,c\,d\,n\right )}{2\,b\,d}\right )-x^2\,\left (\frac {\left (\frac {d\,g\,i^2\,\left (8\,A\,a\,d+12\,A\,b\,c+B\,a\,d\,n-B\,b\,c\,n\right )}{4}-\frac {A\,d\,g\,i^2\,\left (12\,a\,d+12\,b\,c\right )}{12}\right )\,\left (12\,a\,d+12\,b\,c\right )}{24\,b\,d}-\frac {g\,i^2\,\left (3\,A\,a^2\,d^2+9\,A\,b^2\,c^2+B\,a^2\,d^2\,n-2\,B\,b^2\,c^2\,n+18\,A\,a\,b\,c\,d+B\,a\,b\,c\,d\,n\right )}{6\,b}+\frac {A\,a\,c\,d\,g\,i^2}{2}\right )+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^4\,g\,i^2\,n-4\,B\,a\,c^3\,d\,g\,i^2\,n\right )}{12\,d^2}+\frac {\ln \left (a+b\,x\right )\,\left (B\,g\,n\,a^4\,d^2\,i^2-4\,B\,g\,n\,a^3\,b\,c\,d\,i^2+6\,B\,g\,n\,a^2\,b^2\,c^2\,i^2\right )}{12\,b^3}+\frac {A\,b\,d^2\,g\,i^2\,x^4}{4} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.31 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx=\frac {g \left (-24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} c^{2} d^{2} x +8 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{3} b c \,d^{3}-12 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{2} b^{2} c^{2} d^{2}-12 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c^{2} d^{2} x^{2}-2 \,\mathrm {log}\left (d x +c \right ) a^{4} d^{4} n -2 \,\mathrm {log}\left (d x +c \right ) b^{4} c^{4} n -6 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} d^{4} x^{4}-8 a^{2} b^{2} d^{4} x^{3}-6 a \,b^{3} d^{4} x^{4}-24 a^{2} b^{2} c \,d^{3} x^{2}-2 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a^{4} d^{4}+8 \,\mathrm {log}\left (d x +c \right ) a^{3} b c \,d^{3} n -12 \,\mathrm {log}\left (d x +c \right ) a^{2} b^{2} c^{2} d^{2} n +8 \,\mathrm {log}\left (d x +c \right ) a \,b^{3} c^{3} d n -8 a^{2} b^{2} c \,d^{3} n x +4 a \,b^{3} c^{2} d^{2} n x -4 a \,b^{3} c \,d^{3} n \,x^{2}-24 a^{2} b^{2} c^{2} d^{2} x -8 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} d^{4} x^{3}+2 a^{3} b \,d^{4} n x -a^{2} b^{2} d^{4} n \,x^{2}-2 a \,b^{3} d^{4} n \,x^{3}+2 b^{4} c^{3} d n x +5 b^{4} c^{2} d^{2} n \,x^{2}+2 b^{4} c \,d^{3} n \,x^{3}-16 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b^{4} c \,d^{3} x^{3}-12 a \,b^{3} c^{2} d^{2} x^{2}-16 a \,b^{3} c \,d^{3} x^{3}-24 \,\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) a \,b^{3} c \,d^{3} x^{2}\right )}{24 b^{2} d^{2}} \] Input:

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

Output:

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