\(\int \frac {(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2}{(a g+b g x)^5} \, dx\) [136]

Optimal result

Integrand size = 34, antiderivative size = 587 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {8 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {8 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {4 B d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {4 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \] Output:

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

Mathematica [C] (verified)

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

Time = 0.98 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {18 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {B \left (18 A (b c-a d)^4+9 B (b c-a d)^4+24 A d (-b c+a d)^3 (a+b x)+28 B d (-b c+a d)^3 (a+b x)+36 A d^2 (b c-a d)^2 (a+b x)^2+78 B d^2 (b c-a d)^2 (a+b x)^2+72 A d^3 (-b c+a d) (a+b x)^3+300 B d^3 (-b c+a d) (a+b x)^3-72 A d^4 (a+b x)^4 \log (a+b x)-300 B d^4 (a+b x)^4 \log (a+b x)+72 B d^4 (a+b x)^4 \log ^2(a+b x)+18 B (b c-a d)^4 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+24 B d (-b c+a d)^3 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+36 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+72 B d^3 (-b c+a d) (a+b x)^3 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-72 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+72 A d^4 (a+b x)^4 \log (c+d x)+300 B d^4 (a+b x)^4 \log (c+d x)-144 B d^4 (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+72 B d^4 (a+b x)^4 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (c+d x)+72 B d^4 (a+b x)^4 \log ^2(c+d x)-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{72 b g^5 (a+b x)^4} \] Input:

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

Output:

-1/72*(18*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (B*(18*A*(b*c - a*d 
)^4 + 9*B*(b*c - a*d)^4 + 24*A*d*(-(b*c) + a*d)^3*(a + b*x) + 28*B*d*(-(b* 
c) + a*d)^3*(a + b*x) + 36*A*d^2*(b*c - a*d)^2*(a + b*x)^2 + 78*B*d^2*(b*c 
 - a*d)^2*(a + b*x)^2 + 72*A*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 300*B*d^3*(- 
(b*c) + a*d)*(a + b*x)^3 - 72*A*d^4*(a + b*x)^4*Log[a + b*x] - 300*B*d^4*( 
a + b*x)^4*Log[a + b*x] + 72*B*d^4*(a + b*x)^4*Log[a + b*x]^2 + 18*B*(b*c 
- a*d)^4*Log[(e*(a + b*x)^2)/(c + d*x)^2] + 24*B*d*(-(b*c) + a*d)^3*(a + b 
*x)*Log[(e*(a + b*x)^2)/(c + d*x)^2] + 36*B*d^2*(b*c - a*d)^2*(a + b*x)^2* 
Log[(e*(a + b*x)^2)/(c + d*x)^2] + 72*B*d^3*(-(b*c) + a*d)*(a + b*x)^3*Log 
[(e*(a + b*x)^2)/(c + d*x)^2] - 72*B*d^4*(a + b*x)^4*Log[a + b*x]*Log[(e*( 
a + b*x)^2)/(c + d*x)^2] + 72*A*d^4*(a + b*x)^4*Log[c + d*x] + 300*B*d^4*( 
a + b*x)^4*Log[c + d*x] - 144*B*d^4*(a + b*x)^4*Log[(d*(a + b*x))/(-(b*c) 
+ a*d)]*Log[c + d*x] + 72*B*d^4*(a + b*x)^4*Log[(e*(a + b*x)^2)/(c + d*x)^ 
2]*Log[c + d*x] + 72*B*d^4*(a + b*x)^4*Log[c + d*x]^2 - 144*B*d^4*(a + b*x 
)^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 144*B*d^4*(a + b*x)^4*Po 
lyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 144*B*d^4*(a + b*x)^4*PolyLog[2, 
(b*(c + d*x))/(b*c - a*d)]))/(b*c - a*d)^4)/(b*g^5*(a + b*x)^4)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2950, 2795, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{4 (a+b x)^4}-\frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 (a+b x)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{(a+b x)^3}+\frac {4 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 (a+b x)^3}+\frac {d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{a+b x}+\frac {4 B d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{a+b x}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 (a+b x)^2}-\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(a+b x)^2}-\frac {b^3 B^2 (c+d x)^4}{8 (a+b x)^4}+\frac {8 b^2 B^2 d (c+d x)^3}{9 (a+b x)^3}+\frac {8 B^2 d^3 (c+d x)}{a+b x}-\frac {3 b B^2 d^2 (c+d x)^2}{(a+b x)^2}}{g^5 (b c-a d)^4}\)

Input:

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

Output:

((8*B^2*d^3*(c + d*x))/(a + b*x) - (3*b*B^2*d^2*(c + d*x)^2)/(a + b*x)^2 + 
 (8*b^2*B^2*d*(c + d*x)^3)/(9*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(8*(a + 
 b*x)^4) + (4*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(a 
 + b*x) - (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])) 
/(a + b*x)^2 + (4*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x) 
^2]))/(3*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + 
 d*x)^2]))/(4*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c 
+ d*x)^2])^2)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/ 
(c + d*x)^2])^2)/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b 
*x)^2)/(c + d*x)^2])^2)/(a + b*x)^3 - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + 
b*x)^2)/(c + d*x)^2])^2)/(4*(a + b*x)^4))/((b*c - a*d)^4*g^5)
 

Defintions of rubi rules used

rule 2009

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

rule 2795

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + 
(e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ 
c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b 
, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 
] && IntegerQ[m] && IntegerQ[r]))
 

rule 2950

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( 
m + 1)*(g/b)^m   Subst[Int[x^m*((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] & 
& EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E 
qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1392\) vs. \(2(575)=1150\).

Time = 2.81 (sec) , antiderivative size = 1393, normalized size of antiderivative = 2.37

method	result	size
orering	\(\text {Expression too large to display}\)	\(1393\)
derivativedivides	\(\text {Expression too large to display}\)	\(1486\)
default	\(\text {Expression too large to display}\)	\(1486\)
norman	\(\text {Expression too large to display}\)	\(1816\)
parallelrisch	\(\text {Expression too large to display}\)	\(2110\)
risch	\(\text {Expression too large to display}\)	\(2235\)
parts	\(\text {Expression too large to display}\)	\(2235\)

Input:

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

Output:

-1/576*(b*x+a)*(10500*b^4*d^5*x^5+35976*a*b^3*d^5*x^4+16524*b^4*c*d^4*x^4+ 
43314*a^2*b^2*d^5*x^3+57276*a*b^3*c*d^4*x^3+4410*b^4*c^2*d^3*x^3+19886*a^3 
*b*d^5*x^2+70284*a^2*b^2*c*d^4*x^2+15630*a*b^3*c^2*d^3*x^2-800*b^4*c^3*d^2 
*x^2+1499*a^4*d^5*x+33776*a^3*b*c*d^4*x+19620*a^2*b^2*c^2*d^3*x-2660*a*b^3 
*c^3*d^2*x+265*b^4*c^4*d*x+1499*a^4*c*d^4+13890*a^3*b*c^2*d^3-7350*a^2*b^2 
*c^3*d^2+3010*a*b^3*c^4*d-549*b^4*c^5)/(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c 
^2*d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)*(A+B*ln(e*(b*x+a)^2/(d*x+ 
c)^2))^2/(b*g*x+a*g)^5-1/576*(b*x+a)^2*(d*x+c)*(3900*b^4*d^4*x^4+13092*a*b 
^3*d^4*x^3+2508*b^4*c*d^3*x^3+15282*a^2*b^2*d^4*x^2+8712*a*b^3*c*d^3*x^2-5 
94*b^4*c^2*d^2*x^2+6640*a^3*b*d^4*x+10644*a^2*b^2*c*d^3*x-1932*a*b^3*c^2*d 
^2*x+248*b^4*c^3*d*x+415*a^4*d^4+4980*a^3*b*c*d^3-2148*a^2*b^2*c^2*d^2+788 
*a*b^3*c^3*d-135*b^4*c^4)/b/(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a 
^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)*(2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/( 
b*g*x+a*g)^5*B*(2*e*(b*x+a)/(d*x+c)^2*b-2*e*(b*x+a)^2/(d*x+c)^3*d)/e/(b*x+ 
a)^2*(d*x+c)^2-5*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^6*b*g)-1/57 
6/b*(300*b^3*d^3*x^3+978*a*b^2*d^3*x^2-78*b^3*c*d^2*x^2+1084*a^2*b*d^3*x-2 
12*a*b^2*c*d^2*x+28*b^3*c^2*d*x+415*a^3*d^3-161*a^2*b*c*d^2+55*a*b^2*c^2*d 
-9*b^3*c^3)/(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2*d^3-10*a^2*b^3*c^3*d^2+5 
*a*b^4*c^4*d-b^5*c^5)*(d*x+c)^2*(b*x+a)^3*(2*B^2*(2*e*(b*x+a)/(d*x+c)^2*b- 
2*e*(b*x+a)^2/(d*x+c)^3*d)^2/e^2/(b*x+a)^4*(d*x+c)^4/(b*g*x+a*g)^5-20*(...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 1084, normalized size of antiderivative = 1.85 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/72*(9*(2*A^2 + 2*A*B + B^2)*b^4*c^4 - 8*(9*A^2 + 12*A*B + 8*B^2)*a*b^3* 
c^3*d + 108*(A^2 + 2*A*B + 2*B^2)*a^2*b^2*c^2*d^2 - 72*(A^2 + 4*A*B + 8*B^ 
2)*a^3*b*c*d^3 + (18*A^2 + 150*A*B + 415*B^2)*a^4*d^4 - 12*((6*A*B + 25*B^ 
2)*b^4*c*d^3 - (6*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((6*A*B + 13*B^2)*b^4*c 
^2*d^2 - 16*(3*A*B + 11*B^2)*a*b^3*c*d^3 + (42*A*B + 163*B^2)*a^2*b^2*d^4) 
*x^2 - 18*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x^2 + 
 4*B^2*a^3*b*d^4*x - 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)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c 
*d*x + c^2))^2 - 4*((6*A*B + 7*B^2)*b^4*c^3*d - 12*(3*A*B + 5*B^2)*a*b^3*c 
^2*d^2 + 108*(A*B + 3*B^2)*a^2*b^2*c*d^3 - (78*A*B + 271*B^2)*a^3*b*d^4)*x 
 - 6*((6*A*B + 25*B^2)*b^4*d^4*x^4 - 3*(2*A*B + B^2)*b^4*c^4 + 8*(3*A*B + 
2*B^2)*a*b^3*c^3*d - 36*(A*B + B^2)*a^2*b^2*c^2*d^2 + 24*(A*B + 2*B^2)*a^3 
*b*c*d^3 + 4*(3*B^2*b^4*c*d^3 + 2*(3*A*B + 11*B^2)*a*b^3*d^4)*x^3 - 6*(B^2 
*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 - 6*(A*B + 3*B^2)*a^2*b^2*d^4)*x^2 + 4*(B 
^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 6*(A*B + 2*B^2 
)*a^3*b*d^4)*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c 
^2)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^ 
4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 
4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d 
+ 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2383 vs. \(2 (558) = 1116\).

Time = 106.16 (sec) , antiderivative size = 2383, normalized size of antiderivative = 4.06 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-B*d**4*(6*A + 25*B)*log(x + (6*A*B*a*d**5 + 6*A*B*b*c*d**4 + 25*B**2*a*d* 
*5 + 25*B**2*b*c*d**4 - B*a**5*d**9*(6*A + 25*B)/(a*d - b*c)**4 + 5*B*a**4 
*b*c*d**8*(6*A + 25*B)/(a*d - b*c)**4 - 10*B*a**3*b**2*c**2*d**7*(6*A + 25 
*B)/(a*d - b*c)**4 + 10*B*a**2*b**3*c**3*d**6*(6*A + 25*B)/(a*d - b*c)**4 
- 5*B*a*b**4*c**4*d**5*(6*A + 25*B)/(a*d - b*c)**4 + B*b**5*c**5*d**4*(6*A 
 + 25*B)/(a*d - b*c)**4)/(12*A*B*b*d**5 + 50*B**2*b*d**5))/(6*b*g**5*(a*d 
- b*c)**4) + B*d**4*(6*A + 25*B)*log(x + (6*A*B*a*d**5 + 6*A*B*b*c*d**4 + 
25*B**2*a*d**5 + 25*B**2*b*c*d**4 + B*a**5*d**9*(6*A + 25*B)/(a*d - b*c)** 
4 - 5*B*a**4*b*c*d**8*(6*A + 25*B)/(a*d - b*c)**4 + 10*B*a**3*b**2*c**2*d* 
*7*(6*A + 25*B)/(a*d - b*c)**4 - 10*B*a**2*b**3*c**3*d**6*(6*A + 25*B)/(a* 
d - b*c)**4 + 5*B*a*b**4*c**4*d**5*(6*A + 25*B)/(a*d - b*c)**4 - B*b**5*c* 
*5*d**4*(6*A + 25*B)/(a*d - b*c)**4)/(12*A*B*b*d**5 + 50*B**2*b*d**5))/(6* 
b*g**5*(a*d - b*c)**4) + (4*B**2*a**3*c*d**3 + 4*B**2*a**3*d**4*x - 6*B**2 
*a**2*b*c**2*d**2 + 6*B**2*a**2*b*d**4*x**2 + 4*B**2*a*b**2*c**3*d + 4*B** 
2*a*b**2*d**4*x**3 - B**2*b**3*c**4 + B**2*b**3*d**4*x**4)*log(e*(a + b*x) 
**2/(c + d*x)**2)**2/(4*a**8*d**4*g**5 - 16*a**7*b*c*d**3*g**5 + 16*a**7*b 
*d**4*g**5*x + 24*a**6*b**2*c**2*d**2*g**5 - 64*a**6*b**2*c*d**3*g**5*x + 
24*a**6*b**2*d**4*g**5*x**2 - 16*a**5*b**3*c**3*d*g**5 + 96*a**5*b**3*c**2 
*d**2*g**5*x - 96*a**5*b**3*c*d**3*g**5*x**2 + 16*a**5*b**3*d**4*g**5*x**3 
 + 4*a**4*b**4*c**4*g**5 - 64*a**4*b**4*c**3*d*g**5*x + 144*a**4*b**4*c...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2279 vs. \(2 (575) = 1150\).

Time = 0.23 (sec) , antiderivative size = 2279, normalized size of antiderivative = 3.88 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

1/72*(6*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 2 
5*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 
 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5* 
d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4* 
d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^ 
3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6* 
b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b* 
d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2* 
d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 
4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log 
(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) 
 + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - (9*b^4*c^4 - 64*a*b^3*c^3*d + 216*a^ 
2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4 
)*x^3 + 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b 
^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4 
)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 
 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d 
^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b^4*d^4*x^4 + 4*a*b^3*d^4 
*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25* 
b^4*d^4*x^4 + 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x...
 

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 874, normalized size of antiderivative = 1.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(B^2*d^4/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4* 
a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) - B^2/((b*g*x + a*g)^4*b*g))*log(b^2*e/ 
(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2 
/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2 
))^2 + 1/12*(12*B^2*d^3/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2* 
g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 6*B^2*d^2/((b^2*c^2*g - 2*a*b*c*d* 
g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^2) + 4*B^2*d/((b*g*x + a*g)^3*(b*c - a* 
d)*b*g^2) - 3*(2*A*B*b^3*g^3 + B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4))*log 
(b^2*e/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2* 
d^2*g^2/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g 
) + d^2)) - 1/6*(6*A*B*d^4 + 25*B^2*d^4)*log(-b*c*g/(b*g*x + a*g) + a*d*g/ 
(b*g*x + a*g) - d)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^ 
5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) + 1/6*(6*A*B*d^3 + 25*B^2*d^3)/(( 
b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x 
+ a*g)*b*g) - 1/12*(6*A*B*b*d^2 + 13*B^2*b*d^2)/((b^2*c^2*g - 2*a*b*c*d*g 
+ a^2*d^2*g)*(b*g*x + a*g)^2*b^2*g^2) + 1/18*(6*A*B*b^2*d*g + 7*B^2*b^2*d* 
g)/((b*g*x + a*g)^3*(b*c - a*d)*b^3*g^3) - 1/8*(2*A^2*b^3*g^3 + 2*A*B*b^3* 
g^3 + B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)
 

Mupad [B] (verification not implemented)

Time = 32.24 (sec) , antiderivative size = 1883, normalized size of antiderivative = 3.21 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

(B*d^4*atan((B*d^4*(6*A + 25*B)*(6*b^5*c^4*g^5 - 6*a^4*b*d^4*g^5 - 12*a*b^ 
4*c^3*d*g^5 + 12*a^3*b^2*c*d^3*g^5)*1i)/(6*b*g^5*(a*d - b*c)^4*(25*B^2*d^4 
 + 6*A*B*d^4)) + (B*d^5*x*(6*A + 25*B)*(b^4*c^3*g^5 - a^3*b*d^3*g^5 - 3*a* 
b^3*c^2*d*g^5 + 3*a^2*b^2*c*d^2*g^5)*2i)/(g^5*(a*d - b*c)^4*(25*B^2*d^4 + 
6*A*B*d^4)))*(6*A + 25*B)*1i)/(3*b*g^5*(a*d - b*c)^4) - log((e*(a + b*x)^2 
)/(c + d*x)^2)^2*(B^2/(4*b^2*g^5*(4*a^3*x + a^4/b + b^3*x^4 + 6*a^2*b*x^2 
+ 4*a*b^2*x^3)) - (B^2*d^4)/(4*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^ 
2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))) - (log((e*(a + b*x)^2)/(c + d*x)^2)*( 
(A*B)/(2*b^2*d*g^5) + (B^2*d^4*(a*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6 
*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2* 
d - 10*a^2*b*c*d^2)/(6*b*d^4)) + (4*a^4*d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 
 - 5*a*b^3*c^3*d - 10*a^3*b*c*d^3)/(2*b*d^5)))/(2*b*g^5*(a^4*d^4 + b^4*c^4 
 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^4*x^2*(b*( 
b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2) 
) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(3*d^3) + (a*(a*d - b*c))/d^2) - a*( 
(b^2*c - a*b*d)/(2*d^2) - (b*(a*d - b*c))/d^2) + (b^3*c^2 + 4*a^2*b*d^2 - 
5*a*b^2*c*d)/(2*d^3)))/(2*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4 
*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (B^2*d^4*x^3*(b*((b^2*c - a*b*d)/(2*d^2) 
- (b*(a*d - b*c))/d^2) + (b^3*c - a*b^2*d)/(2*d^2)))/(2*b*g^5*(a^4*d^4 + b 
^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^4...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 2755, normalized size of antiderivative = 4.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(72*log(a + b*x)*a**6*b*d**4 + 288*log(a + b*x)*a**5*b**2*d**4*x + 264*log 
(a + b*x)*a**5*b**2*d**4 + 36*log(a + b*x)*a**4*b**3*c*d**3 + 432*log(a + 
b*x)*a**4*b**3*d**4*x**2 + 1056*log(a + b*x)*a**4*b**3*d**4*x + 144*log(a 
+ b*x)*a**3*b**4*c*d**3*x + 288*log(a + b*x)*a**3*b**4*d**4*x**3 + 1584*lo 
g(a + b*x)*a**3*b**4*d**4*x**2 + 216*log(a + b*x)*a**2*b**5*c*d**3*x**2 + 
72*log(a + b*x)*a**2*b**5*d**4*x**4 + 1056*log(a + b*x)*a**2*b**5*d**4*x** 
3 + 144*log(a + b*x)*a*b**6*c*d**3*x**3 + 264*log(a + b*x)*a*b**6*d**4*x** 
4 + 36*log(a + b*x)*b**7*c*d**3*x**4 - 72*log(c + d*x)*a**6*b*d**4 - 288*l 
og(c + d*x)*a**5*b**2*d**4*x - 264*log(c + d*x)*a**5*b**2*d**4 - 36*log(c 
+ d*x)*a**4*b**3*c*d**3 - 432*log(c + d*x)*a**4*b**3*d**4*x**2 - 1056*log( 
c + d*x)*a**4*b**3*d**4*x - 144*log(c + d*x)*a**3*b**4*c*d**3*x - 288*log( 
c + d*x)*a**3*b**4*d**4*x**3 - 1584*log(c + d*x)*a**3*b**4*d**4*x**2 - 216 
*log(c + d*x)*a**2*b**5*c*d**3*x**2 - 72*log(c + d*x)*a**2*b**5*d**4*x**4 
- 1056*log(c + d*x)*a**2*b**5*d**4*x**3 - 144*log(c + d*x)*a*b**6*c*d**3*x 
**3 - 264*log(c + d*x)*a*b**6*d**4*x**4 - 36*log(c + d*x)*b**7*c*d**3*x**4 
 + 72*log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2)) 
**2*a**4*b**3*c*d**3 + 72*log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2 
*c*d*x + d**2*x**2))**2*a**4*b**3*d**4*x - 108*log((a**2*e + 2*a*b*e*x + b 
**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))**2*a**3*b**4*c**2*d**2 + 108*log 
((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*c*d*x + d**2*x**2))**2*a*...