\(\int \frac {(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2}{(a g+b g x)^4} \, dx\) [135]

Optimal result

Integrand size = 34, antiderivative size = 429 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {8 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {2 b B^2 d (c+d x)^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {8 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}-\frac {4 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}+\frac {2 b B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {4 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)}+\frac {b d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^3 g^4 (a+b x)^2}-\frac {b^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 (b c-a d)^3 g^4 (a+b x)^3} \] Output:

-8*B^2*d^2*(d*x+c)/(-a*d+b*c)^3/g^4/(b*x+a)+2*b*B^2*d*(d*x+c)^2/(-a*d+b*c) 
^3/g^4/(b*x+a)^2-8/27*b^2*B^2*(d*x+c)^3/(-a*d+b*c)^3/g^4/(b*x+a)^3-4*B*d^2 
*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^3/g^4/(b*x+a)+2*b*B*d* 
(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^3/g^4/(b*x+a)^2-4/9*b 
^2*B*(d*x+c)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^3/g^4/(b*x+a)^3- 
d^2*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^3/g^4/(b*x+a)+b*d 
*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^3/g^4/(b*x+a)^2-1/ 
3*b^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^3/g^4/(b*x+a) 
^3
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

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

Time = 0.67 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.39 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {9 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {2 B \left (6 A (b c-a d)^3+4 B (b c-a d)^3-9 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 (a+b x)+18 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) (a+b x)^2+18 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 (a+b x)^3 \log (a+b x)-18 B d^3 (a+b x)^3 \log ^2(a+b x)+6 B (b c-a d)^3 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-9 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+18 B d^2 (b c-a d) (a+b x)^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+18 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-18 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 (a+b x)^3 \log (c+d x)+36 B d^3 (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-18 B d^3 (a+b x)^3 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (c+d x)-18 B d^3 (a+b x)^3 \log ^2(c+d x)+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^3}}{27 b g^4 (a+b x)^3} \] Input:

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

Output:

-1/27*(9*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (2*B*(6*A*(b*c - a*d 
)^3 + 4*B*(b*c - a*d)^3 - 9*A*d*(b*c - a*d)^2*(a + b*x) - 15*B*d*(b*c - a* 
d)^2*(a + b*x) + 18*A*d^2*(b*c - a*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*( 
a + b*x)^2 + 18*A*d^3*(a + b*x)^3*Log[a + b*x] + 66*B*d^3*(a + b*x)^3*Log[ 
a + b*x] - 18*B*d^3*(a + b*x)^3*Log[a + b*x]^2 + 6*B*(b*c - a*d)^3*Log[(e* 
(a + b*x)^2)/(c + d*x)^2] - 9*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x) 
^2)/(c + d*x)^2] + 18*B*d^2*(b*c - a*d)*(a + b*x)^2*Log[(e*(a + b*x)^2)/(c 
 + d*x)^2] + 18*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(e*(a + b*x)^2)/(c + d* 
x)^2] - 18*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d^3*(a + b*x)^3*Log[c + d 
*x] + 36*B*d^3*(a + b*x)^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] 
- 18*B*d^3*(a + b*x)^3*Log[(e*(a + b*x)^2)/(c + d*x)^2]*Log[c + d*x] - 18* 
B*d^3*(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[( 
b*(c + d*x))/(b*c - a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/ 
(-(b*c) + a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d 
)]))/(b*c - a*d)^3)/(b*g^4*(a + b*x)^3)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 326, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{3 (a+b x)^3}-\frac {4 b^2 B (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{9 (a+b x)^3}-\frac {d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{a+b x}-\frac {4 B d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{a+b x}+\frac {b d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{(a+b x)^2}+\frac {2 b B d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(a+b x)^2}-\frac {8 b^2 B^2 (c+d x)^3}{27 (a+b x)^3}-\frac {8 B^2 d^2 (c+d x)}{a+b x}+\frac {2 b B^2 d (c+d x)^2}{(a+b x)^2}}{g^4 (b c-a d)^3}\)

Input:

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

Output:

((-8*B^2*d^2*(c + d*x))/(a + b*x) + (2*b*B^2*d*(c + d*x)^2)/(a + b*x)^2 - 
(8*b^2*B^2*(c + d*x)^3)/(27*(a + b*x)^3) - (4*B*d^2*(c + d*x)*(A + B*Log[( 
e*(a + b*x)^2)/(c + d*x)^2]))/(a + b*x) + (2*b*B*d*(c + d*x)^2*(A + B*Log[ 
(e*(a + b*x)^2)/(c + d*x)^2]))/(a + b*x)^2 - (4*b^2*B*(c + d*x)^3*(A + B*L 
og[(e*(a + b*x)^2)/(c + d*x)^2]))/(9*(a + b*x)^3) - (d^2*(c + d*x)*(A + B* 
Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(a + b*x) + (b*d*(c + d*x)^2*(A + B*L 
og[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(a + b*x)^2 - (b^2*(c + d*x)^3*(A + B* 
Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*(a + b*x)^3))/((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 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. \(1001\) vs. \(2(423)=846\).

Time = 1.96 (sec) , antiderivative size = 1002, normalized size of antiderivative = 2.34

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1002\)
derivativedivides	\(\text {Expression too large to display}\)	\(1019\)
default	\(\text {Expression too large to display}\)	\(1019\)
norman	\(\text {Expression too large to display}\)	\(1054\)
orering	\(\text {Expression too large to display}\)	\(1138\)
risch	\(\text {Expression too large to display}\)	\(1309\)
parts	\(\text {Expression too large to display}\)	\(1309\)

Input:

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

Output:

-1/54*(588*B^2*x*a^2*b^5*d^4+60*B^2*x*b^7*c^2*d^2+264*B^2*x^2*a*b^6*d^4-26 
4*B^2*x^2*b^7*c*d^3-54*A^2*a^2*b^5*c*d^3+54*A^2*a*b^6*c^2*d^2-132*B^2*x^3* 
ln(e*(b*x+a)^2/(d*x+c)^2)*b^7*d^4-18*B^2*ln(e*(b*x+a)^2/(d*x+c)^2)^2*b^7*c 
^3*d-24*B^2*ln(e*(b*x+a)^2/(d*x+c)^2)*b^7*c^3*d-36*A*B*ln(e*(b*x+a)^2/(d*x 
+c)^2)*b^7*c^3*d-216*B^2*ln(e*(b*x+a)^2/(d*x+c)^2)*a^2*b^5*c*d^3+108*B^2*l 
n(e*(b*x+a)^2/(d*x+c)^2)*a*b^6*c^2*d^2+18*A^2*a^3*b^4*d^4-18*A^2*b^7*c^3*d 
+340*B^2*a^3*b^4*d^4-16*B^2*b^7*c^3*d+132*A*B*a^3*b^4*d^4-24*A*B*b^7*c^3*d 
-432*B^2*a^2*b^5*c*d^3+108*B^2*a*b^6*c^2*d^2-108*A*B*x^2*ln(e*(b*x+a)^2/(d 
*x+c)^2)*a*b^6*d^4-216*A*B*a^2*b^5*c*d^3+108*A*B*a*b^6*c^2*d^2+72*A*B*x^2* 
a*b^6*d^4-72*A*B*x^2*b^7*c*d^3+180*A*B*x*a^2*b^5*d^4+36*A*B*x*b^7*c^2*d^2- 
648*B^2*x*a*b^6*c*d^3-18*B^2*x^3*ln(e*(b*x+a)^2/(d*x+c)^2)^2*b^7*d^4-216*A 
*B*x*a*b^6*c*d^3-36*A*B*x^3*ln(e*(b*x+a)^2/(d*x+c)^2)*b^7*d^4-54*B^2*x^2*l 
n(e*(b*x+a)^2/(d*x+c)^2)^2*a*b^6*d^4-324*B^2*x^2*ln(e*(b*x+a)^2/(d*x+c)^2) 
*a*b^6*d^4-72*B^2*x^2*ln(e*(b*x+a)^2/(d*x+c)^2)*b^7*c*d^3-54*B^2*x*ln(e*(b 
*x+a)^2/(d*x+c)^2)^2*a^2*b^5*d^4-216*B^2*x*ln(e*(b*x+a)^2/(d*x+c)^2)*a^2*b 
^5*d^4+36*B^2*x*ln(e*(b*x+a)^2/(d*x+c)^2)*b^7*c^2*d^2-54*B^2*ln(e*(b*x+a)^ 
2/(d*x+c)^2)^2*a^2*b^5*c*d^3+54*B^2*ln(e*(b*x+a)^2/(d*x+c)^2)^2*a*b^6*c^2* 
d^2-108*A*B*x*ln(e*(b*x+a)^2/(d*x+c)^2)*a^2*b^5*d^4-216*B^2*x*ln(e*(b*x+a) 
^2/(d*x+c)^2)*a*b^6*c*d^3-108*A*B*ln(e*(b*x+a)^2/(d*x+c)^2)*a^2*b^5*c*d^3+ 
108*A*B*ln(e*(b*x+a)^2/(d*x+c)^2)*a*b^6*c^2*d^2)/g^4/(b*x+a)^3/(a*d-b*c...
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 719, normalized size of antiderivative = 1.68 \[ \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)^4} \, dx=-\frac {{\left (9 \, A^{2} + 12 \, A B + 8 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a b^{2} c^{2} d + 27 \, {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a^{2} b c d^{2} - {\left (9 \, A^{2} + 66 \, A B + 170 \, B^{2}\right )} a^{3} d^{3} + 12 \, {\left ({\left (3 \, A B + 11 \, B^{2}\right )} b^{3} c d^{2} - {\left (3 \, A B + 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 9 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} - 6 \, {\left ({\left (3 \, A B + 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (A B + 3 \, B^{2}\right )} a b^{2} c d^{2} + {\left (15 \, A B + 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (3 \, A B + 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + {\left (3 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 9 \, {\left (A B + B^{2}\right )} a b^{2} c^{2} d + 9 \, {\left (A B + 2 \, B^{2}\right )} a^{2} b c d^{2} + 3 \, {\left (2 \, B^{2} b^{3} c d^{2} + 3 \, {\left (A B + 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} - 3 \, {\left (A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{27 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \] Input:

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

Output:

-1/27*((9*A^2 + 12*A*B + 8*B^2)*b^3*c^3 - 27*(A^2 + 2*A*B + 2*B^2)*a*b^2*c 
^2*d + 27*(A^2 + 4*A*B + 8*B^2)*a^2*b*c*d^2 - (9*A^2 + 66*A*B + 170*B^2)*a 
^3*d^3 + 12*((3*A*B + 11*B^2)*b^3*c*d^2 - (3*A*B + 11*B^2)*a*b^2*d^3)*x^2 
+ 9*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*b^3*c 
^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*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 - 6*((3*A*B + 5*B^2)*b^3*c^2*d - 18*(A* 
B + 3*B^2)*a*b^2*c*d^2 + (15*A*B + 49*B^2)*a^2*b*d^3)*x + 6*((3*A*B + 11*B 
^2)*b^3*d^3*x^3 + (3*A*B + 2*B^2)*b^3*c^3 - 9*(A*B + B^2)*a*b^2*c^2*d + 9* 
(A*B + 2*B^2)*a^2*b*c*d^2 + 3*(2*B^2*b^3*c*d^2 + 3*(A*B + 3*B^2)*a*b^2*d^3 
)*x^2 - 3*(B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 3*(A*B + 2*B^2)*a^2*b*d^3)* 
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^7*c 
^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3 
 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3*(a^2*b^5*c 
^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 
 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1561 vs. \(2 (406) = 812\).

Time = 12.59 (sec) , antiderivative size = 1561, normalized size of antiderivative = 3.64 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-4*B*d**3*(3*A + 11*B)*log(x + (12*A*B*a*d**4 + 12*A*B*b*c*d**3 + 44*B**2* 
a*d**4 + 44*B**2*b*c*d**3 - 4*B*a**4*d**7*(3*A + 11*B)/(a*d - b*c)**3 + 16 
*B*a**3*b*c*d**6*(3*A + 11*B)/(a*d - b*c)**3 - 24*B*a**2*b**2*c**2*d**5*(3 
*A + 11*B)/(a*d - b*c)**3 + 16*B*a*b**3*c**3*d**4*(3*A + 11*B)/(a*d - b*c) 
**3 - 4*B*b**4*c**4*d**3*(3*A + 11*B)/(a*d - b*c)**3)/(24*A*B*b*d**4 + 88* 
B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) + 4*B*d**3*(3*A + 11*B)*log(x + (1 
2*A*B*a*d**4 + 12*A*B*b*c*d**3 + 44*B**2*a*d**4 + 44*B**2*b*c*d**3 + 4*B*a 
**4*d**7*(3*A + 11*B)/(a*d - b*c)**3 - 16*B*a**3*b*c*d**6*(3*A + 11*B)/(a* 
d - b*c)**3 + 24*B*a**2*b**2*c**2*d**5*(3*A + 11*B)/(a*d - b*c)**3 - 16*B* 
a*b**3*c**3*d**4*(3*A + 11*B)/(a*d - b*c)**3 + 4*B*b**4*c**4*d**3*(3*A + 1 
1*B)/(a*d - b*c)**3)/(24*A*B*b*d**4 + 88*B**2*b*d**4))/(9*b*g**4*(a*d - b* 
c)**3) + (3*B**2*a**2*c*d**2 + 3*B**2*a**2*d**3*x - 3*B**2*a*b*c**2*d + 3* 
B**2*a*b*d**3*x**2 + B**2*b**2*c**3 + B**2*b**2*d**3*x**3)*log(e*(a + b*x) 
**2/(c + d*x)**2)**2/(3*a**6*d**3*g**4 - 9*a**5*b*c*d**2*g**4 + 9*a**5*b*d 
**3*g**4*x + 9*a**4*b**2*c**2*d*g**4 - 27*a**4*b**2*c*d**2*g**4*x + 9*a**4 
*b**2*d**3*g**4*x**2 - 3*a**3*b**3*c**3*g**4 + 27*a**3*b**3*c**2*d*g**4*x 
- 27*a**3*b**3*c*d**2*g**4*x**2 + 3*a**3*b**3*d**3*g**4*x**3 - 9*a**2*b**4 
*c**3*g**4*x + 27*a**2*b**4*c**2*d*g**4*x**2 - 9*a**2*b**4*c*d**2*g**4*x** 
3 - 9*a*b**5*c**3*g**4*x**2 + 9*a*b**5*c**2*d*g**4*x**3 - 3*b**6*c**3*g**4 
*x**3) + (-6*A*B*a**2*d**2 + 12*A*B*a*b*c*d - 6*A*B*b**2*c**2 - 22*B**2...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1575 vs. \(2 (423) = 846\).

Time = 0.15 (sec) , antiderivative size = 1575, normalized size of antiderivative = 3.67 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-2/27*(3*((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))*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)) + (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) 
)/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3 
*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^ 
3*g^4)*x^3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 
- a^4*b^3*d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4* 
b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2 - 2/9*A*B*((6*b^2*d^2*x^2 + 2*...
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 29.07 (sec) , antiderivative size = 1069, normalized size of antiderivative = 2.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

((9*A^2*a^2*d^2 + 9*A^2*b^2*c^2 + 170*B^2*a^2*d^2 + 8*B^2*b^2*c^2 + 66*A*B 
*a^2*d^2 + 12*A*B*b^2*c^2 - 18*A^2*a*b*c*d - 46*B^2*a*b*c*d - 42*A*B*a*b*c 
*d)/(3*(a*d - b*c)) + (2*x*(49*B^2*a*b*d^2 - 5*B^2*b^2*c*d + 15*A*B*a*b*d^ 
2 - 3*A*B*b^2*c*d))/(a*d - b*c) + (4*d*x^2*(11*B^2*b^2*d + 3*A*B*b^2*d))/( 
a*d - b*c))/(x*(27*a^2*b^3*c*g^4 - 27*a^3*b^2*d*g^4) - x^2*(27*a^2*b^3*d*g 
^4 - 27*a*b^4*c*g^4) + x^3*(9*b^5*c*g^4 - 9*a*b^4*d*g^4) + 9*a^3*b^2*c*g^4 
 - 9*a^4*b*d*g^4) - log((e*(a + b*x)^2)/(c + d*x)^2)^2*(B^2/(3*b^2*g^4*(3* 
a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2)) - (B^2*d^3)/(3*b*g^4*(a^3*d^3 - b^3* 
c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (log((e*(a + b*x)^2)/(c + d*x)^2) 
*((2*A*B)/(3*b^2*d*g^4) + (2*B^2*d^3*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d) 
/(3*b*d^3) + (2*a*(a*d - b*c))/(3*b*d^2)) + (2*(3*a^3*d^3 - b^3*c^3 + 4*a* 
b^2*c^2*d - 6*a^2*b*c*d^2))/(3*b*d^4)))/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a* 
b^2*c^2*d - 3*a^2*b*c*d^2)) - (2*B^2*d^3*x^2*((2*(b^2*c - a*b*d))/(3*d^2) 
- (4*b*(a*d - b*c))/(3*d^2)))/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d 
- 3*a^2*b*c*d^2)) + (2*B^2*d^3*x*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(3* 
b*d^3) + (2*a*(a*d - b*c))/(3*b*d^2)) + (2*(3*a^2*d^2 + b^2*c^2 - 4*a*b*c* 
d))/(3*d^3) + (4*a*(a*d - b*c))/(3*d^2)))/(3*b*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*a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + ( 
3*a*b*x^2)/d) - (B*d^3*atan((B*d^3*((b^4*c^3*g^4 + a^3*b*d^3*g^4 - a*b^3*c 
^2*d*g^4 - a^2*b^2*c*d^2*g^4)/(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b^2*c*...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1896, normalized size of antiderivative = 4.42 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(36*log(a + b*x)*a**5*b*d**3 + 108*log(a + b*x)*a**4*b**2*d**3*x + 108*log 
(a + b*x)*a**4*b**2*d**3 + 24*log(a + b*x)*a**3*b**3*c*d**2 + 108*log(a + 
b*x)*a**3*b**3*d**3*x**2 + 324*log(a + b*x)*a**3*b**3*d**3*x + 72*log(a + 
b*x)*a**2*b**4*c*d**2*x + 36*log(a + b*x)*a**2*b**4*d**3*x**3 + 324*log(a 
+ b*x)*a**2*b**4*d**3*x**2 + 72*log(a + b*x)*a*b**5*c*d**2*x**2 + 108*log( 
a + b*x)*a*b**5*d**3*x**3 + 24*log(a + b*x)*b**6*c*d**2*x**3 - 36*log(c + 
d*x)*a**5*b*d**3 - 108*log(c + d*x)*a**4*b**2*d**3*x - 108*log(c + d*x)*a* 
*4*b**2*d**3 - 24*log(c + d*x)*a**3*b**3*c*d**2 - 108*log(c + d*x)*a**3*b* 
*3*d**3*x**2 - 324*log(c + d*x)*a**3*b**3*d**3*x - 72*log(c + d*x)*a**2*b* 
*4*c*d**2*x - 36*log(c + d*x)*a**2*b**4*d**3*x**3 - 324*log(c + d*x)*a**2* 
b**4*d**3*x**2 - 72*log(c + d*x)*a*b**5*c*d**2*x**2 - 108*log(c + d*x)*a*b 
**5*d**3*x**3 - 24*log(c + d*x)*b**6*c*d**2*x**3 + 27*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**3*c*d**2 + 27* 
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**3*d**3*x - 27*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**2*b**4*c**2*d + 27*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**2*b**4*d**3*x**2 + 9*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*b**5*c**3 + 
9*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*b**5*d**3*x**3 - 18*log((a**2*e + 2*a*b*e*x + b**2*e*x**2)/(c**2 + 2*...