3.1.46 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(c g+d g x)^5} \, dx\) [46]

3.1.46.1 Optimal result

Integrand size = 35, antiderivative size = 536 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=-\frac {B^2 d^3 n^2 (a+b x)^4}{32 (b c-a d)^4 g^5 (c+d x)^4}+\frac {2 b B^2 d^2 n^2 (a+b x)^3}{9 (b c-a d)^4 g^5 (c+d x)^3}-\frac {3 b^2 B^2 d n^2 (a+b x)^2}{4 (b c-a d)^4 g^5 (c+d x)^2}+\frac {2 b^3 B^2 n^2 (a+b x)}{(b c-a d)^4 g^5 (c+d x)}+\frac {B d^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 (b c-a d)^4 g^5 (c+d x)^4}-\frac {2 b B d^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^4 g^5 (c+d x)^3}+\frac {3 b^2 B d n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^4 g^5 (c+d x)^2}-\frac {2 b^3 B n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 g^5 (c+d x)}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d g^5 (c+d x)^4}+\frac {b^4 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {a+b x}{c+d x}\right )}{2 d (b c-a d)^4 g^5}-\frac {b^4 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{4 d (b c-a d)^4 g^5} \]

output

-1/32*B^2*d^3*n^2*(b*x+a)^4/(-a*d+b*c)^4/g^5/(d*x+c)^4+2/9*b*B^2*d^2*n^2*( 
b*x+a)^3/(-a*d+b*c)^4/g^5/(d*x+c)^3-3/4*b^2*B^2*d*n^2*(b*x+a)^2/(-a*d+b*c) 
^4/g^5/(d*x+c)^2+2*b^3*B^2*n^2*(b*x+a)/(-a*d+b*c)^4/g^5/(d*x+c)+1/8*B*d^3* 
n*(b*x+a)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(d*x+c)^4-2/3 
*b*B*d^2*n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(d*x 
+c)^3+3/2*b^2*B*d*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4 
/g^5/(d*x+c)^2-2*b^3*B*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c 
)^4/g^5/(d*x+c)-1/4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/d/g^5/(d*x+c)^4+1/2* 
b^4*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)^4 
/g^5-1/4*b^4*B^2*n^2*ln((b*x+a)/(d*x+c))^2/d/(-a*d+b*c)^4/g^5
 

3.1.46.2 Mathematica [C] (verified)

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

Time = 0.51 (sec) , antiderivative size = 700, normalized size of antiderivative = 1.31 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=\frac {-72 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n \left (36 A (b c-a d)^4-9 B (b c-a d)^4 n+48 A b (b c-a d)^3 (c+d x)-28 b B (b c-a d)^3 n (c+d x)+72 A b^2 (b c-a d)^2 (c+d x)^2-78 b^2 B (b c-a d)^2 n (c+d x)^2+144 A b^3 (b c-a d) (c+d x)^3-300 b^3 B (b c-a d) n (c+d x)^3+144 A b^4 (c+d x)^4 \log (a+b x)-300 b^4 B n (c+d x)^4 \log (a+b x)-72 b^4 B n (c+d x)^4 \log ^2(a+b x)+36 B (b c-a d)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+48 b B (b c-a d)^3 (c+d x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+72 b^2 B (b c-a d)^2 (c+d x)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+144 b^3 B (b c-a d) (c+d x)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+144 b^4 B (c+d x)^4 \log (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-144 A b^4 (c+d x)^4 \log (c+d x)+300 b^4 B n (c+d x)^4 \log (c+d x)+144 b^4 B n (c+d x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-144 b^4 B (c+d x)^4 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log (c+d x)-72 b^4 B n (c+d x)^4 \log ^2(c+d x)+144 b^4 B n (c+d x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+144 b^4 B n (c+d x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+144 b^4 B n (c+d x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{288 d g^5 (c+d x)^4} \]

input

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^5,x]
 

output

(-72*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(36*A*(b*c - a*d)^4 - 
 9*B*(b*c - a*d)^4*n + 48*A*b*(b*c - a*d)^3*(c + d*x) - 28*b*B*(b*c - a*d) 
^3*n*(c + d*x) + 72*A*b^2*(b*c - a*d)^2*(c + d*x)^2 - 78*b^2*B*(b*c - a*d) 
^2*n*(c + d*x)^2 + 144*A*b^3*(b*c - a*d)*(c + d*x)^3 - 300*b^3*B*(b*c - a* 
d)*n*(c + d*x)^3 + 144*A*b^4*(c + d*x)^4*Log[a + b*x] - 300*b^4*B*n*(c + d 
*x)^4*Log[a + b*x] - 72*b^4*B*n*(c + d*x)^4*Log[a + b*x]^2 + 36*B*(b*c - a 
*d)^4*Log[e*((a + b*x)/(c + d*x))^n] + 48*b*B*(b*c - a*d)^3*(c + d*x)*Log[ 
e*((a + b*x)/(c + d*x))^n] + 72*b^2*B*(b*c - a*d)^2*(c + d*x)^2*Log[e*((a 
+ b*x)/(c + d*x))^n] + 144*b^3*B*(b*c - a*d)*(c + d*x)^3*Log[e*((a + b*x)/ 
(c + d*x))^n] + 144*b^4*B*(c + d*x)^4*Log[a + b*x]*Log[e*((a + b*x)/(c + d 
*x))^n] - 144*A*b^4*(c + d*x)^4*Log[c + d*x] + 300*b^4*B*n*(c + d*x)^4*Log 
[c + d*x] + 144*b^4*B*n*(c + d*x)^4*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[ 
c + d*x] - 144*b^4*B*(c + d*x)^4*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d* 
x] - 72*b^4*B*n*(c + d*x)^4*Log[c + d*x]^2 + 144*b^4*B*n*(c + d*x)^4*Log[a 
 + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 144*b^4*B*n*(c + d*x)^4*PolyLog[2 
, (d*(a + b*x))/(-(b*c) + a*d)] + 144*b^4*B*n*(c + d*x)^4*PolyLog[2, (b*(c 
 + d*x))/(b*c - a*d)]))/(b*c - a*d)^4)/(288*d*g^5*(c + d*x)^4)
 

3.1.46.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.74, 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.

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

\(\Big \downarrow \) 2951

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

\(\Big \downarrow \) 2756

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

\(\Big \downarrow \) 2772

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {B n \left (b^4 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {4 b^3 d (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+\frac {3 b^2 d^2 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}+\frac {d^4 (a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 (c+d x)^4}-\frac {4 b d^3 (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (c+d x)^3}-B n \left (\frac {1}{2} b^4 \log ^2\left (\frac {a+b x}{c+d x}\right )-\frac {4 b^3 d (a+b x)}{c+d x}+\frac {3 b^2 d^2 (a+b x)^2}{2 (c+d x)^2}+\frac {d^4 (a+b x)^4}{16 (c+d x)^4}-\frac {4 b d^3 (a+b x)^3}{9 (c+d x)^3}\right )\right )}{2 d}-\frac {\left (b-\frac {d (a+b x)}{c+d x}\right )^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d}}{g^5 (b c-a d)^4}\)

input

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^5,x]
 

output

(-1/4*((b - (d*(a + b*x))/(c + d*x))^4*(A + B*Log[e*((a + b*x)/(c + d*x))^ 
n])^2)/d + (B*n*((d^4*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/ 
(4*(c + d*x)^4) - (4*b*d^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^ 
n]))/(3*(c + d*x)^3) + (3*b^2*d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + 
 d*x))^n]))/(c + d*x)^2 - (4*b^3*d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + 
d*x))^n]))/(c + d*x) + b^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + 
 b*x)/(c + d*x)] - B*n*((d^4*(a + b*x)^4)/(16*(c + d*x)^4) - (4*b*d^3*(a + 
 b*x)^3)/(9*(c + d*x)^3) + (3*b^2*d^2*(a + b*x)^2)/(2*(c + d*x)^2) - (4*b^ 
3*d*(a + b*x))/(c + d*x) + (b^4*Log[(a + b*x)/(c + d*x)]^2)/2)))/(2*d))/(( 
b*c - a*d)^4*g^5)
 

3.1.46.3.1 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])
 

3.1.46.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2325\) vs. \(2(518)=1036\).

Time = 42.18 (sec) , antiderivative size = 2326, normalized size of antiderivative = 4.34

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2326\)

input

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^5,x,method=_RETURNVERBOS 
E)
 

output

1/288*(-1056*A*B*x^3*a*b^4*c^7*d^2*n^2+432*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^ 
n)^2*a*b^4*c^8*d*n+72*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b^2*c^6*d^3*n^ 
2-576*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c^7*d^2*n^2+2160*A*B*x^3*a 
^2*b^3*c^6*d^3*n^2+9*B^2*x^4*a^5*c^2*d^7*n^3+36*B^2*x^3*a^5*c^3*d^6*n^3+72 
*A^2*x^4*a^5*c^2*d^7*n+54*B^2*x^2*a^5*c^4*d^5*n^3+288*A^2*x^3*a^5*c^3*d^6* 
n+36*B^2*x*a^5*c^5*d^4*n^3+576*B^2*x*a*b^4*c^9*n^3+432*A^2*x^2*a^5*c^4*d^5 
*n-72*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^5*c^6*d^3*n+288*B^2*ln(e*((b*x+a)/ 
(d*x+c))^n)^2*a^2*b^3*c^9*n+36*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^5*c^6*d^3*n 
^2-576*B^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c^9*n^2+288*A^2*x*a^5*c^5*d^4 
*n+288*A^2*x*a*b^4*c^9*n-64*B^2*x^4*a^4*b*c^3*d^6*n^3+216*B^2*x^4*a^3*b^2* 
c^4*d^5*n^3-576*B^2*x^4*a^2*b^3*c^5*d^4*n^3+415*B^2*x^4*a*b^4*c^6*d^3*n^3- 
36*A*B*x^4*a^5*c^2*d^7*n^2-256*B^2*x^3*a^4*b*c^4*d^5*n^3+864*B^2*x^3*a^3*b 
^2*c^5*d^4*n^3-2004*B^2*x^3*a^2*b^3*c^6*d^3*n^3+1360*B^2*x^3*a*b^4*c^7*d^2 
*n^3-1152*A^2*x*a^2*b^3*c^8*d*n-144*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^5*c^6* 
d^3*n+576*A*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^3*c^9*n+144*A*B*x^4*ln(e*((b 
*x+a)/(d*x+c))^n)*a*b^4*c^6*d^3*n+576*A*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a* 
b^4*c^7*d^2*n+864*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^4*c^8*d*n-288*A^2* 
x^4*a^4*b*c^3*d^6*n+432*A^2*x^4*a^3*b^2*c^4*d^5*n-288*A^2*x^4*a^2*b^3*c^5* 
d^4*n+72*A^2*x^4*a*b^4*c^6*d^3*n-144*A*B*x^3*a^5*c^3*d^6*n^2-384*B^2*x^2*a 
^4*b*c^5*d^4*n^3+1218*B^2*x^2*a^3*b^2*c^6*d^3*n^3-2400*B^2*x^2*a^2*b^3*...
 

3.1.46.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1768 vs. \(2 (518) = 1036\).

Time = 0.32 (sec) , antiderivative size = 1768, normalized size of antiderivative = 3.30 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=\text {Too large to display} \]

input

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^5,x, algorithm="f 
ricas")
 

output

-1/288*(72*A^2*b^4*c^4 - 288*A^2*a*b^3*c^3*d + 432*A^2*a^2*b^2*c^2*d^2 - 2 
88*A^2*a^3*b*c*d^3 + 72*A^2*a^4*d^4 + 12*(25*(B^2*b^4*c*d^3 - B^2*a*b^3*d^ 
4)*n^2 - 12*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*n)*x^3 + (415*B^2*b^4*c^4 - 57 
6*B^2*a*b^3*c^3*d + 216*B^2*a^2*b^2*c^2*d^2 - 64*B^2*a^3*b*c*d^3 + 9*B^2*a 
^4*d^4)*n^2 + 6*((163*B^2*b^4*c^2*d^2 - 176*B^2*a*b^3*c*d^3 + 13*B^2*a^2*b 
^2*d^4)*n^2 - 12*(7*A*B*b^4*c^2*d^2 - 8*A*B*a*b^3*c*d^3 + A*B*a^2*b^2*d^4) 
*n)*x^2 + 72*(B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4* 
B^2*a^3*b*c*d^3 + B^2*a^4*d^4)*log(e)^2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2* 
b^4*c*d^3*n^2*x^3 + 6*B^2*b^4*c^2*d^2*n^2*x^2 + 4*B^2*b^4*c^3*d*n^2*x + (4 
*B^2*a*b^3*c^3*d - 6*B^2*a^2*b^2*c^2*d^2 + 4*B^2*a^3*b*c*d^3 - B^2*a^4*d^4 
)*n^2)*log((b*x + a)/(d*x + c))^2 - 12*(25*A*B*b^4*c^4 - 48*A*B*a*b^3*c^3* 
d + 36*A*B*a^2*b^2*c^2*d^2 - 16*A*B*a^3*b*c*d^3 + 3*A*B*a^4*d^4)*n + 4*((2 
71*B^2*b^4*c^3*d - 324*B^2*a*b^3*c^2*d^2 + 60*B^2*a^2*b^2*c*d^3 - 7*B^2*a^ 
3*b*d^4)*n^2 - 12*(13*A*B*b^4*c^3*d - 18*A*B*a*b^3*c^2*d^2 + 6*A*B*a^2*b^2 
*c*d^3 - A*B*a^3*b*d^4)*n)*x + 12*(12*A*B*b^4*c^4 - 48*A*B*a*b^3*c^3*d + 7 
2*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 12*A*B*a^4*d^4 - 12*(B^2*b^4* 
c*d^3 - B^2*a*b^3*d^4)*n*x^3 - 6*(7*B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 
B^2*a^2*b^2*d^4)*n*x^2 - 4*(13*B^2*b^4*c^3*d - 18*B^2*a*b^3*c^2*d^2 + 6*B^ 
2*a^2*b^2*c*d^3 - B^2*a^3*b*d^4)*n*x - (25*B^2*b^4*c^4 - 48*B^2*a*b^3*c^3* 
d + 36*B^2*a^2*b^2*c^2*d^2 - 16*B^2*a^3*b*c*d^3 + 3*B^2*a^4*d^4)*n - 12...
 

3.1.46.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=\text {Timed out} \]

input

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(d*g*x+c*g)**5,x)
 

output

Timed out
 

3.1.46.7 Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2138 vs. \(2 (518) = 1036\).

Time = 0.33 (sec) , antiderivative size = 2138, normalized size of antiderivative = 3.99 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=\text {Too large to display} \]

input

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^5,x, algorithm="m 
axima")
 

output

1/24*A*B*n*((12*b^3*d^3*x^3 + 25*b^3*c^3 - 23*a*b^2*c^2*d + 13*a^2*b*c*d^2 
 - 3*a^3*d^3 + 6*(7*b^3*c*d^2 - a*b^2*d^3)*x^2 + 4*(13*b^3*c^2*d - 5*a*b^2 
*c*d^2 + a^2*b*d^3)*x)/((b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 - a 
^3*d^8)*g^5*x^4 + 4*(b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - a^3 
*c*d^7)*g^5*x^3 + 6*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - a^3 
*c^2*d^6)*g^5*x^2 + 4*(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 - a 
^3*c^3*d^5)*g^5*x + (b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^3*c 
^4*d^4)*g^5) + 12*b^4*log(b*x + a)/((b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b 
^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*g^5) - 12*b^4*log(d*x + c)/((b^4*c^4 
*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*g^5)) 
+ 1/288*(12*n*((12*b^3*d^3*x^3 + 25*b^3*c^3 - 23*a*b^2*c^2*d + 13*a^2*b*c* 
d^2 - 3*a^3*d^3 + 6*(7*b^3*c*d^2 - a*b^2*d^3)*x^2 + 4*(13*b^3*c^2*d - 5*a* 
b^2*c*d^2 + a^2*b*d^3)*x)/((b^3*c^3*d^5 - 3*a*b^2*c^2*d^6 + 3*a^2*b*c*d^7 
- a^3*d^8)*g^5*x^4 + 4*(b^3*c^4*d^4 - 3*a*b^2*c^3*d^5 + 3*a^2*b*c^2*d^6 - 
a^3*c*d^7)*g^5*x^3 + 6*(b^3*c^5*d^3 - 3*a*b^2*c^4*d^4 + 3*a^2*b*c^3*d^5 - 
a^3*c^2*d^6)*g^5*x^2 + 4*(b^3*c^6*d^2 - 3*a*b^2*c^5*d^3 + 3*a^2*b*c^4*d^4 
- a^3*c^3*d^5)*g^5*x + (b^3*c^7*d - 3*a*b^2*c^6*d^2 + 3*a^2*b*c^5*d^3 - a^ 
3*c^4*d^4)*g^5) + 12*b^4*log(b*x + a)/((b^4*c^4*d - 4*a*b^3*c^3*d^2 + 6*a^ 
2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*g^5) - 12*b^4*log(d*x + c)/((b^4* 
c^4*d - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*...
 

3.1.46.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1265 vs. \(2 (518) = 1036\).

Time = 1.69 (sec) , antiderivative size = 1265, normalized size of antiderivative = 2.36 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=\text {Too large to display} \]

input

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^5,x, algorithm="g 
iac")
 

output

1/288*(72*(4*(b*x + a)*B^2*b^3*n^2/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a 
^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)) - 6*(b*x + a)^2*B^2*b^2*d*n^2/((b 
^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c 
)^2) + 4*(b*x + a)^3*B^2*b*d^2*n^2/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a 
^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)^3) - (b*x + a)^4*B^2*d^3*n^2/((b^3 
*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)^ 
4))*log((b*x + a)/(d*x + c))^2 + 12*(3*(B^2*d^3*n^2 - 4*B^2*d^3*n*log(e) - 
 4*A*B*d^3*n)*(b*x + a)^4/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2*b*c*d^ 
2*g^5 - a^3*d^3*g^5)*(d*x + c)^4) - 16*(B^2*b*d^2*n^2 - 3*B^2*b*d^2*n*log( 
e) - 3*A*B*b*d^2*n)*(b*x + a)^3/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2* 
b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)^3) + 36*(B^2*b^2*d*n^2 - 2*B^2*b^2*d* 
n*log(e) - 2*A*B*b^2*d*n)*(b*x + a)^2/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 
3*a^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)^2) - 48*(B^2*b^3*n^2 - B^2*b^3* 
n*log(e) - A*B*b^3*n)*(b*x + a)/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2* 
b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)))*log((b*x + a)/(d*x + c)) - 9*(B^2*d 
^3*n^2 - 4*B^2*d^3*n*log(e) + 8*B^2*d^3*log(e)^2 - 4*A*B*d^3*n + 16*A*B*d^ 
3*log(e) + 8*A^2*d^3)*(b*x + a)^4/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^ 
2*b*c*d^2*g^5 - a^3*d^3*g^5)*(d*x + c)^4) + 32*(2*B^2*b*d^2*n^2 - 6*B^2*b* 
d^2*n*log(e) + 9*B^2*b*d^2*log(e)^2 - 6*A*B*b*d^2*n + 18*A*B*b*d^2*log(e) 
+ 9*A^2*b*d^2)*(b*x + a)^3/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2*b*...
 

3.1.46.9 Mupad [B] (verification not implemented)

Time = 6.34 (sec) , antiderivative size = 1765, normalized size of antiderivative = 3.29 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^5} \, dx=\text {Too large to display} \]

input

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(c*g + d*g*x)^5,x)
 

output

(B*b^4*n*atan((B*b^4*n*(12*A - 25*B*n)*(24*a^4*d^5*g^5 - 24*b^4*c^4*d*g^5 
- 48*a^3*b*c*d^4*g^5 + 48*a*b^3*c^3*d^2*g^5)*1i)/(24*d*g^5*(25*B^2*b^4*n^2 
 - 12*A*B*b^4*n)*(a*d - b*c)^4) + (B*b^5*n*x*(12*A - 25*B*n)*(a^3*d^4*g^5 
- b^3*c^3*d*g^5 - 3*a^2*b*c*d^3*g^5 + 3*a*b^2*c^2*d^2*g^5)*2i)/(g^5*(25*B^ 
2*b^4*n^2 - 12*A*B*b^4*n)*(a*d - b*c)^4))*(12*A - 25*B*n)*1i)/(12*d*g^5*(a 
*d - b*c)^4) - ((72*A^2*a^3*d^3 - 72*A^2*b^3*c^3 + 9*B^2*a^3*d^3*n^2 - 415 
*B^2*b^3*c^3*n^2 + 216*A^2*a*b^2*c^2*d - 216*A^2*a^2*b*c*d^2 - 36*A*B*a^3* 
d^3*n + 300*A*B*b^3*c^3*n + 161*B^2*a*b^2*c^2*d*n^2 - 55*B^2*a^2*b*c*d^2*n 
^2 - 276*A*B*a*b^2*c^2*d*n + 156*A*B*a^2*b*c*d^2*n)/(12*(a*d - b*c)) + (x^ 
2*(13*B^2*a*b^2*d^3*n^2 - 163*B^2*b^3*c*d^2*n^2 - 12*A*B*a*b^2*d^3*n + 84* 
A*B*b^3*c*d^2*n))/(2*(a*d - b*c)) - (x*(7*B^2*a^2*b*d^3*n^2 + 271*B^2*b^3* 
c^2*d*n^2 - 53*B^2*a*b^2*c*d^2*n^2 - 12*A*B*a^2*b*d^3*n - 156*A*B*b^3*c^2* 
d*n + 60*A*B*a*b^2*c*d^2*n))/(3*(a*d - b*c)) - (b*x^3*(25*B^2*b^2*d^3*n^2 
- 12*A*B*b^2*d^3*n))/(a*d - b*c))/(x*(96*a^2*c^3*d^4*g^5 + 96*b^2*c^5*d^2* 
g^5 - 192*a*b*c^4*d^3*g^5) + x^3*(96*a^2*c*d^6*g^5 + 96*b^2*c^3*d^4*g^5 - 
192*a*b*c^2*d^5*g^5) + x^4*(24*a^2*d^7*g^5 + 24*b^2*c^2*d^5*g^5 - 48*a*b*c 
*d^6*g^5) + x^2*(144*a^2*c^2*d^5*g^5 + 144*b^2*c^4*d^3*g^5 - 288*a*b*c^3*d 
^4*g^5) + 24*b^2*c^6*d*g^5 + 24*a^2*c^4*d^3*g^5 - 48*a*b*c^5*d^2*g^5) - lo 
g(e*((a + b*x)/(c + d*x))^n)^2*(B^2/(4*d*(c^4*g^5 + d^4*g^5*x^4 + 4*c*d^3* 
g^5*x^3 + 6*c^2*d^2*g^5*x^2 + 4*c^3*d*g^5*x)) - (B^2*b^4)/(4*d*g^5*(a^4...