\(\int \frac {A+B \log (e (\frac {a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)} \, dx\) [142]

Optimal result

Integrand size = 43, antiderivative size = 389 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {3 b B d^2 n (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d n (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 B n (c+d x)^3}{9 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^4 i}+\frac {B d^3 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^4 i} \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.71 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.33 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\frac {-\frac {12 A (b c-a d)^3}{(a+b x)^3}-\frac {4 B (b c-a d)^3 n}{(a+b x)^3}+\frac {18 A d (b c-a d)^2}{(a+b x)^2}+\frac {15 B d (b c-a d)^2 n}{(a+b x)^2}+\frac {36 A d^2 (-b c+a d)}{a+b x}+\frac {66 B d^2 (-b c+a d) n}{a+b x}-36 A d^3 \log (a+b x)-66 B d^3 n \log (a+b x)+18 B d^3 n \log ^2(a+b x)-\frac {12 B (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac {18 B d (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2}+\frac {36 B d^2 (-b c+a d) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x}-36 B d^3 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 A d^3 \log (c+d x)+66 B d^3 n \log (c+d x)-36 B d^3 n \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+36 B d^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)+18 B d^3 n \log ^2(c+d x)-36 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-36 B d^3 n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-36 B d^3 n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^4 g^4 i} \] Input:

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

Output:

((-12*A*(b*c - a*d)^3)/(a + b*x)^3 - (4*B*(b*c - a*d)^3*n)/(a + b*x)^3 + ( 
18*A*d*(b*c - a*d)^2)/(a + b*x)^2 + (15*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + 
 (36*A*d^2*(-(b*c) + a*d))/(a + b*x) + (66*B*d^2*(-(b*c) + a*d)*n)/(a + b* 
x) - 36*A*d^3*Log[a + b*x] - 66*B*d^3*n*Log[a + b*x] + 18*B*d^3*n*Log[a + 
b*x]^2 - (12*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^3 + 
 (18*B*d*(b*c - a*d)^2*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^2 + (36*B 
*d^2*(-(b*c) + a*d)*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x) - 36*B*d^3*L 
og[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n] + 36*A*d^3*Log[c + d*x] + 66*B* 
d^3*n*Log[c + d*x] - 36*B*d^3*n*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + 
d*x] + 36*B*d^3*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x] + 18*B*d^3*n*L 
og[c + d*x]^2 - 36*B*d^3*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 3 
6*B*d^3*n*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 36*B*d^3*n*PolyLog[2, 
 (b*(c + d*x))/(b*c - a*d)])/(36*(b*c - a*d)^4*g^4*i)
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.71, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2772, 27, 2010, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

\(\displaystyle \frac {-B n \int -\frac {(c+d x)^4 \left (2 b^3-\frac {9 d (a+b x) b^2}{c+d x}+\frac {18 d^2 (a+b x)^2 b}{(c+d x)^2}+\frac {6 d^3 (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^3}\right )}{6 (a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\left (d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}}{g^4 i (b c-a d)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{6} B n \int \frac {(c+d x)^4 \left (2 b^3-\frac {9 d (a+b x) b^2}{c+d x}+\frac {18 d^2 (a+b x)^2 b}{(c+d x)^2}+\frac {6 d^3 (a+b x)^3 \log \left (\frac {a+b x}{c+d x}\right )}{(c+d x)^3}\right )}{(a+b x)^4}d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\left (d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}}{g^4 i (b c-a d)^4}\)

\(\Big \downarrow \) 2010

\(\displaystyle \frac {\frac {1}{6} B n \int \left (\frac {b \left (2 b^2-\frac {9 d (a+b x) b}{c+d x}+\frac {18 d^2 (a+b x)^2}{(c+d x)^2}\right ) (c+d x)^4}{(a+b x)^4}+\frac {6 d^3 \log \left (\frac {a+b x}{c+d x}\right ) (c+d x)}{a+b x}\right )d\frac {a+b x}{c+d x}-\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\left (d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}}{g^4 i (b c-a d)^4}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^2}-\left (d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )\right )-\frac {3 b d^2 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {1}{6} B n \left (-\frac {2 b^3 (c+d x)^3}{3 (a+b x)^3}+\frac {9 b^2 d (c+d x)^2}{2 (a+b x)^2}+3 d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {18 b d^2 (c+d x)}{a+b x}\right )}{g^4 i (b c-a d)^4}\)

Input:

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

Output:

((-3*b*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + ( 
3*b^2*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^2 
) - (b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^ 
3) - d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)] + 
 (B*n*((-18*b*d^2*(c + d*x))/(a + b*x) + (9*b^2*d*(c + d*x)^2)/(2*(a + b*x 
)^2) - (2*b^3*(c + d*x)^3)/(3*(a + b*x)^3) + 3*d^3*Log[(a + b*x)/(c + d*x) 
]^2))/6)/((b*c - a*d)^4*g^4*i)
 

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 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

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. \(1089\) vs. \(2(379)=758\).

Time = 27.76 (sec) , antiderivative size = 1090, normalized size of antiderivative = 2.80

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1090\)

Input:

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

Output:

-1/36*(18*B*ln(e*((b*x+a)/(d*x+c))^n)^2*a^8*c^2*d^3+36*A*ln(e*((b*x+a)/(d* 
x+c))^n)*a^8*c^2*d^3+108*A*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^2*d^3+1 
62*A*x^2*a^6*b^2*c^2*d^3*n-288*A*x^2*a^5*b^3*c^3*d^2*n+162*A*x^2*a^4*b^4*c 
^4*d*n+54*B*x*ln(e*((b*x+a)/(d*x+c))^n)^2*a^7*b*c^2*d^3+108*B*x*a^7*b*c^2* 
d^3*n^2-162*B*x*a^6*b^2*c^3*d^2*n^2+66*B*x*a^5*b^3*c^4*d*n^2+108*A*x*ln(e* 
((b*x+a)/(d*x+c))^n)*a^7*b*c^2*d^3+108*A*x*a^7*b*c^2*d^3*n-216*A*x*a^6*b^2 
*c^3*d^2*n+144*A*x*a^5*b^3*c^4*d*n+108*B*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b*c 
^3*d^2*n-54*B*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^4*d*n+18*B*x^3*ln(e*((b* 
x+a)/(d*x+c))^n)^2*a^5*b^3*c^2*d^3+85*B*x^3*a^5*b^3*c^2*d^3*n^2-108*B*x^3* 
a^4*b^4*c^3*d^2*n^2+27*B*x^3*a^3*b^5*c^4*d*n^2+36*A*x^3*ln(e*((b*x+a)/(d*x 
+c))^n)*a^5*b^3*c^2*d^3+66*A*x^3*a^5*b^3*c^2*d^3*n-108*A*x^3*a^4*b^4*c^3*d 
^2*n+54*A*x^3*a^3*b^5*c^4*d*n+54*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b^2 
*c^2*d^3+189*B*x^2*a^6*b^2*c^2*d^3*n^2-258*B*x^2*a^5*b^3*c^3*d^2*n^2+81*B* 
x^2*a^4*b^4*c^4*d*n^2+66*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^3*c^2*d^3*n 
+162*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^2*d^3*n+36*B*x^2*ln(e*((b*x 
+a)/(d*x+c))^n)*a^5*b^3*c^3*d^2*n+108*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^7*b* 
c^2*d^3*n+108*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b^2*c^3*d^2*n-18*B*x*ln(e* 
((b*x+a)/(d*x+c))^n)*a^5*b^3*c^4*d*n-4*B*x^3*a^2*b^6*c^5*n^2-12*A*x^3*a^2* 
b^6*c^5*n-12*B*x^2*a^3*b^5*c^5*n^2-36*A*x^2*a^3*b^5*c^5*n-12*B*x*a^4*b^4*c 
^5*n^2-36*A*x*a^4*b^4*c^5*n+12*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*b^3*c^5*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (379) = 758\).

Time = 0.10 (sec) , antiderivative size = 859, normalized size of antiderivative = 2.21 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/36*(12*A*b^3*c^3 - 54*A*a*b^2*c^2*d + 108*A*a^2*b*c*d^2 - 66*A*a^3*d^3 
+ 6*(6*A*b^3*c*d^2 - 6*A*a*b^2*d^3 + 11*(B*b^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 
 + 18*(B*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2*b*d^3*n*x + B*a^3*d 
^3*n)*log((b*x + a)/(d*x + c))^2 + (4*B*b^3*c^3 - 27*B*a*b^2*c^2*d + 108*B 
*a^2*b*c*d^2 - 85*B*a^3*d^3)*n - 3*(6*A*b^3*c^2*d - 36*A*a*b^2*c*d^2 + 30* 
A*a^2*b*d^3 + (5*B*b^3*c^2*d - 54*B*a*b^2*c*d^2 + 49*B*a^2*b*d^3)*n)*x + 6 
*(2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 - 11*B*a^3*d^3 + 6*(B*b 
^3*c*d^2 - B*a*b^2*d^3)*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b 
*d^3)*x + 6*(B*b^3*d^3*x^3 + 3*B*a*b^2*d^3*x^2 + 3*B*a^2*b*d^3*x + B*a^3*d 
^3)*log((b*x + a)/(d*x + c)))*log(e) + 6*(6*A*a^3*d^3 + (11*B*b^3*d^3*n + 
6*A*b^3*d^3)*x^3 + 3*(6*A*a*b^2*d^3 + (2*B*b^3*c*d^2 + 9*B*a*b^2*d^3)*n)*x 
^2 + (2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2)*n + 3*(6*A*a^2*b*d 
^3 - (B*b^3*c^2*d - 6*B*a*b^2*c*d^2 - 6*B*a^2*b*d^3)*n)*x)*log((b*x + a)/( 
d*x + c)))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 
 + a^4*b^3*d^4)*g^4*i*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2 
*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*i*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b 
^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*i*x + (a^3 
*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)* 
g^4*i)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1472 vs. \(2 (379) = 758\).

Time = 0.13 (sec) , antiderivative size = 1472, normalized size of antiderivative = 3.78 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*B*((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^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)* 
g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3 
)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d 
^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^ 
4*i) + 6*d^3*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 
4*a^3*b*c*d^3 + a^4*d^4)*g^4*i) - 6*d^3*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c 
^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i))*log(e*(b*x/(d* 
x + c) + a/(d*x + c))^n) - 1/36*(4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c* 
d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a* 
b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 
3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d 
 - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 
3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x 
^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^ 
2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))*B*n/(a^3*b^4*c^4*g^4*i - 
4*a^4*b^3*c^3*d*g^4*i + 6*a^5*b^2*c^2*d^2*g^4*i - 4*a^6*b*c*d^3*g^4*i + a^ 
7*d^4*g^4*i + (b^7*c^4*g^4*i - 4*a*b^6*c^3*d*g^4*i + 6*a^2*b^5*c^2*d^2*g^4 
*i - 4*a^3*b^4*c*d^3*g^4*i + a^4*b^3*d^4*g^4*i)*x^3 + 3*(a*b^6*c^4*g^4*i - 
 4*a^2*b^5*c^3*d*g^4*i + 6*a^3*b^4*c^2*d^2*g^4*i - 4*a^4*b^3*c*d^3*g^4*...
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 138.57 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.60 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b n - \frac {3 \, {\left (b x + a\right )} B d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}} + \frac {4 \, B b n - \frac {9 \, {\left (b x + a\right )} B d n}{d x + c} + 12 \, B b \log \left (e\right ) - \frac {18 \, {\left (b x + a\right )} B d \log \left (e\right )}{d x + c} + 12 \, A b - \frac {18 \, {\left (b x + a\right )} A d}{d x + c}}{\frac {{\left (b x + a\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x + a\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}^{2} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 28.44 (sec) , antiderivative size = 986, normalized size of antiderivative = 2.53 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

((66*A*a^2*d^2 + 12*A*b^2*c^2 + 85*B*a^2*d^2*n + 4*B*b^2*c^2*n - 42*A*a*b* 
c*d - 23*B*a*b*c*d*n)/(6*(a*d - b*c)) + (x*(30*A*a*b*d^2 - 6*A*b^2*c*d + 4 
9*B*a*b*d^2*n - 5*B*b^2*c*d*n))/(2*(a*d - b*c)) + (d*x^2*(6*A*b^2*d + 11*B 
*b^2*d*n))/(a*d - b*c))/(x*(18*a^4*b*d^2*g^4*i + 18*a^2*b^3*c^2*g^4*i - 36 
*a^3*b^2*c*d*g^4*i) + x^2*(18*a*b^4*c^2*g^4*i + 18*a^3*b^2*d^2*g^4*i - 36* 
a^2*b^3*c*d*g^4*i) + x^3*(6*b^5*c^2*g^4*i + 6*a^2*b^3*d^2*g^4*i - 12*a*b^4 
*c*d*g^4*i) + 6*a^5*d^2*g^4*i + 6*a^3*b^2*c^2*g^4*i - 12*a^4*b*c*d*g^4*i) 
+ (d^3*atan((d^3*((a^4*d^4*g^4*i - b^4*c^4*g^4*i + 2*a*b^3*c^3*d*g^4*i - 2 
*a^3*b*c*d^3*g^4*i)/(a^3*d^3*g^4*i - b^3*c^3*g^4*i + 3*a*b^2*c^2*d*g^4*i - 
 3*a^2*b*c*d^2*g^4*i) + 2*b*d*x)*(A + (11*B*n)/6)*(a^3*d^3*g^4*i - b^3*c^3 
*g^4*i + 3*a*b^2*c^2*d*g^4*i - 3*a^2*b*c*d^2*g^4*i)*6i)/(g^4*i*(6*A*d^3 + 
11*B*d^3*n)*(a*d - b*c)^4))*(A + (11*B*n)/6)*2i)/(g^4*i*(a*d - b*c)^4) - ( 
B*d^3*log(e*((a + b*x)/(c + d*x))^n)^2)/(2*g^4*i*n*(a*d - b*c)*(a^3*d^3 - 
b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B*d^3*log(e*((a + b*x)/(c + d 
*x))^n)*(x*(b*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*d^2) + (a*g^4*i*n*(a 
*d - b*c))/(3*d)) + (2*a*b*g^4*i*n*(a*d - b*c))/(3*d) + (b*g^4*i*n*(a*d - 
b*c)*(3*a*d - b*c))/(3*d^2)) + a*((g^4*i*n*(a*d - b*c)*(3*a*d - b*c))/(6*d 
^2) + (a*g^4*i*n*(a*d - b*c))/(3*d)) + (g^4*i*n*(a*d - b*c)*(3*a^2*d^2 + b 
^2*c^2 - 3*a*b*c*d))/(3*d^3) + (b^2*g^4*i*n*x^2*(a*d - b*c))/d))/(g^4*i*n* 
(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^3*g^...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1414, normalized size of antiderivative = 3.63 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a g+b g x)^4 (c i+d i x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(36*log(a + b*x)*a**5*d**3*n + 54*log(a + b*x)*a**4*b*d**3*n**2 + 108*l 
og(a + b*x)*a**4*b*d**3*n*x + 12*log(a + b*x)*a**3*b**2*c*d**2*n**2 + 162* 
log(a + b*x)*a**3*b**2*d**3*n**2*x + 108*log(a + b*x)*a**3*b**2*d**3*n*x** 
2 + 36*log(a + b*x)*a**2*b**3*c*d**2*n**2*x + 162*log(a + b*x)*a**2*b**3*d 
**3*n**2*x**2 + 36*log(a + b*x)*a**2*b**3*d**3*n*x**3 + 36*log(a + b*x)*a* 
b**4*c*d**2*n**2*x**2 + 54*log(a + b*x)*a*b**4*d**3*n**2*x**3 + 12*log(a + 
 b*x)*b**5*c*d**2*n**2*x**3 - 36*log(c + d*x)*a**5*d**3*n - 54*log(c + d*x 
)*a**4*b*d**3*n**2 - 108*log(c + d*x)*a**4*b*d**3*n*x - 12*log(c + d*x)*a* 
*3*b**2*c*d**2*n**2 - 162*log(c + d*x)*a**3*b**2*d**3*n**2*x - 108*log(c + 
 d*x)*a**3*b**2*d**3*n*x**2 - 36*log(c + d*x)*a**2*b**3*c*d**2*n**2*x - 16 
2*log(c + d*x)*a**2*b**3*d**3*n**2*x**2 - 36*log(c + d*x)*a**2*b**3*d**3*n 
*x**3 - 36*log(c + d*x)*a*b**4*c*d**2*n**2*x**2 - 54*log(c + d*x)*a*b**4*d 
**3*n**2*x**3 - 12*log(c + d*x)*b**5*c*d**2*n**2*x**3 + 18*log(((a + b*x)* 
*n*e)/(c + d*x)**n)**2*a**4*b*d**3 + 54*log(((a + b*x)**n*e)/(c + d*x)**n) 
**2*a**3*b**2*d**3*x + 54*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**2*b**3* 
d**3*x**2 + 18*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*d**3*x**3 - 54 
*log(((a + b*x)**n*e)/(c + d*x)**n)*a**4*b*d**3*n + 96*log(((a + b*x)**n*e 
)/(c + d*x)**n)*a**3*b**2*c*d**2*n - 54*log(((a + b*x)**n*e)/(c + d*x)**n) 
*a**3*b**2*d**3*n*x - 54*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3*c**2 
*d*n + 72*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3*c*d**2*n*x + 12*...