\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^5} \, dx\) [105]

Optimal result

Integrand size = 32, antiderivative size = 575 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {2 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{32 (b c-a d)^4 g^5 (a+b x)^4}+\frac {2 B d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \] Output:

2*B^2*d^3*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3/4*b*B^2*d^2*(d*x+c)^2/(-a*d+b 
*c)^4/g^5/(b*x+a)^2+2/9*b^2*B^2*d*(d*x+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/3 
2*b^3*B^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+2*B*d^3*(d*x+c)*(A+B*ln(e*( 
b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*B*d^2*(d*x+c)^2*(A+B*ln(e* 
(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^2+2/3*b^2*B*d*(d*x+c)^3*(A+B*ln 
(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/8*b^3*B*(d*x+c)^4*(A+B*l 
n(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b* 
x+a)/(d*x+c)))^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)/(d*x+c)))^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)/(d*x+c)))^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)/(d*x+c)))^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 = 0.90 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {72 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (36 A (b c-a d)^4+9 B (b c-a d)^4+48 A d (-b c+a d)^3 (a+b x)+28 B d (-b c+a d)^3 (a+b x)+72 A d^2 (b c-a d)^2 (a+b x)^2+78 B d^2 (b c-a d)^2 (a+b x)^2+144 A d^3 (-b c+a d) (a+b x)^3+300 B d^3 (-b c+a d) (a+b x)^3-144 A d^4 (a+b x)^4 \log (a+b x)-300 B d^4 (a+b x)^4 \log (a+b x)+72 B d^4 (a+b x)^4 \log ^2(a+b x)+36 B (b c-a d)^4 \log \left (\frac {e (a+b x)}{c+d x}\right )+48 B d (-b c+a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+72 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+144 B d^3 (-b c+a d) (a+b x)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+144 A d^4 (a+b x)^4 \log (c+d x)+300 B d^4 (a+b x)^4 \log (c+d x)-144 B d^4 (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+144 B d^4 (a+b x)^4 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)+72 B d^4 (a+b x)^4 \log ^2(c+d x)-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{288 b g^5 (a+b x)^4} \] Input:

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

Output:

-1/288*(72*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(36*A*(b*c - a*d)^4 
 + 9*B*(b*c - a*d)^4 + 48*A*d*(-(b*c) + a*d)^3*(a + b*x) + 28*B*d*(-(b*c) 
+ a*d)^3*(a + b*x) + 72*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 + 144*A*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 300*B*d^3*(-(b 
*c) + a*d)*(a + b*x)^3 - 144*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 + 36*B*(b*c - 
 a*d)^4*Log[(e*(a + b*x))/(c + d*x)] + 48*B*d*(-(b*c) + a*d)^3*(a + b*x)*L 
og[(e*(a + b*x))/(c + d*x)] + 72*B*d^2*(b*c - a*d)^2*(a + b*x)^2*Log[(e*(a 
 + b*x))/(c + d*x)] + 144*B*d^3*(-(b*c) + a*d)*(a + b*x)^3*Log[(e*(a + b*x 
))/(c + d*x)] - 144*B*d^4*(a + b*x)^4*Log[a + b*x]*Log[(e*(a + b*x))/(c + 
d*x)] + 144*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] + 144*B*d^4*(a + b*x)^4*Log[(e*(a + b*x))/(c + d*x)]*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*PolyLog[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 = 0.60 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, 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.

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

\(\Big \downarrow \) 2950

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\frac {b^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}-\frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 (a+b x)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a+b x)^3}+\frac {2 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}+\frac {2 B d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}-\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {b^3 B^2 (c+d x)^4}{32 (a+b x)^4}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (a+b x)^3}+\frac {2 B^2 d^3 (c+d x)}{a+b x}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (a+b x)^2}}{g^5 (b c-a d)^4}\)

Input:

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

Output:

((2*B^2*d^3*(c + d*x))/(a + b*x) - (3*b*B^2*d^2*(c + d*x)^2)/(4*(a + b*x)^ 
2) + (2*b^2*B^2*d*(c + d*x)^3)/(9*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(32 
*(a + b*x)^4) + (2*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/( 
a + b*x) - (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2 
*(a + b*x)^2) + (2*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
))/(3*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x) 
]))/(8*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^ 
2)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 
)/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
)^2)/(a + b*x)^3 - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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. \(1178\) vs. \(2(559)=1118\).

Time = 2.01 (sec) , antiderivative size = 1179, normalized size of antiderivative = 2.05

method	result	size
parts	\(\text {Expression too large to display}\)	\(1179\)
orering	\(\text {Expression too large to display}\)	\(1304\)
derivativedivides	\(\text {Expression too large to display}\)	\(1393\)
default	\(\text {Expression too large to display}\)	\(1393\)
norman	\(\text {Expression too large to display}\)	\(1796\)
parallelrisch	\(\text {Expression too large to display}\)	\(2035\)
risch	\(\text {Expression too large to display}\)	\(3080\)

Input:

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

Output:

-1/4*A^2/g^5/(b*x+a)^4/b-B^2/g^5/d^2*(a*d-b*c)*e*(d^5/(a*d-b*c)^5*(-1/(b*e 
/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/(b*e/d+(a*d- 
b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/(b*e/d+(a*d-b*c)*e/d/( 
d*x+c)))-3*d^4/(a*d-b*c)^5*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b* 
e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+ 
(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)+3*d^3/(a*d-b*c 
)^5*b^2*e^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))^2-2/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+ 
c))-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)-d^2/(a*d-b*c)^5*b^3*e^3*(-1/4/(b 
*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/8/(b*e/d 
+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/32/(b*e/d+(a*d 
-b*c)*e/d/(d*x+c))^4))-2*B*A/g^5/d^2*(a*d-b*c)*e*(d^5/(a*d-b*c)^5*(-1/(b*e 
/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b* 
c)*e/d/(d*x+c)))-3*d^4/(a*d-b*c)^5*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c)) 
^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)+3* 
d^3/(a*d-b*c)^5*b^2*e^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a* 
d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)-d^2/(a*d-b*c)^5*b 
^3*e^3*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c 
))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 1035, normalized size of antiderivative = 1.80 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/288*(9*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 32*(9*A^2 + 6*A*B + 2*B^2)*a*b^3 
*c^3*d + 216*(2*A^2 + 2*A*B + B^2)*a^2*b^2*c^2*d^2 - 288*(A^2 + 2*A*B + 2* 
B^2)*a^3*b*c*d^3 + (72*A^2 + 300*A*B + 415*B^2)*a^4*d^4 - 12*((12*A*B + 25 
*B^2)*b^4*c*d^3 - (12*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((12*A*B + 13*B^2)* 
b^4*c^2*d^2 - 16*(6*A*B + 11*B^2)*a*b^3*c*d^3 + (84*A*B + 163*B^2)*a^2*b^2 
*d^4)*x^2 - 72*(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*e*x + a*e)/(d*x + c))^2 - 4*((12*A*B + 
 7*B^2)*b^4*c^3*d - 12*(6*A*B + 5*B^2)*a*b^3*c^2*d^2 + 108*(2*A*B + 3*B^2) 
*a^2*b^2*c*d^3 - (156*A*B + 271*B^2)*a^3*b*d^4)*x - 12*((12*A*B + 25*B^2)* 
b^4*d^4*x^4 - 3*(4*A*B + B^2)*b^4*c^4 + 16*(3*A*B + B^2)*a*b^3*c^3*d - 36* 
(2*A*B + B^2)*a^2*b^2*c^2*d^2 + 48*(A*B + B^2)*a^3*b*c*d^3 + 4*(3*B^2*b^4* 
c*d^3 + 2*(6*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*(2*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 + 12*(A*B + B^2)*a^3*b*d^4)*x)*log((b* 
e*x + a*e)/(d*x + c)))/((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^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*...
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2377 vs. \(2 (534) = 1068\).

Time = 104.99 (sec) , antiderivative size = 2377, normalized size of antiderivative = 4.13 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-B*d**4*(12*A + 25*B)*log(x + (12*A*B*a*d**5 + 12*A*B*b*c*d**4 + 25*B**2*a 
*d**5 + 25*B**2*b*c*d**4 - B*a**5*d**9*(12*A + 25*B)/(a*d - b*c)**4 + 5*B* 
a**4*b*c*d**8*(12*A + 25*B)/(a*d - b*c)**4 - 10*B*a**3*b**2*c**2*d**7*(12* 
A + 25*B)/(a*d - b*c)**4 + 10*B*a**2*b**3*c**3*d**6*(12*A + 25*B)/(a*d - b 
*c)**4 - 5*B*a*b**4*c**4*d**5*(12*A + 25*B)/(a*d - b*c)**4 + B*b**5*c**5*d 
**4*(12*A + 25*B)/(a*d - b*c)**4)/(24*A*B*b*d**5 + 50*B**2*b*d**5))/(24*b* 
g**5*(a*d - b*c)**4) + B*d**4*(12*A + 25*B)*log(x + (12*A*B*a*d**5 + 12*A* 
B*b*c*d**4 + 25*B**2*a*d**5 + 25*B**2*b*c*d**4 + B*a**5*d**9*(12*A + 25*B) 
/(a*d - b*c)**4 - 5*B*a**4*b*c*d**8*(12*A + 25*B)/(a*d - b*c)**4 + 10*B*a* 
*3*b**2*c**2*d**7*(12*A + 25*B)/(a*d - b*c)**4 - 10*B*a**2*b**3*c**3*d**6* 
(12*A + 25*B)/(a*d - b*c)**4 + 5*B*a*b**4*c**4*d**5*(12*A + 25*B)/(a*d - b 
*c)**4 - B*b**5*c**5*d**4*(12*A + 25*B)/(a*d - b*c)**4)/(24*A*B*b*d**5 + 5 
0*B**2*b*d**5))/(24*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)/(c + d*x))**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 + 1...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2123 vs. \(2 (559) = 1118\).

Time = 0.21 (sec) , antiderivative size = 2123, normalized size of antiderivative = 3.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

1/288*(12*((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 + 
 25*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))*l 
og(b*e*x/(d*x + c) + a*e/(d*x + c)) - (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 + 
25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*...
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 1014, normalized size of antiderivative = 1.76 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/288*(72*(B^2*b^3*e^5 - 4*(b*e*x + a*e)*B^2*b^2*d*e^4/(d*x + c) + 6*(b*e 
*x + a*e)^2*B^2*b*d^2*e^3/(d*x + c)^2 - 4*(b*e*x + a*e)^3*B^2*d^3*e^2/(d*x 
 + c)^3)*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x 
+ c)^4 - 3*(b*e*x + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4 
*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*e*x + a*e)^4*a^3*d^3*g^5/(d*x + c)^4) + 
12*(12*A*B*b^3*e^5 + 3*B^2*b^3*e^5 - 48*(b*e*x + a*e)*A*B*b^2*d*e^4/(d*x + 
 c) - 16*(b*e*x + a*e)*B^2*b^2*d*e^4/(d*x + c) + 72*(b*e*x + a*e)^2*A*B*b* 
d^2*e^3/(d*x + c)^2 + 36*(b*e*x + a*e)^2*B^2*b*d^2*e^3/(d*x + c)^2 - 48*(b 
*e*x + a*e)^3*A*B*d^3*e^2/(d*x + c)^3 - 48*(b*e*x + a*e)^3*B^2*d^3*e^2/(d* 
x + c)^3)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x + 
 c)^4 - 3*(b*e*x + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4* 
a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*e*x + a*e)^4*a^3*d^3*g^5/(d*x + c)^4) + ( 
72*A^2*b^3*e^5 + 36*A*B*b^3*e^5 + 9*B^2*b^3*e^5 - 288*(b*e*x + a*e)*A^2*b^ 
2*d*e^4/(d*x + c) - 192*(b*e*x + a*e)*A*B*b^2*d*e^4/(d*x + c) - 64*(b*e*x 
+ a*e)*B^2*b^2*d*e^4/(d*x + c) + 432*(b*e*x + a*e)^2*A^2*b*d^2*e^3/(d*x + 
c)^2 + 432*(b*e*x + a*e)^2*A*B*b*d^2*e^3/(d*x + c)^2 + 216*(b*e*x + a*e)^2 
*B^2*b*d^2*e^3/(d*x + c)^2 - 288*(b*e*x + a*e)^3*A^2*d^3*e^2/(d*x + c)^3 - 
 576*(b*e*x + a*e)^3*A*B*d^3*e^2/(d*x + c)^3 - 576*(b*e*x + a*e)^3*B^2*d^3 
*e^2/(d*x + c)^3)/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*e*x + a* 
e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4*a^2*b*c*d^2*g^5/(d...
 

Mupad [B] (verification not implemented)

Time = 32.45 (sec) , antiderivative size = 1881, normalized size of antiderivative = 3.27 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

(B*d^4*atan((B*d^4*(12*A + 25*B)*(24*b^5*c^4*g^5 - 24*a^4*b*d^4*g^5 - 48*a 
*b^4*c^3*d*g^5 + 48*a^3*b^2*c*d^3*g^5)*1i)/(24*b*g^5*(a*d - b*c)^4*(25*B^2 
*d^4 + 12*A*B*d^4)) + (B*d^5*x*(12*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 + 12*A*B*d^4)))*(12*A + 25*B)*1i)/(12*b*g^5*(a*d - b*c)^4) - log((e*(a 
 + b*x))/(c + d*x))^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))/(c + d*x))* 
((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)/( 
12*b*d^3) + (a*(a*d - b*c))/(4*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)/(12*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)/(4*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)/(12*b*d^3) + (a*(a*d - b*c))/(4*b* 
d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*d^3) + (a*(a*d - b*c))/(2*d^2 
)) - a*((b^2*c - a*b*d)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c^2 + 4* 
a^2*b*d^2 - 5*a*b^2*c*d)/(4*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)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c - a*b^2*d)/(4*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*...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 2134, normalized size of antiderivative = 3.71 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(144*log(a + b*x)*a**6*b*d**4 + 576*log(a + b*x)*a**5*b**2*d**4*x + 264*lo 
g(a + b*x)*a**5*b**2*d**4 + 36*log(a + b*x)*a**4*b**3*c*d**3 + 864*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 + 576*log(a + b*x)*a**3*b**4*d**4*x**3 + 1584*l 
og(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 + 
 144*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 - 144*log(c + d*x)*a**6*b*d**4 - 57 
6*log(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 - 864*log(c + d*x)*a**4*b**3*d**4*x**2 - 1056*l 
og(c + d*x)*a**4*b**3*d**4*x - 144*log(c + d*x)*a**3*b**4*c*d**3*x - 576*l 
og(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 - 144*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 + 288*log((a*e + b*e*x)/(c + d*x))**2*a**4*b**3*c*d**3 + 288*log((a*e 
 + b*e*x)/(c + d*x))**2*a**4*b**3*d**4*x - 432*log((a*e + b*e*x)/(c + d*x) 
)**2*a**3*b**4*c**2*d**2 + 432*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**4*d 
**4*x**2 + 288*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**5*c**3*d + 288*log( 
(a*e + b*e*x)/(c + d*x))**2*a**2*b**5*d**4*x**3 - 72*log((a*e + b*e*x)/...