\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c g+d g x)^4} \, dx\) [45]

Optimal result

Integrand size = 35, antiderivative size = 429 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx=\frac {2 B^2 d^2 n^2 (a+b x)^3}{27 (b c-a d)^3 g^4 (c+d x)^3}-\frac {b B^2 d n^2 (a+b x)^2}{2 (b c-a d)^3 g^4 (c+d x)^2}+\frac {2 b^2 B^2 n^2 (a+b x)}{(b c-a d)^3 g^4 (c+d x)}-\frac {2 B d^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^3 g^4 (c+d x)^3}+\frac {b B d n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^4 (c+d x)^2}-\frac {2 b^2 B n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 g^4 (c+d x)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{3 d (b c-a d)^3 g^4}-\frac {b^3 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{3 d (b c-a d)^3 g^4} \] Output:

2/27*B^2*d^2*n^2*(b*x+a)^3/(-a*d+b*c)^3/g^4/(d*x+c)^3-1/2*b*B^2*d*n^2*(b*x 
+a)^2/(-a*d+b*c)^3/g^4/(d*x+c)^2+2*b^2*B^2*n^2*(b*x+a)/(-a*d+b*c)^3/g^4/(d 
*x+c)-2/9*B*d^2*n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g 
^4/(d*x+c)^3+b*B*d*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^ 
3/g^4/(d*x+c)^2-2*b^2*B*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b* 
c)^3/g^4/(d*x+c)-1/3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/d/g^4/(d*x+c)^3+2/3 
*b^3*B*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((b*x+a)/(d*x+c))/d/(-a*d+b*c)^ 
3/g^4-1/3*b^3*B^2*n^2*ln((b*x+a)/(d*x+c))^2/d/(-a*d+b*c)^3/g^4
 

Mathematica [C] (verified)

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

Time = 0.47 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.43 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx=\frac {-18 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n \left (12 A (b c-a d)^3-4 B (b c-a d)^3 n+18 A b (b c-a d)^2 (c+d x)-15 b B (b c-a d)^2 n (c+d x)+36 A b^2 (b c-a d) (c+d x)^2-66 b^2 B (b c-a d) n (c+d x)^2+36 A b^3 (c+d x)^3 \log (a+b x)-66 b^3 B n (c+d x)^3 \log (a+b x)-18 b^3 B n (c+d x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+18 b B (b c-a d)^2 (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 b^2 B (b c-a d) (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+36 b^3 B (c+d x)^3 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-36 A b^3 (c+d x)^3 \log (c+d x)+66 b^3 B n (c+d x)^3 \log (c+d x)+36 b^3 B n (c+d x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 b^3 B (c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)-18 b^3 B n (c+d x)^3 \log ^2(c+d x)+36 b^3 B n (c+d x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 b^3 B n (c+d x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 b^3 B n (c+d x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3}}{54 d g^4 (c+d x)^3} \] Input:

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^4,x]
 

Output:

(-18*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(12*A*(b*c - a*d)^3 - 
 4*B*(b*c - a*d)^3*n + 18*A*b*(b*c - a*d)^2*(c + d*x) - 15*b*B*(b*c - a*d) 
^2*n*(c + d*x) + 36*A*b^2*(b*c - a*d)*(c + d*x)^2 - 66*b^2*B*(b*c - a*d)*n 
*(c + d*x)^2 + 36*A*b^3*(c + d*x)^3*Log[a + b*x] - 66*b^3*B*n*(c + d*x)^3* 
Log[a + b*x] - 18*b^3*B*n*(c + d*x)^3*Log[a + b*x]^2 + 12*B*(b*c - a*d)^3* 
Log[e*((a + b*x)/(c + d*x))^n] + 18*b*B*(b*c - a*d)^2*(c + d*x)*Log[e*((a 
+ b*x)/(c + d*x))^n] + 36*b^2*B*(b*c - a*d)*(c + d*x)^2*Log[e*((a + b*x)/( 
c + d*x))^n] + 36*b^3*B*(c + d*x)^3*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x 
))^n] - 36*A*b^3*(c + d*x)^3*Log[c + d*x] + 66*b^3*B*n*(c + d*x)^3*Log[c + 
 d*x] + 36*b^3*B*n*(c + d*x)^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d 
*x] - 36*b^3*B*(c + d*x)^3*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x] - 1 
8*b^3*B*n*(c + d*x)^3*Log[c + d*x]^2 + 36*b^3*B*n*(c + d*x)^3*Log[a + b*x] 
*Log[(b*(c + d*x))/(b*c - a*d)] + 36*b^3*B*n*(c + d*x)^3*PolyLog[2, (d*(a 
+ b*x))/(-(b*c) + a*d)] + 36*b^3*B*n*(c + d*x)^3*PolyLog[2, (b*(c + d*x))/ 
(b*c - a*d)]))/(b*c - a*d)^3)/(54*d*g^4*(c + d*x)^3)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.77, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2951, 2756, 2772, 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.

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

\(\Big \downarrow \) 2951

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^4,x]
 

Output:

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

Defintions of rubi rules used

rule 2009

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

rule 2756

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), 
x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] 
- Simp[b*n*(p/(e*(q + 1)))   Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 
 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, 
 -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

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 2951

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^(m + 
1)*(g/d)^m   Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x], x, (a + 
b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && NeQ[b*c 
- a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, - 
1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1150\) vs. \(2(417)=834\).

Time = 11.02 (sec) , antiderivative size = 1151, normalized size of antiderivative = 2.68

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1151\)

Input:

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x,method=_RETURNVERBOS 
E)
 

Output:

-1/54*(-36*A*B*x^2*a*b^3*d^7*n^2+36*A*B*x^2*b^4*c*d^6*n^2+54*B^2*x*ln(e*(( 
b*x+a)/(d*x+c))^n)^2*b^4*c^2*d^5*n+18*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2* 
b^2*d^7*n^2-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d^5*n^2+162*B^2*x* 
a*b^3*c*d^6*n^3+18*A*B*x*a^2*b^2*d^7*n^2+90*A*B*x*b^4*c^2*d^5*n^2-54*B^2*l 
n(e*((b*x+a)/(d*x+c))^n)^2*a^2*b^2*c*d^6*n+54*B^2*ln(e*((b*x+a)/(d*x+c))^n 
)^2*a*b^3*c^2*d^5*n+54*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c*d^6*n^2-108 
*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*c^2*d^5*n^2+36*A*B*ln(e*((b*x+a)/(d*x 
+c))^n)*a^3*b*d^7*n+36*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*b^4*d^7*n+54*B^2* 
x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^4*c*d^6*n-36*B^2*x^2*ln(e*((b*x+a)/(d*x+ 
c))^n)*a*b^3*d^7*n^2-162*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^6*n^2+5 
4*A*B*a^2*b^2*c*d^6*n^2-108*A*B*a*b^3*c^2*d^5*n^2+4*B^2*a^3*b*d^7*n^3-85*B 
^2*b^4*c^3*d^4*n^3+18*A^2*a^3*b*d^7*n-18*A^2*b^4*c^3*d^4*n+108*A*B*x^2*ln( 
e*((b*x+a)/(d*x+c))^n)*b^4*c*d^6*n-108*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b 
^3*c*d^6*n^2+108*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d^5*n-108*A*B*x*a 
*b^3*c*d^6*n^2-108*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^2*c*d^6*n+108*A*B*l 
n(e*((b*x+a)/(d*x+c))^n)*a*b^3*c^2*d^5*n-27*B^2*a^2*b^2*c*d^6*n^3+108*B^2* 
a*b^3*c^2*d^5*n^3-12*A*B*a^3*b*d^7*n^2+66*A*B*b^4*c^3*d^4*n^2-54*A^2*a^2*b 
^2*c*d^6*n+54*A^2*a*b^3*c^2*d^5*n-12*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b*d 
^7*n^2+18*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*b^4*d^7*n-66*B^2*x^3*ln(e*(( 
b*x+a)/(d*x+c))^n)*b^4*d^7*n^2+66*B^2*x^2*a*b^3*d^7*n^3-66*B^2*x^2*b^4*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1167 vs. \(2 (417) = 834\).

Time = 0.10 (sec) , antiderivative size = 1167, normalized size of antiderivative = 2.72 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x, algorithm="f 
ricas")
 

Output:

-1/54*(18*A^2*b^3*c^3 - 54*A^2*a*b^2*c^2*d + 54*A^2*a^2*b*c*d^2 - 18*A^2*a 
^3*d^3 + (85*B^2*b^3*c^3 - 108*B^2*a*b^2*c^2*d + 27*B^2*a^2*b*c*d^2 - 4*B^ 
2*a^3*d^3)*n^2 + 6*(11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 - 6*(A*B*b^3*c* 
d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2* 
a^2*b*c*d^2 - B^2*a^3*d^3)*log(e)^2 - 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*b^3* 
c*d^2*n^2*x^2 + 3*B^2*b^3*c^2*d*n^2*x + (3*B^2*a*b^2*c^2*d - 3*B^2*a^2*b*c 
*d^2 + B^2*a^3*d^3)*n^2)*log((b*x + a)/(d*x + c))^2 - 6*(11*A*B*b^3*c^3 - 
18*A*B*a*b^2*c^2*d + 9*A*B*a^2*b*c*d^2 - 2*A*B*a^3*d^3)*n + 3*((49*B^2*b^3 
*c^2*d - 54*B^2*a*b^2*c*d^2 + 5*B^2*a^2*b*d^3)*n^2 - 6*(5*A*B*b^3*c^2*d - 
6*A*B*a*b^2*c*d^2 + A*B*a^2*b*d^3)*n)*x + 6*(6*A*B*b^3*c^3 - 18*A*B*a*b^2* 
c^2*d + 18*A*B*a^2*b*c*d^2 - 6*A*B*a^3*d^3 - 6*(B^2*b^3*c*d^2 - B^2*a*b^2* 
d^3)*n*x^2 - 3*(5*B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3)*n*x - 
 (11*B^2*b^3*c^3 - 18*B^2*a*b^2*c^2*d + 9*B^2*a^2*b*c*d^2 - 2*B^2*a^3*d^3) 
*n - 6*(B^2*b^3*d^3*n*x^3 + 3*B^2*b^3*c*d^2*n*x^2 + 3*B^2*b^3*c^2*d*n*x + 
(3*B^2*a*b^2*c^2*d - 3*B^2*a^2*b*c*d^2 + B^2*a^3*d^3)*n)*log((b*x + a)/(d* 
x + c)))*log(e) + 6*((11*B^2*b^3*d^3*n^2 - 6*A*B*b^3*d^3*n)*x^3 + (18*B^2* 
a*b^2*c^2*d - 9*B^2*a^2*b*c*d^2 + 2*B^2*a^3*d^3)*n^2 - 3*(6*A*B*b^3*c*d^2* 
n - (9*B^2*b^3*c*d^2 + 2*B^2*a*b^2*d^3)*n^2)*x^2 - 6*(3*A*B*a*b^2*c^2*d - 
3*A*B*a^2*b*c*d^2 + A*B*a^3*d^3)*n - 3*(6*A*B*b^3*c^2*d*n - (6*B^2*b^3*c^2 
*d + 6*B^2*a*b^2*c*d^2 - B^2*a^2*b*d^3)*n^2)*x)*log((b*x + a)/(d*x + c)...
 

Sympy [F]

\[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx=\frac {\int \frac {A^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{g^{4}} \] Input:

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(d*g*x+c*g)**4,x)
 

Output:

(Integral(A**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d** 
4*x**4), x) + Integral(B**2*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(c* 
*4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Inte 
gral(2*A*B*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**4 + 4*c**3*d*x + 6* 
c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x))/g**4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1435 vs. \(2 (417) = 834\).

Time = 0.12 (sec) , antiderivative size = 1435, normalized size of antiderivative = 3.34 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x, algorithm="m 
axima")
 

Output:

1/9*A*B*n*((6*b^2*d^2*x^2 + 11*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2 + 3*(5*b^2* 
c*d - a*b*d^2)*x)/((b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*g^4*x^3 + 3*(b^2* 
c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*g^4*x^2 + 3*(b^2*c^4*d^2 - 2*a*b*c^3* 
d^3 + a^2*c^2*d^4)*g^4*x + (b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*g^4) 
+ 6*b^3*log(b*x + a)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d 
^4)*g^4) - 6*b^3*log(d*x + c)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^ 
3 - a^3*d^4)*g^4)) + 1/54*(6*n*((6*b^2*d^2*x^2 + 11*b^2*c^2 - 7*a*b*c*d + 
2*a^2*d^2 + 3*(5*b^2*c*d - a*b*d^2)*x)/((b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d 
^6)*g^4*x^3 + 3*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*g^4*x^2 + 3*(b^2 
*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*g^4*x + (b^2*c^5*d - 2*a*b*c^4*d^2 
 + a^2*c^3*d^3)*g^4) + 6*b^3*log(b*x + a)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 
3*a^2*b*c*d^3 - a^3*d^4)*g^4) - 6*b^3*log(d*x + c)/((b^3*c^3*d - 3*a*b^2*c 
^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*g^4))*log(e*(b*x/(d*x + c) + a/(d*x + c) 
)^n) - (85*b^3*c^3 - 108*a*b^2*c^2*d + 27*a^2*b*c*d^2 - 4*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*b^3*c*d^2*x^2 + 3*b^3*c^2*d 
*x + b^3*c^3)*log(b*x + a)^2 + 18*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c 
^2*d*x + b^3*c^3)*log(d*x + c)^2 + 3*(49*b^3*c^2*d - 54*a*b^2*c*d^2 + 5*a^ 
2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)* 
log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*b^3*c*d^2*x^2 + 33*b^3*c^2*d*x + 11* 
b^3*c^3 + 6*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*l...
 

Giac [A] (verification not implemented)

Time = 1.11 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.81 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx =\text {Too large to display} \] Input:

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x, algorithm="g 
iac")
 

Output:

1/54*(18*(3*(b*x + a)*B^2*b^2*n^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2* 
g^4)*(d*x + c)) - 3*(b*x + a)^2*B^2*b*d*n^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 
+ a^2*d^2*g^4)*(d*x + c)^2) + (b*x + a)^3*B^2*d^2*n^2/((b^2*c^2*g^4 - 2*a* 
b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^3))*log((b*x + a)/(d*x + c))^2 - 6*(2*( 
B^2*d^2*n^2 - 3*B^2*d^2*n*log(e) - 3*A*B*d^2*n)*(b*x + a)^3/((b^2*c^2*g^4 
- 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^3) - 9*(B^2*b*d*n^2 - 2*B^2*b*d*n 
*log(e) - 2*A*B*b*d*n)*(b*x + a)^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2 
*g^4)*(d*x + c)^2) + 18*(B^2*b^2*n^2 - B^2*b^2*n*log(e) - A*B*b^2*n)*(b*x 
+ a)/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)))*log((b*x + a 
)/(d*x + c)) + 2*(2*B^2*d^2*n^2 - 6*B^2*d^2*n*log(e) + 9*B^2*d^2*log(e)^2 
- 6*A*B*d^2*n + 18*A*B*d^2*log(e) + 9*A^2*d^2)*(b*x + a)^3/((b^2*c^2*g^4 - 
 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^3) - 27*(B^2*b*d*n^2 - 2*B^2*b*d*n 
*log(e) + 2*B^2*b*d*log(e)^2 - 2*A*B*b*d*n + 4*A*B*b*d*log(e) + 2*A^2*b*d) 
*(b*x + a)^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^2) + 5 
4*(2*B^2*b^2*n^2 - 2*B^2*b^2*n*log(e) + B^2*b^2*log(e)^2 - 2*A*B*b^2*n + 2 
*A*B*b^2*log(e) + A^2*b^2)*(b*x + a)/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d 
^2*g^4)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 28.49 (sec) , antiderivative size = 1040, normalized size of antiderivative = 2.42 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx =\text {Too large to display} \] Input:

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(c*g + d*g*x)^4,x)
 

Output:

- log(e*((a + b*x)/(c + d*x))^n)^2*(B^2/(3*d*(c^3*g^4 + d^3*g^4*x^3 + 3*c* 
d^2*g^4*x^2 + 3*c^2*d*g^4*x)) + (B^2*b^3)/(3*d*g^4*(a^3*d^3 - b^3*c^3 + 3* 
a*b^2*c^2*d - 3*a^2*b*c*d^2))) - ((18*A^2*a^2*d^2 + 18*A^2*b^2*c^2 + 4*B^2 
*a^2*d^2*n^2 + 85*B^2*b^2*c^2*n^2 - 36*A^2*a*b*c*d - 12*A*B*a^2*d^2*n - 66 
*A*B*b^2*c^2*n - 23*B^2*a*b*c*d*n^2 + 42*A*B*a*b*c*d*n)/(6*(a*d - b*c)) - 
(x*(5*B^2*a*b*d^2*n^2 - 49*B^2*b^2*c*d*n^2 - 6*A*B*a*b*d^2*n + 30*A*B*b^2* 
c*d*n))/(2*(a*d - b*c)) + (b*x^2*(11*B^2*b*d^2*n^2 - 6*A*B*b*d^2*n))/(a*d 
- b*c))/(x*(27*a*c^2*d^3*g^4 - 27*b*c^3*d^2*g^4) - x^2*(27*b*c^2*d^3*g^4 - 
 27*a*c*d^4*g^4) + x^3*(9*a*d^5*g^4 - 9*b*c*d^4*g^4) + 9*a*c^3*d^2*g^4 - 9 
*b*c^4*d*g^4) - log(e*((a + b*x)/(c + d*x))^n)*((2*A*B)/(3*c^3*d*g^4 + 3*d 
^4*g^4*x^3 + 9*c^2*d^2*g^4*x + 9*c*d^3*g^4*x^2) + (2*B^2*b^3*(x*(d*((d*g^4 
*n*(a*d - b*c)*(a*d - 3*b*c))/(2*b^2) - (c*d*g^4*n*(a*d - b*c))/b) - (2*c* 
d^2*g^4*n*(a*d - b*c))/b + (d^2*g^4*n*(a*d - b*c)*(a*d - 3*b*c))/b^2) + c* 
((d*g^4*n*(a*d - b*c)*(a*d - 3*b*c))/(2*b^2) - (c*d*g^4*n*(a*d - b*c))/b) 
- (d*g^4*n*(a*d - b*c)*(a^2*d^2 + 3*b^2*c^2 - 3*a*b*c*d))/b^3 - (3*d^3*g^4 
*n*x^2*(a*d - b*c))/b))/(3*d*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^ 
2*b*c*d^2)*(3*c^3*d*g^4 + 3*d^4*g^4*x^3 + 9*c^2*d^2*g^4*x + 9*c*d^3*g^4*x^ 
2))) - (B*b^3*n*atan((B*b^3*n*(6*A - 11*B*n)*((a^3*d^4*g^4 + b^3*c^3*d*g^4 
 - a^2*b*c*d^3*g^4 - a*b^2*c^2*d^2*g^4)/(a^2*d^3*g^4 + b^2*c^2*d*g^4 - 2*a 
*b*c*d^2*g^4) + 2*b*d*x)*(a^2*d^3*g^4 + b^2*c^2*d*g^4 - 2*a*b*c*d^2*g^4...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1589, normalized size of antiderivative = 3.70 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx =\text {Too large to display} \] Input:

int((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x)
 

Output:

( - 36*log(a + b*x)*a*b**4*c**4*n + 12*log(a + b*x)*a*b**4*c**3*d*n**2 - 1 
08*log(a + b*x)*a*b**4*c**3*d*n*x + 36*log(a + b*x)*a*b**4*c**2*d**2*n**2* 
x - 108*log(a + b*x)*a*b**4*c**2*d**2*n*x**2 + 36*log(a + b*x)*a*b**4*c*d* 
*3*n**2*x**2 - 36*log(a + b*x)*a*b**4*c*d**3*n*x**3 + 12*log(a + b*x)*a*b* 
*4*d**4*n**2*x**3 + 54*log(a + b*x)*b**5*c**4*n**2 + 162*log(a + b*x)*b**5 
*c**3*d*n**2*x + 162*log(a + b*x)*b**5*c**2*d**2*n**2*x**2 + 54*log(a + b* 
x)*b**5*c*d**3*n**2*x**3 + 36*log(c + d*x)*a*b**4*c**4*n - 12*log(c + d*x) 
*a*b**4*c**3*d*n**2 + 108*log(c + d*x)*a*b**4*c**3*d*n*x - 36*log(c + d*x) 
*a*b**4*c**2*d**2*n**2*x + 108*log(c + d*x)*a*b**4*c**2*d**2*n*x**2 - 36*l 
og(c + d*x)*a*b**4*c*d**3*n**2*x**2 + 36*log(c + d*x)*a*b**4*c*d**3*n*x**3 
 - 12*log(c + d*x)*a*b**4*d**4*n**2*x**3 - 54*log(c + d*x)*b**5*c**4*n**2 
- 162*log(c + d*x)*b**5*c**3*d*n**2*x - 162*log(c + d*x)*b**5*c**2*d**2*n* 
*2*x**2 - 54*log(c + d*x)*b**5*c*d**3*n**2*x**3 - 18*log(((a + b*x)**n*e)/ 
(c + d*x)**n)**2*a**3*b**2*c*d**3 + 54*log(((a + b*x)**n*e)/(c + d*x)**n)* 
*2*a**2*b**3*c**2*d**2 - 54*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c 
**3*d - 54*log(((a + b*x)**n*e)/(c + d*x)**n)**2*b**5*c**3*d*x - 54*log((( 
a + b*x)**n*e)/(c + d*x)**n)**2*b**5*c**2*d**2*x**2 - 18*log(((a + b*x)**n 
*e)/(c + d*x)**n)**2*b**5*c*d**3*x**3 - 36*log(((a + b*x)**n*e)/(c + d*x)* 
*n)*a**4*b*c*d**3 + 108*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**2*c**2* 
d**2 + 12*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**2*c*d**3*n - 108*l...