\(\int \frac {(c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^4} \, dx\) [115]

Optimal result

Integrand size = 41, antiderivative size = 181 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\frac {B d i n (c+d x)^2}{4 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b B i n (c+d x)^3}{9 (b c-a d)^2 g^4 (a+b x)^3}+\frac {d i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^2 g^4 (a+b x)^2}-\frac {b i (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^2 g^4 (a+b x)^3} \] Output:

1/4*B*d*i*n*(d*x+c)^2/(-a*d+b*c)^2/g^4/(b*x+a)^2-1/9*b*B*i*n*(d*x+c)^3/(-a 
*d+b*c)^2/g^4/(b*x+a)^3+1/2*d*i*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/ 
(-a*d+b*c)^2/g^4/(b*x+a)^2-1/3*b*i*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n 
))/(-a*d+b*c)^2/g^4/(b*x+a)^3
 

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.08 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {i \left (\frac {12 A b c}{(a+b x)^3}-\frac {12 a A d}{(a+b x)^3}+\frac {4 b B c n}{(a+b x)^3}-\frac {4 a B d n}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}+\frac {3 B d n}{(a+b x)^2}-\frac {6 B d^2 n}{(b c-a d) (a+b x)}-\frac {6 B d^3 n \log (a+b x)}{(b c-a d)^2}+\frac {6 B (2 b c+a d+3 b d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac {6 B d^3 n \log (c+d x)}{(b c-a d)^2}\right )}{36 b^2 g^4} \] Input:

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

Output:

-1/36*(i*((12*A*b*c)/(a + b*x)^3 - (12*a*A*d)/(a + b*x)^3 + (4*b*B*c*n)/(a 
 + b*x)^3 - (4*a*B*d*n)/(a + b*x)^3 + (18*A*d)/(a + b*x)^2 + (3*B*d*n)/(a 
+ b*x)^2 - (6*B*d^2*n)/((b*c - a*d)*(a + b*x)) - (6*B*d^3*n*Log[a + b*x])/ 
(b*c - a*d)^2 + (6*B*(2*b*c + a*d + 3*b*d*x)*Log[e*((a + b*x)/(c + d*x))^n 
])/(a + b*x)^3 + (6*B*d^3*n*Log[c + d*x])/(b*c - a*d)^2))/(b^2*g^4)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2961, 2772, 27, 53, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 53

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(i*((B*n*((3*d*(c + d*x)^2)/(2*(a + b*x)^2) - (2*b*(c + d*x)^3)/(3*(a + b* 
x)^3)))/6 + (d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + 
 b*x)^2) - (b*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + 
b*x)^3)))/((b*c - a*d)^2*g^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 53

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, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 2009

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

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^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, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

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. \(454\) vs. \(2(173)=346\).

Time = 10.16 (sec) , antiderivative size = 455, normalized size of antiderivative = 2.51

method	result	size
parallelrisch	\(-\frac {5 B \,a^{3} b^{3} d^{4} i \,n^{2}+4 B \,b^{6} c^{3} d i \,n^{2}+6 A \,a^{3} b^{3} d^{4} i n +12 A \,b^{6} c^{3} d i n -36 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c \,d^{3} i n -18 B x a \,b^{5} c \,d^{3} i \,n^{2}-36 A x a \,b^{5} c \,d^{3} i n -18 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} c^{2} d^{2} i n -9 B a \,b^{5} c^{2} d^{2} i \,n^{2}-18 A a \,b^{5} c^{2} d^{2} i n -6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} d^{4} i n +6 B \,x^{2} a \,b^{5} d^{4} i \,n^{2}-6 B \,x^{2} b^{6} c \,d^{3} i \,n^{2}+15 B x \,a^{2} b^{4} d^{4} i \,n^{2}+3 B x \,b^{6} c^{2} d^{2} i \,n^{2}+18 A x \,a^{2} b^{4} d^{4} i n +18 A x \,b^{6} c^{2} d^{2} i n +12 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{3} d i n -18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{5} d^{4} i n +18 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{6} c^{2} d^{2} i n}{36 g^{4} \left (b x +a \right )^{3} n \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right ) b^{5} d}\)	\(455\)

Input:

int((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x,method=_RE 
TURNVERBOSE)
 

Output:

-1/36*(5*B*a^3*b^3*d^4*i*n^2+4*B*b^6*c^3*d*i*n^2+6*A*a^3*b^3*d^4*i*n+12*A* 
b^6*c^3*d*i*n-36*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*c*d^3*i*n-18*B*x*a*b^ 
5*c*d^3*i*n^2-36*A*x*a*b^5*c*d^3*i*n-18*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5* 
c^2*d^2*i*n-9*B*a*b^5*c^2*d^2*i*n^2-18*A*a*b^5*c^2*d^2*i*n-6*B*x^3*ln(e*(( 
b*x+a)/(d*x+c))^n)*b^6*d^4*i*n+6*B*x^2*a*b^5*d^4*i*n^2-6*B*x^2*b^6*c*d^3*i 
*n^2+15*B*x*a^2*b^4*d^4*i*n^2+3*B*x*b^6*c^2*d^2*i*n^2+18*A*x*a^2*b^4*d^4*i 
*n+18*A*x*b^6*c^2*d^2*i*n+12*B*ln(e*((b*x+a)/(d*x+c))^n)*b^6*c^3*d*i*n-18* 
B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^5*d^4*i*n+18*B*x*ln(e*((b*x+a)/(d*x+c) 
)^n)*b^6*c^2*d^2*i*n)/g^4/(b*x+a)^3/n/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^5/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (173) = 346\).

Time = 0.10 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.64 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\frac {6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i n x^{2} - {\left (4 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 5 \, B a^{3} d^{3}\right )} i n - 6 \, {\left (2 \, A b^{3} c^{3} - 3 \, A a b^{2} c^{2} d + A a^{3} d^{3}\right )} i - 3 \, {\left ({\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} i n + 6 \, {\left (A b^{3} c^{2} d - 2 \, A a b^{2} c d^{2} + A a^{2} b d^{3}\right )} i\right )} x - 6 \, {\left (3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} i x + {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + B a^{3} d^{3}\right )} i\right )} \log \left (e\right ) + 6 \, {\left (B b^{3} d^{3} i n x^{3} + 3 \, B a b^{2} d^{3} i n x^{2} - 3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i n x - {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, al 
gorithm="fricas")
 

Output:

1/36*(6*(B*b^3*c*d^2 - B*a*b^2*d^3)*i*n*x^2 - (4*B*b^3*c^3 - 9*B*a*b^2*c^2 
*d + 5*B*a^3*d^3)*i*n - 6*(2*A*b^3*c^3 - 3*A*a*b^2*c^2*d + A*a^3*d^3)*i - 
3*((B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*i*n + 6*(A*b^3*c^2*d - 
2*A*a*b^2*c*d^2 + A*a^2*b*d^3)*i)*x - 6*(3*(B*b^3*c^2*d - 2*B*a*b^2*c*d^2 
+ B*a^2*b*d^3)*i*x + (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d + B*a^3*d^3)*i)*log(e) 
 + 6*(B*b^3*d^3*i*n*x^3 + 3*B*a*b^2*d^3*i*n*x^2 - 3*(B*b^3*c^2*d - 2*B*a*b 
^2*c*d^2)*i*n*x - (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d)*i*n)*log((b*x + a)/(d*x 
+ c)))/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a 
^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b 
^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 945 vs. \(2 (173) = 346\).

Time = 0.09 (sec) , antiderivative size = 945, normalized size of antiderivative = 5.22 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, al 
gorithm="maxima")
 

Output:

-1/18*B*c*i*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^ 
2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*( 
a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3* 
b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g 
^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3 
*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c* 
d^2 - a^3*b*d^3)*g^4)) - 1/36*B*d*i*n*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3 
*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2 
*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 
 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a 
^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*( 
3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^ 
2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a 
*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/6*(3*b*x + a)*B*d*i* 
log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3* 
a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/6*(3*b*x + a)*A*d*i/(b^5*g^4*x^3 + 3*a*b^ 
4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*B*c*i*log(e*(b*x/(d*x + c 
) + a/(d*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3 
*b*g^4) - 1/3*A*c*i/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3 
*b*g^4)
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.88 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.29 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b i n - \frac {3 \, {\left (b x + a\right )} B d i n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}} + \frac {4 \, B b i n - \frac {9 \, {\left (b x + a\right )} B d i n}{d x + c} + 12 \, B b i \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B d i \log \left (e\right )}{d x + c} + 12 \, A b i - \frac {18 \, {\left (b x + a\right )} A d i}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, al 
gorithm="giac")
 

Output:

-1/36*(6*(2*B*b*i*n - 3*(b*x + a)*B*d*i*n/(d*x + c))*log((b*x + a)/(d*x + 
c))/((b*x + a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3) + 
(4*B*b*i*n - 9*(b*x + a)*B*d*i*n/(d*x + c) + 12*B*b*i*log(e) - 18*(b*x + a 
)*B*d*i*log(e)/(d*x + c) + 12*A*b*i - 18*(b*x + a)*A*d*i/(d*x + c))/((b*x 
+ a)^3*b*c*g^4/(d*x + c)^3 - (b*x + a)^3*a*d*g^4/(d*x + c)^3))*(b*c/(b*c - 
 a*d)^2 - a*d/(b*c - a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 26.50 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.07 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx=-\frac {\frac {6\,A\,a^2\,d^2\,i-12\,A\,b^2\,c^2\,i+5\,B\,a^2\,d^2\,i\,n-4\,B\,b^2\,c^2\,i\,n+6\,A\,a\,b\,c\,d\,i+5\,B\,a\,b\,c\,d\,i\,n}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (6\,A\,a\,b\,d^2\,i-6\,A\,b^2\,c\,d\,i-B\,b^2\,c\,d\,i\,n+5\,B\,a\,b\,d^2\,i\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,d^2\,i\,n\,x^2}{a\,d-b\,c}}{6\,a^3\,b^2\,g^4+18\,a^2\,b^3\,g^4\,x+18\,a\,b^4\,g^4\,x^2+6\,b^5\,g^4\,x^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {B\,c\,i}{3\,b}+\frac {B\,a\,d\,i}{6\,b^2}+\frac {B\,d\,i\,x}{2\,b}\right )}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B\,d^3\,i\,n\,\mathrm {atanh}\left (\frac {6\,b^4\,c^2\,g^4-6\,a^2\,b^2\,d^2\,g^4}{6\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{3\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \] Input:

int(((c*i + d*i*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n)))/(a*g + b*g*x)^4 
,x)
 

Output:

- ((6*A*a^2*d^2*i - 12*A*b^2*c^2*i + 5*B*a^2*d^2*i*n - 4*B*b^2*c^2*i*n + 6 
*A*a*b*c*d*i + 5*B*a*b*c*d*i*n)/(6*(a*d - b*c)) + (x*(6*A*a*b*d^2*i - 6*A* 
b^2*c*d*i - B*b^2*c*d*i*n + 5*B*a*b*d^2*i*n))/(2*(a*d - b*c)) + (B*b^2*d^2 
*i*n*x^2)/(a*d - b*c))/(6*a^3*b^2*g^4 + 6*b^5*g^4*x^3 + 18*a^2*b^3*g^4*x + 
 18*a*b^4*g^4*x^2) - (log(e*((a + b*x)/(c + d*x))^n)*((B*c*i)/(3*b) + (B*a 
*d*i)/(6*b^2) + (B*d*i*x)/(2*b)))/(a^3*g^4 + b^3*g^4*x^3 + 3*a*b^2*g^4*x^2 
 + 3*a^2*b*g^4*x) - (B*d^3*i*n*atanh((6*b^4*c^2*g^4 - 6*a^2*b^2*d^2*g^4)/( 
6*b^2*g^4*(a*d - b*c)^2) - (2*b*d*x)/(a*d - b*c)))/(3*b^2*g^4*(a*d - b*c)^ 
2)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 937, normalized size of antiderivative = 5.18 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x)
 

Output:

(i*(12*log(a + b*x)*a**4*b**2*c*d**2*n - 6*log(a + b*x)*a**3*b**3*c**2*d*n 
 + 36*log(a + b*x)*a**3*b**3*c*d**2*n*x - 18*log(a + b*x)*a**2*b**4*c**2*d 
*n*x + 36*log(a + b*x)*a**2*b**4*c*d**2*n*x**2 - 18*log(a + b*x)*a*b**5*c* 
*2*d*n*x**2 + 12*log(a + b*x)*a*b**5*c*d**2*n*x**3 - 6*log(a + b*x)*b**6*c 
**2*d*n*x**3 - 12*log(c + d*x)*a**4*b**2*c*d**2*n + 6*log(c + d*x)*a**3*b* 
*3*c**2*d*n - 36*log(c + d*x)*a**3*b**3*c*d**2*n*x + 18*log(c + d*x)*a**2* 
b**4*c**2*d*n*x - 36*log(c + d*x)*a**2*b**4*c*d**2*n*x**2 + 18*log(c + d*x 
)*a*b**5*c**2*d*n*x**2 - 12*log(c + d*x)*a*b**5*c*d**2*n*x**3 + 6*log(c + 
d*x)*b**6*c**2*d*n*x**3 - 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b**2* 
c*d**2 + 24*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**3*c**2*d + 18*log(( 
(a + b*x)**n*e)/(c + d*x)**n)*a**3*b**3*d**3*x**2 - 12*log(((a + b*x)**n*e 
)/(c + d*x)**n)*a**2*b**4*c**3 - 36*log(((a + b*x)**n*e)/(c + d*x)**n)*a** 
2*b**4*c*d**2*x**2 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**4*d**3*x 
**3 + 18*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**5*c**2*d*x**2 - 12*log((( 
a + b*x)**n*e)/(c + d*x)**n)*a*b**5*c*d**2*x**3 + 6*log(((a + b*x)**n*e)/( 
c + d*x)**n)*b**6*c**2*d*x**3 - 6*a**6*d**3 - 3*a**5*b*d**3*n - 18*a**5*b* 
d**3*x + 18*a**4*b**2*c**2*d - 2*a**4*b**2*c*d**2*n + 36*a**4*b**2*c*d**2* 
x - 9*a**4*b**2*d**3*n*x - 12*a**3*b**3*c**3 + 9*a**3*b**3*c**2*d*n - 18*a 
**3*b**3*c**2*d*x + 12*a**3*b**3*c*d**2*n*x - 4*a**2*b**4*c**3*n - 3*a**2* 
b**4*c**2*d*n*x + 2*a**2*b**4*d**3*n*x**3 - 2*a*b**5*c*d**2*n*x**3))/(3...