\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^6} \, dx\) [73]

Optimal result

Integrand size = 42, antiderivative size = 463 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=-\frac {2 B^2 d^2 i^2 (c+d x)^3}{27 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B^2 d i^2 (c+d x)^4}{16 (b c-a d)^3 g^6 (a+b x)^4}-\frac {2 b^2 B^2 i^2 (c+d x)^5}{125 (b c-a d)^3 g^6 (a+b x)^5}-\frac {2 B d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 (b c-a d)^3 g^6 (a+b x)^4}-\frac {2 b^2 B i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 (b c-a d)^3 g^6 (a+b x)^5} \] Output:

-2/27*B^2*d^2*i^2*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/16*b*B^2*d*i^2*(d 
*x+c)^4/(-a*d+b*c)^3/g^6/(b*x+a)^4-2/125*b^2*B^2*i^2*(d*x+c)^5/(-a*d+b*c)^ 
3/g^6/(b*x+a)^5-2/9*B*d^2*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+ 
b*c)^3/g^6/(b*x+a)^3+1/4*b*B*d*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/( 
-a*d+b*c)^3/g^6/(b*x+a)^4-2/25*b^2*B*i^2*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+ 
c)))/(-a*d+b*c)^3/g^6/(b*x+a)^5-1/3*d^2*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d 
*x+c)))^2/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/2*b*d*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+ 
a)/(d*x+c)))^2/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/5*b^2*i^2*(d*x+c)^5*(A+B*ln(e* 
(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^6/(b*x+a)^5
 

Mathematica [C] (verified)

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

Time = 3.20 (sec) , antiderivative size = 2010, normalized size of antiderivative = 4.34 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Result too large to show} \] Input:

Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + 
b*g*x)^6,x]
 

Output:

-1/54000*(i^2*(10800*(b*c - a*d)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 
+ 27000*d*(b*c - a*d)^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 
 18000*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x) 
])^2 + 1000*B*d^2*(a + b*x)^2*(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3 - 18 
*A*d*(b*c - a*d)^2*(a + b*x) - 15*B*d*(b*c - a*d)^2*(a + b*x) + 36*A*d^2*( 
b*c - a*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*(a + b*x)^2 + 36*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 + 12*B*(b*c - a*d)^3*Log[(e*(a + b*x))/(c + d*x)] - 18 
*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 36*B*d^2*(b*c 
- a*d)*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] + 36*B*d^3*(a + b*x)^3*Log 
[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] - 36*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] - 36*B*d^3*(a + b*x)^3*Log[(e*(a + b*x))/ 
(c + d*x)]*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*PolyLo 
g[2, (b*(c + d*x))/(b*c - a*d)]) + 375*B*d*(a + b*x)*(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*(-(...
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.73, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 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.

\(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^6} \, dx\)

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

\(\displaystyle \frac {i^2 \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^6}{(a+b x)^6}-\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^5}{(a+b x)^5}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^4}{(a+b x)^4}\right )d\frac {a+b x}{c+d x}}{g^6 (b c-a d)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^2 \left (-\frac {b^2 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 (a+b x)^5}-\frac {2 b^2 B (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{25 (a+b x)^5}-\frac {d^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 (a+b x)^3}-\frac {2 B d^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 (a+b x)^3}+\frac {b d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^4}+\frac {b B d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 (a+b x)^4}-\frac {2 b^2 B^2 (c+d x)^5}{125 (a+b x)^5}-\frac {2 B^2 d^2 (c+d x)^3}{27 (a+b x)^3}+\frac {b B^2 d (c+d x)^4}{16 (a+b x)^4}\right )}{g^6 (b c-a d)^3}\)

Input:

Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x) 
^6,x]
 

Output:

(i^2*((-2*B^2*d^2*(c + d*x)^3)/(27*(a + b*x)^3) + (b*B^2*d*(c + d*x)^4)/(1 
6*(a + b*x)^4) - (2*b^2*B^2*(c + d*x)^5)/(125*(a + b*x)^5) - (2*B*d^2*(c + 
 d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(a + b*x)^3) + (b*B*d*(c 
+ d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(4*(a + b*x)^4) - (2*b^2*B* 
(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(25*(a + b*x)^5) - (d^2* 
(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(a + b*x)^3) + (b*d 
*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b*x)^4) - (b^ 
2*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(a + b*x)^5)))/(( 
b*c - a*d)^3*g^6)
 

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 2962

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ 
)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + 
 B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; 
 FreeQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGt 
Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I 
ntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(967\) vs. \(2(445)=890\).

Time = 2.51 (sec) , antiderivative size = 968, normalized size of antiderivative = 2.09

method	result	size
parts	\(\text {Expression too large to display}\)	\(968\)
derivativedivides	\(\text {Expression too large to display}\)	\(1082\)
default	\(\text {Expression too large to display}\)	\(1082\)
orering	\(\text {Expression too large to display}\)	\(1819\)
norman	\(\text {Expression too large to display}\)	\(1977\)
parallelrisch	\(\text {Expression too large to display}\)	\(2248\)
risch	\(\text {Expression too large to display}\)	\(5135\)

Input:

int((d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x,method=_RE 
TURNVERBOSE)
 

Output:

i^2*A^2/g^6*(-1/5*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^3/(b*x+a)^5+1/2*d*(a*d-b*c 
)/b^3/(b*x+a)^4-1/3*d^2/b^3/(b*x+a)^3)-i^2*B^2/g^6/d^4*(a*d-b*c)^3*e^3*(d^ 
6/(a*d-b*c)^6*(-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)-2*d^5/(a*d-b*c)^6*b*e*(-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)+d^4/(a*d-b*c)^6*b^2*e^2*(-1/5/(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/25/(b*e/d+(a*d-b*c)*e/d/(d*x 
+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/125/(b*e/d+(a*d-b*c)*e/d/(d*x+c)) 
^5))-2*i^2*A*B/g^6/d^4*(a*d-b*c)^3*e^3*(d^6/(a*d-b*c)^6*(-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)-2*d^5/(a*d-b*c)^6*b*e*(-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)+d^4 
/(a*d-b*c)^6*b^2*e^2*(-1/5/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5*ln(b*e/d+(a*d-b 
*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (445) = 890\).

Time = 0.11 (sec) , antiderivative size = 1323, normalized size of antiderivative = 2.86 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x, al 
gorithm="fricas")
 

Output:

-1/54000*(60*((60*A*B + 47*B^2)*b^5*c*d^4 - (60*A*B + 47*B^2)*a*b^4*d^5)*i 
^2*x^4 - 30*((60*A*B - 13*B^2)*b^5*c^2*d^3 - 50*(12*A*B + 7*B^2)*a*b^4*c*d 
^4 + 3*(180*A*B + 121*B^2)*a^2*b^3*d^5)*i^2*x^3 + 10*(2*(900*A^2 + 60*A*B 
- 43*B^2)*b^5*c^3*d^2 - 75*(72*A^2 + 12*A*B - 5*B^2)*a*b^4*c^2*d^3 + 600*( 
9*A^2 + 6*A*B + 2*B^2)*a^2*b^3*c*d^4 - (1800*A^2 + 2820*A*B + 1489*B^2)*a^ 
3*b^2*d^5)*i^2*x^2 + 5*(27*(200*A^2 + 60*A*B + 7*B^2)*b^5*c^4*d - 100*(144 
*A^2 + 60*A*B + 11*B^2)*a*b^4*c^3*d^2 + 1200*(9*A^2 + 6*A*B + 2*B^2)*a^2*b 
^3*c^2*d^3 - (1800*A^2 + 2820*A*B + 1489*B^2)*a^4*b*d^5)*i^2*x + (432*(25* 
A^2 + 10*A*B + 2*B^2)*b^5*c^5 - 3375*(8*A^2 + 4*A*B + B^2)*a*b^4*c^4*d + 2 
000*(9*A^2 + 6*A*B + 2*B^2)*a^2*b^3*c^3*d^2 - (1800*A^2 + 2820*A*B + 1489* 
B^2)*a^5*d^5)*i^2 + 1800*(B^2*b^5*d^5*i^2*x^5 + 5*B^2*a*b^4*d^5*i^2*x^4 + 
10*B^2*a^2*b^3*d^5*i^2*x^3 + 10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2*d^3 + 3 
*B^2*a^2*b^3*c*d^4)*i^2*x^2 + 5*(3*B^2*b^5*c^4*d - 8*B^2*a*b^4*c^3*d^2 + 6 
*B^2*a^2*b^3*c^2*d^3)*i^2*x + (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2 
*a^2*b^3*c^3*d^2)*i^2)*log((b*e*x + a*e)/(d*x + c))^2 + 60*((60*A*B + 47*B 
^2)*b^5*d^5*i^2*x^5 + 5*(12*B^2*b^5*c*d^4 + 5*(12*A*B + 7*B^2)*a*b^4*d^5)* 
i^2*x^4 - 10*(3*B^2*b^5*c^2*d^3 - 30*B^2*a*b^4*c*d^4 - 20*(3*A*B + B^2)*a^ 
2*b^3*d^5)*i^2*x^3 + 10*(2*(30*A*B + B^2)*b^5*c^3*d^2 - 15*(12*A*B + B^2)* 
a*b^4*c^2*d^3 + 60*(3*A*B + B^2)*a^2*b^3*c*d^4)*i^2*x^2 + 5*(9*(20*A*B + 3 
*B^2)*b^5*c^4*d - 20*(24*A*B + 5*B^2)*a*b^4*c^3*d^2 + 120*(3*A*B + B^2)...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10880 vs. \(2 (445) = 890\).

Time = 0.90 (sec) , antiderivative size = 10880, normalized size of antiderivative = 23.50 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x, al 
gorithm="maxima")
 

Output:

-1/10*(5*b*x + a)*B^2*c*d*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^7* 
g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^ 
4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(10*b^2*x^2 + 5*a*b*x + a^2)*B^2*d^2*i^2 
*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 1 
0*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 
1/9000*(60*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^ 
2*d^2 - 163*a^3*b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 
 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3*b^4*c^3*d 
 - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a 
*b^9*c^3*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 
5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5 
*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 
- 4*a^5*b^5*c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3 
*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2 + 5*(a^4*b 
^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d 
^4)*g^6*x + (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2 
*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b^5*c^4*d + 
 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^ 
6) - 60*d^5*log(d*x + c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 
10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6))*log(b*e*x/(d*x ...
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.68 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x, al 
gorithm="giac")
 

Output:

-1/54000*(1800*(6*B^2*b^2*e^6*i^2 - 15*(b*e*x + a*e)*B^2*b*d*e^5*i^2/(d*x 
+ c) + 10*(b*e*x + a*e)^2*B^2*d^2*e^4*i^2/(d*x + c)^2)*log((b*e*x + a*e)/( 
d*x + c))^2/((b*e*x + a*e)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*e*x + a*e)^5*a 
*b*c*d*g^6/(d*x + c)^5 + (b*e*x + a*e)^5*a^2*d^2*g^6/(d*x + c)^5) + 60*(36 
0*A*B*b^2*e^6*i^2 + 72*B^2*b^2*e^6*i^2 - 900*(b*e*x + a*e)*A*B*b*d*e^5*i^2 
/(d*x + c) - 225*(b*e*x + a*e)*B^2*b*d*e^5*i^2/(d*x + c) + 600*(b*e*x + a* 
e)^2*A*B*d^2*e^4*i^2/(d*x + c)^2 + 200*(b*e*x + a*e)^2*B^2*d^2*e^4*i^2/(d* 
x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^5*b^2*c^2*g^6/(d*x + 
 c)^5 - 2*(b*e*x + a*e)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*e*x + a*e)^5*a^2*d^ 
2*g^6/(d*x + c)^5) + (10800*A^2*b^2*e^6*i^2 + 4320*A*B*b^2*e^6*i^2 + 864*B 
^2*b^2*e^6*i^2 - 27000*(b*e*x + a*e)*A^2*b*d*e^5*i^2/(d*x + c) - 13500*(b* 
e*x + a*e)*A*B*b*d*e^5*i^2/(d*x + c) - 3375*(b*e*x + a*e)*B^2*b*d*e^5*i^2/ 
(d*x + c) + 18000*(b*e*x + a*e)^2*A^2*d^2*e^4*i^2/(d*x + c)^2 + 12000*(b*e 
*x + a*e)^2*A*B*d^2*e^4*i^2/(d*x + c)^2 + 4000*(b*e*x + a*e)^2*B^2*d^2*e^4 
*i^2/(d*x + c)^2)/((b*e*x + a*e)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*e*x + a* 
e)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*e*x + a*e)^5*a^2*d^2*g^6/(d*x + c)^5))*( 
b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))
 

Mupad [B] (verification not implemented)

Time = 35.88 (sec) , antiderivative size = 3434, normalized size of antiderivative = 7.42 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x) 
^6,x)
 

Output:

((1800*A^2*a^4*d^4*i^2 + 10800*A^2*b^4*c^4*i^2 + 1489*B^2*a^4*d^4*i^2 + 86 
4*B^2*b^4*c^4*i^2 + 2820*A*B*a^4*d^4*i^2 + 4320*A*B*b^4*c^4*i^2 - 16200*A^ 
2*a*b^3*c^3*d*i^2 + 1800*A^2*a^3*b*c*d^3*i^2 - 2511*B^2*a*b^3*c^3*d*i^2 + 
1489*B^2*a^3*b*c*d^3*i^2 + 1800*A^2*a^2*b^2*c^2*d^2*i^2 + 1489*B^2*a^2*b^2 
*c^2*d^2*i^2 + 2820*A*B*a^2*b^2*c^2*d^2*i^2 - 9180*A*B*a*b^3*c^3*d*i^2 + 2 
820*A*B*a^3*b*c*d^3*i^2)/(60*(a*d - b*c)) + (x^3*(363*B^2*a*b^3*d^4*i^2 + 
13*B^2*b^4*c*d^3*i^2 + 540*A*B*a*b^3*d^4*i^2 - 60*A*B*b^4*c*d^3*i^2))/(2*( 
a*d - b*c)) + (x*(1800*A^2*a^3*b*d^4*i^2 + 1489*B^2*a^3*b*d^4*i^2 + 5400*A 
^2*b^4*c^3*d*i^2 + 189*B^2*b^4*c^3*d*i^2 - 9000*A^2*a*b^3*c^2*d^2*i^2 + 18 
00*A^2*a^2*b^2*c*d^3*i^2 - 911*B^2*a*b^3*c^2*d^2*i^2 + 1489*B^2*a^2*b^2*c* 
d^3*i^2 + 2820*A*B*a^3*b*d^4*i^2 + 1620*A*B*b^4*c^3*d*i^2 - 4380*A*B*a*b^3 
*c^2*d^2*i^2 + 2820*A*B*a^2*b^2*c*d^3*i^2))/(12*(a*d - b*c)) + (x^2*(1800* 
A^2*a^2*b^2*d^4*i^2 + 1489*B^2*a^2*b^2*d^4*i^2 + 1800*A^2*b^4*c^2*d^2*i^2 
- 86*B^2*b^4*c^2*d^2*i^2 - 3600*A^2*a*b^3*c*d^3*i^2 + 289*B^2*a*b^3*c*d^3* 
i^2 + 2820*A*B*a^2*b^2*d^4*i^2 + 120*A*B*b^4*c^2*d^2*i^2 - 780*A*B*a*b^3*c 
*d^3*i^2))/(6*(a*d - b*c)) + (d*x^4*(47*B^2*b^4*d^3*i^2 + 60*A*B*b^4*d^3*i 
^2))/(a*d - b*c))/(x*(4500*a^4*b^5*c*g^6 - 4500*a^5*b^4*d*g^6) - x^4*(4500 
*a^2*b^7*d*g^6 - 4500*a*b^8*c*g^6) + x^5*(900*b^9*c*g^6 - 900*a*b^8*d*g^6) 
 + x^2*(9000*a^3*b^6*c*g^6 - 9000*a^4*b^5*d*g^6) + x^3*(9000*a^2*b^7*c*g^6 
 - 9000*a^3*b^6*d*g^6) + 900*a^5*b^4*c*g^6 - 900*a^6*b^3*d*g^6) - log((...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 3022, normalized size of antiderivative = 6.53 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x)
 

Output:

( - 3600*log(a + b*x)*a**7*b*d**5 - 18000*log(a + b*x)*a**6*b**2*d**5*x - 
2100*log(a + b*x)*a**6*b**2*d**5 - 720*log(a + b*x)*a**5*b**3*c*d**4 - 360 
00*log(a + b*x)*a**5*b**3*d**5*x**2 - 10500*log(a + b*x)*a**5*b**3*d**5*x 
- 3600*log(a + b*x)*a**4*b**4*c*d**4*x - 36000*log(a + b*x)*a**4*b**4*d**5 
*x**3 - 21000*log(a + b*x)*a**4*b**4*d**5*x**2 - 7200*log(a + b*x)*a**3*b* 
*5*c*d**4*x**2 - 18000*log(a + b*x)*a**3*b**5*d**5*x**4 - 21000*log(a + b* 
x)*a**3*b**5*d**5*x**3 - 7200*log(a + b*x)*a**2*b**6*c*d**4*x**3 - 3600*lo 
g(a + b*x)*a**2*b**6*d**5*x**5 - 10500*log(a + b*x)*a**2*b**6*d**5*x**4 - 
3600*log(a + b*x)*a*b**7*c*d**4*x**4 - 2100*log(a + b*x)*a*b**7*d**5*x**5 
- 720*log(a + b*x)*b**8*c*d**4*x**5 + 3600*log(c + d*x)*a**7*b*d**5 + 1800 
0*log(c + d*x)*a**6*b**2*d**5*x + 2100*log(c + d*x)*a**6*b**2*d**5 + 720*l 
og(c + d*x)*a**5*b**3*c*d**4 + 36000*log(c + d*x)*a**5*b**3*d**5*x**2 + 10 
500*log(c + d*x)*a**5*b**3*d**5*x + 3600*log(c + d*x)*a**4*b**4*c*d**4*x + 
 36000*log(c + d*x)*a**4*b**4*d**5*x**3 + 21000*log(c + d*x)*a**4*b**4*d** 
5*x**2 + 7200*log(c + d*x)*a**3*b**5*c*d**4*x**2 + 18000*log(c + d*x)*a**3 
*b**5*d**5*x**4 + 21000*log(c + d*x)*a**3*b**5*d**5*x**3 + 7200*log(c + d* 
x)*a**2*b**6*c*d**4*x**3 + 3600*log(c + d*x)*a**2*b**6*d**5*x**5 + 10500*l 
og(c + d*x)*a**2*b**6*d**5*x**4 + 3600*log(c + d*x)*a*b**7*c*d**4*x**4 + 2 
100*log(c + d*x)*a*b**7*d**5*x**5 + 720*log(c + d*x)*b**8*c*d**4*x**5 - 18 
000*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**5*c**3*d**2 - 54000*log((a*...