\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^7} \, dx\) [83]

Optimal result

Integrand size = 42, antiderivative size = 463 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=-\frac {B^2 d^2 i^3 (c+d x)^4}{32 (b c-a d)^3 g^7 (a+b x)^4}+\frac {4 b B^2 d i^3 (c+d x)^5}{125 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B^2 i^3 (c+d x)^6}{108 (b c-a d)^3 g^7 (a+b x)^6}-\frac {B d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^3 g^7 (a+b x)^4}+\frac {4 b B d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 B i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{18 (b c-a d)^3 g^7 (a+b x)^6}-\frac {d^2 i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^3 g^7 (a+b x)^4}+\frac {2 b d i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 (b c-a d)^3 g^7 (a+b x)^5}-\frac {b^2 i^3 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{6 (b c-a d)^3 g^7 (a+b x)^6} \] Output:

-1/32*B^2*d^2*i^3*(d*x+c)^4/(-a*d+b*c)^3/g^7/(b*x+a)^4+4/125*b*B^2*d*i^3*( 
d*x+c)^5/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/108*b^2*B^2*i^3*(d*x+c)^6/(-a*d+b*c) 
^3/g^7/(b*x+a)^6-1/8*B*d^2*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d 
+b*c)^3/g^7/(b*x+a)^4+4/25*b*B*d*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c))) 
/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/18*b^2*B*i^3*(d*x+c)^6*(A+B*ln(e*(b*x+a)/(d* 
x+c)))/(-a*d+b*c)^3/g^7/(b*x+a)^6-1/4*d^2*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/ 
(d*x+c)))^2/(-a*d+b*c)^3/g^7/(b*x+a)^4+2/5*b*d*i^3*(d*x+c)^5*(A+B*ln(e*(b* 
x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^7/(b*x+a)^5-1/6*b^2*i^3*(d*x+c)^6*(A+B*ln( 
e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^7/(b*x+a)^6
 

Mathematica [C] (verified)

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

Time = 5.83 (sec) , antiderivative size = 2606, normalized size of antiderivative = 5.63 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Result too large to show} \] Input:

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + 
b*g*x)^7,x]
 

Output:

(i^3*(-6000*A*B*(b*c - a*d)^6 - 1000*B^2*(b*c - a*d)^6 + 25920*a*A*B*d*(-( 
b*c) + a*d)^5 + 5184*a*B^2*d*(-(b*c) + a*d)^5 - 25920*A*b*B*d*(b*c - a*d)^ 
5*x - 5184*b*B^2*d*(b*c - a*d)^5*x + 32400*a*A*B*d^2*(b*c - a*d)^4*(a + b* 
x) + 14580*a*B^2*d^2*(b*c - a*d)^4*(a + b*x) + 7200*A*B*d*(b*c - a*d)^5*(a 
 + b*x) + 2640*B^2*d*(b*c - a*d)^5*(a + b*x) + 32400*A*b*B*d^2*(b*c - a*d) 
^4*x*(a + b*x) + 14580*b*B^2*d^2*(b*c - a*d)^4*x*(a + b*x) - 49500*A*B*d^2 
*(b*c - a*d)^4*(a + b*x)^2 - 15675*B^2*d^2*(b*c - a*d)^4*(a + b*x)^2 + 432 
00*a*A*B*d^3*(-(b*c) + a*d)^3*(a + b*x)^2 + 33840*a*B^2*d^3*(-(b*c) + a*d) 
^3*(a + b*x)^2 - 43200*A*b*B*d^3*(b*c - a*d)^3*x*(a + b*x)^2 - 33840*b*B^2 
*d^3*(b*c - a*d)^3*x*(a + b*x)^2 + 64800*a*A*B*d^4*(b*c - a*d)^2*(a + b*x) 
^3 + 83160*a*B^2*d^4*(b*c - a*d)^2*(a + b*x)^3 + 42000*A*B*d^3*(b*c - a*d) 
^3*(a + b*x)^3 + 34900*B^2*d^3*(b*c - a*d)^3*(a + b*x)^3 + 64800*A*b*B*d^4 
*(b*c - a*d)^2*x*(a + b*x)^3 + 83160*b*B^2*d^4*(b*c - a*d)^2*x*(a + b*x)^3 
 - 63000*A*B*d^4*(b*c - a*d)^2*(a + b*x)^4 - 83850*B^2*d^4*(b*c - a*d)^2*( 
a + b*x)^4 + 129600*a*A*B*d^5*(-(b*c) + a*d)*(a + b*x)^4 + 295920*a*B^2*d^ 
5*(-(b*c) + a*d)*(a + b*x)^4 - 129600*A*b*B*d^5*(b*c - a*d)*x*(a + b*x)^4 
- 295920*b*B^2*d^5*(b*c - a*d)*x*(a + b*x)^4 + 126000*A*B*d^5*(b*c - a*d)* 
(a + b*x)^5 + 293700*B^2*d^5*(b*c - a*d)*(a + b*x)^5 - 129600*a*A*B*d^6*(a 
 + b*x)^5*Log[a + b*x] - 295920*a*B^2*d^6*(a + b*x)^5*Log[a + b*x] - 12960 
0*A*b*B*d^6*x*(a + b*x)^5*Log[a + b*x] - 295920*b*B^2*d^6*x*(a + b*x)^5...
 

Rubi [A] (verified)

Time = 0.60 (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.

\(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^7} \, dx\)

\(\Big \downarrow \) 2962

\(\displaystyle \frac {i^3 \int \frac {(c+d x)^7 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^7}d\frac {a+b x}{c+d x}}{g^7 (b c-a d)^3}\)

\(\Big \downarrow \) 2795

\(\displaystyle \frac {i^3 \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^7}{(a+b x)^7}-\frac {2 b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^6}{(a+b x)^6}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^5}{(a+b x)^5}\right )d\frac {a+b x}{c+d x}}{g^7 (b c-a d)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^3 \left (-\frac {b^2 (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{6 (a+b x)^6}-\frac {b^2 B (c+d x)^6 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{18 (a+b x)^6}-\frac {d^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}-\frac {B d^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 (a+b x)^4}+\frac {2 b d (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 (a+b x)^5}+\frac {4 b B d (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{25 (a+b x)^5}-\frac {b^2 B^2 (c+d x)^6}{108 (a+b x)^6}-\frac {B^2 d^2 (c+d x)^4}{32 (a+b x)^4}+\frac {4 b B^2 d (c+d x)^5}{125 (a+b x)^5}\right )}{g^7 (b c-a d)^3}\)

Input:

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x) 
^7,x]
 

Output:

(i^3*(-1/32*(B^2*d^2*(c + d*x)^4)/(a + b*x)^4 + (4*b*B^2*d*(c + d*x)^5)/(1 
25*(a + b*x)^5) - (b^2*B^2*(c + d*x)^6)/(108*(a + b*x)^6) - (B*d^2*(c + d* 
x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(a + b*x)^4) + (4*b*B*d*(c + 
 d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(25*(a + b*x)^5) - (b^2*B*(c 
 + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(18*(a + b*x)^6) - (d^2*(c 
 + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(a + b*x)^4) + (2*b*d 
*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(a + b*x)^5) - (b^ 
2*(c + d*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(6*(a + b*x)^6)))/(( 
b*c - a*d)^3*g^7)
 

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. \(1018\) vs. \(2(445)=890\).

Time = 3.20 (sec) , antiderivative size = 1019, normalized size of antiderivative = 2.20

method	result	size
parts	\(\text {Expression too large to display}\)	\(1019\)
derivativedivides	\(\text {Expression too large to display}\)	\(1082\)
default	\(\text {Expression too large to display}\)	\(1082\)
orering	\(\text {Expression too large to display}\)	\(2152\)
norman	\(\text {Expression too large to display}\)	\(2555\)
parallelrisch	\(\text {Expression too large to display}\)	\(2803\)
risch	\(\text {Expression too large to display}\)	\(6548\)

Input:

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x,method=_RE 
TURNVERBOSE)
 

Output:

i^3*A^2/g^7*(-3/5*d*(a^2*d^2-2*a*b*c*d+b^2*c^2)/b^4/(b*x+a)^5-1/6*(-a^3*d^ 
3+3*a^2*b*c*d^2-3*a*b^2*c^2*d+b^3*c^3)/b^4/(b*x+a)^6+3/4*d^2*(a*d-b*c)/b^4 
/(b*x+a)^4-1/3*d^3/b^4/(b*x+a)^3)-i^3*B^2/g^7/d^5*(a*d-b*c)^4*e^4*(d^7/(a* 
d-b*c)^7*(-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*d^6/(a*d-b*c)^7*b*e*(-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)+d^5/(a*d-b*c)^7*b^2*e^2*(-1/6/(b*e/d+(a*d-b*c)*e/d/(d*x 
+c))^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/18/(b*e/d+(a*d-b*c)*e/d/(d*x+c) 
)^6*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/108/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^6) 
)-2*i^3*A*B/g^7/d^5*(a*d-b*c)^4*e^4*(d^7/(a*d-b*c)^7*(-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)-2*d^6/(a*d-b*c)^7*b*e*(-1/5/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5*l 
n(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)+d^5/( 
a*d-b*c)^7*b^2*e^2*(-1/6/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^6*ln(b*e/d+(a*d-b*c 
)*e/d/(d*x+c))-1/36/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^6))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1610 vs. \(2 (445) = 890\).

Time = 0.12 (sec) , antiderivative size = 1610, normalized size of antiderivative = 3.48 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x, al 
gorithm="fricas")
 

Output:

-1/108000*(60*((60*A*B + 37*B^2)*b^6*c*d^5 - (60*A*B + 37*B^2)*a*b^5*d^6)* 
i^3*x^5 - 30*((60*A*B - 23*B^2)*b^6*c^2*d^4 - 36*(20*A*B + 9*B^2)*a*b^5*c* 
d^5 + (660*A*B + 347*B^2)*a^2*b^4*d^6)*i^3*x^4 + 20*((1800*A^2 + 60*A*B - 
53*B^2)*b^6*c^3*d^3 - 27*(200*A^2 + 20*A*B - 11*B^2)*a*b^5*c^2*d^4 + 675*( 
8*A^2 + 4*A*B + B^2)*a^2*b^4*c*d^5 - (1800*A^2 + 2220*A*B + 919*B^2)*a^3*b 
^3*d^6)*i^3*x^3 + 15*((5400*A^2 + 1140*A*B + 73*B^2)*b^6*c^4*d^2 - 72*(200 
*A^2 + 60*A*B + 7*B^2)*a*b^5*c^3*d^3 + 1350*(8*A^2 + 4*A*B + B^2)*a^2*b^4* 
c^2*d^4 - (1800*A^2 + 2220*A*B + 919*B^2)*a^4*b^2*d^6)*i^3*x^2 + 6*(8*(135 
0*A^2 + 390*A*B + 53*B^2)*b^6*c^5*d - 45*(600*A^2 + 220*A*B + 39*B^2)*a*b^ 
5*c^4*d^2 + 2250*(8*A^2 + 4*A*B + B^2)*a^2*b^4*c^3*d^3 - (1800*A^2 + 2220* 
A*B + 919*B^2)*a^5*b*d^6)*i^3*x + (1000*(18*A^2 + 6*A*B + B^2)*b^6*c^6 - 1 
728*(25*A^2 + 10*A*B + 2*B^2)*a*b^5*c^5*d + 3375*(8*A^2 + 4*A*B + B^2)*a^2 
*b^4*c^4*d^2 - (1800*A^2 + 2220*A*B + 919*B^2)*a^6*d^6)*i^3 + 1800*(B^2*b^ 
6*d^6*i^3*x^6 + 6*B^2*a*b^5*d^6*i^3*x^5 + 15*B^2*a^2*b^4*d^6*i^3*x^4 + 20* 
(B^2*b^6*c^3*d^3 - 3*B^2*a*b^5*c^2*d^4 + 3*B^2*a^2*b^4*c*d^5)*i^3*x^3 + 15 
*(3*B^2*b^6*c^4*d^2 - 8*B^2*a*b^5*c^3*d^3 + 6*B^2*a^2*b^4*c^2*d^4)*i^3*x^2 
 + 6*(6*B^2*b^6*c^5*d - 15*B^2*a*b^5*c^4*d^2 + 10*B^2*a^2*b^4*c^3*d^3)*i^3 
*x + (10*B^2*b^6*c^6 - 24*B^2*a*b^5*c^5*d + 15*B^2*a^2*b^4*c^4*d^2)*i^3)*l 
og((b*e*x + a*e)/(d*x + c))^2 + 60*((60*A*B + 37*B^2)*b^6*d^6*i^3*x^6 + 6* 
(10*B^2*b^6*c*d^5 + 3*(20*A*B + 9*B^2)*a*b^5*d^6)*i^3*x^5 - 15*(2*B^2*b...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Timed out} \] Input:

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**7,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20330 vs. \(2 (445) = 890\).

Time = 1.83 (sec) , antiderivative size = 20330, normalized size of antiderivative = 43.91 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x, al 
gorithm="maxima")
 

Output:

-1/10*(6*b*x + a)*B^2*c^2*d*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^ 
8*g^7*x^6 + 6*a*b^7*g^7*x^5 + 15*a^2*b^6*g^7*x^4 + 20*a^3*b^5*g^7*x^3 + 15 
*a^4*b^4*g^7*x^2 + 6*a^5*b^3*g^7*x + a^6*b^2*g^7) - 1/20*(15*b^2*x^2 + 6*a 
*b*x + a^2)*B^2*c*d^2*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^9*g^7* 
x^6 + 6*a*b^8*g^7*x^5 + 15*a^2*b^7*g^7*x^4 + 20*a^3*b^6*g^7*x^3 + 15*a^4*b 
^5*g^7*x^2 + 6*a^5*b^4*g^7*x + a^6*b^3*g^7) - 1/60*(20*b^3*x^3 + 15*a*b^2* 
x^2 + 6*a^2*b*x + a^3)*B^2*d^3*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/ 
(b^10*g^7*x^6 + 6*a*b^9*g^7*x^5 + 15*a^2*b^8*g^7*x^4 + 20*a^3*b^7*g^7*x^3 
+ 15*a^4*b^6*g^7*x^2 + 6*a^5*b^5*g^7*x + a^6*b^4*g^7) + 1/10800*(60*((60*b 
^5*d^5*x^5 - 10*b^5*c^5 + 62*a*b^4*c^4*d - 163*a^2*b^3*c^3*d^2 + 237*a^3*b 
^2*c^2*d^3 - 213*a^4*b*c*d^4 + 147*a^5*d^5 - 30*(b^5*c*d^4 - 11*a*b^4*d^5) 
*x^4 + 20*(b^5*c^2*d^3 - 8*a*b^4*c*d^4 + 37*a^2*b^3*d^5)*x^3 - 15*(b^5*c^3 
*d^2 - 7*a*b^4*c^2*d^3 + 23*a^2*b^3*c*d^4 - 57*a^3*b^2*d^5)*x^2 + 6*(2*b^5 
*c^4*d - 13*a*b^4*c^3*d^2 + 37*a^2*b^3*c^2*d^3 - 63*a^3*b^2*c*d^4 + 87*a^4 
*b*d^5)*x)/((b^12*c^5 - 5*a*b^11*c^4*d + 10*a^2*b^10*c^3*d^2 - 10*a^3*b^9* 
c^2*d^3 + 5*a^4*b^8*c*d^4 - a^5*b^7*d^5)*g^7*x^6 + 6*(a*b^11*c^5 - 5*a^2*b 
^10*c^4*d + 10*a^3*b^9*c^3*d^2 - 10*a^4*b^8*c^2*d^3 + 5*a^5*b^7*c*d^4 - a^ 
6*b^6*d^5)*g^7*x^5 + 15*(a^2*b^10*c^5 - 5*a^3*b^9*c^4*d + 10*a^4*b^8*c^3*d 
^2 - 10*a^5*b^7*c^2*d^3 + 5*a^6*b^6*c*d^4 - a^7*b^5*d^5)*g^7*x^4 + 20*(a^3 
*b^9*c^5 - 5*a^4*b^8*c^4*d + 10*a^5*b^7*c^3*d^2 - 10*a^6*b^6*c^2*d^3 + ...
 

Giac [A] (verification not implemented)

Time = 0.50 (sec) , antiderivative size = 780, normalized size of antiderivative = 1.68 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx =\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x, al 
gorithm="giac")
 

Output:

-1/108000*(1800*(10*B^2*b^2*e^7*i^3 - 24*(b*e*x + a*e)*B^2*b*d*e^6*i^3/(d* 
x + c) + 15*(b*e*x + a*e)^2*B^2*d^2*e^5*i^3/(d*x + c)^2)*log((b*e*x + a*e) 
/(d*x + c))^2/((b*e*x + a*e)^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*e*x + a*e)^6 
*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e)^6*a^2*d^2*g^7/(d*x + c)^6) + 60*( 
600*A*B*b^2*e^7*i^3 + 100*B^2*b^2*e^7*i^3 - 1440*(b*e*x + a*e)*A*B*b*d*e^6 
*i^3/(d*x + c) - 288*(b*e*x + a*e)*B^2*b*d*e^6*i^3/(d*x + c) + 900*(b*e*x 
+ a*e)^2*A*B*d^2*e^5*i^3/(d*x + c)^2 + 225*(b*e*x + a*e)^2*B^2*d^2*e^5*i^3 
/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^6*b^2*c^2*g^7/(d 
*x + c)^6 - 2*(b*e*x + a*e)^6*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e)^6*a^ 
2*d^2*g^7/(d*x + c)^6) + (18000*A^2*b^2*e^7*i^3 + 6000*A*B*b^2*e^7*i^3 + 1 
000*B^2*b^2*e^7*i^3 - 43200*(b*e*x + a*e)*A^2*b*d*e^6*i^3/(d*x + c) - 1728 
0*(b*e*x + a*e)*A*B*b*d*e^6*i^3/(d*x + c) - 3456*(b*e*x + a*e)*B^2*b*d*e^6 
*i^3/(d*x + c) + 27000*(b*e*x + a*e)^2*A^2*d^2*e^5*i^3/(d*x + c)^2 + 13500 
*(b*e*x + a*e)^2*A*B*d^2*e^5*i^3/(d*x + c)^2 + 3375*(b*e*x + a*e)^2*B^2*d^ 
2*e^5*i^3/(d*x + c)^2)/((b*e*x + a*e)^6*b^2*c^2*g^7/(d*x + c)^6 - 2*(b*e*x 
 + a*e)^6*a*b*c*d*g^7/(d*x + c)^6 + (b*e*x + a*e)^6*a^2*d^2*g^7/(d*x + c)^ 
6))*(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 = 36.44 (sec) , antiderivative size = 6275, normalized size of antiderivative = 13.55 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx=\text {Too large to display} \] Input:

int(((c*i + d*i*x)^3*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x) 
^7,x)
 

Output:

((1800*A^2*a^5*d^5*i^3 + 18000*A^2*b^5*c^5*i^3 + 919*B^2*a^5*d^5*i^3 + 100 
0*B^2*b^5*c^5*i^3 + 2220*A*B*a^5*d^5*i^3 + 6000*A*B*b^5*c^5*i^3 - 25200*A^ 
2*a*b^4*c^4*d*i^3 + 1800*A^2*a^4*b*c*d^4*i^3 - 2456*B^2*a*b^4*c^4*d*i^3 + 
919*B^2*a^4*b*c*d^4*i^3 + 1800*A^2*a^2*b^3*c^3*d^2*i^3 + 1800*A^2*a^3*b^2* 
c^2*d^3*i^3 + 919*B^2*a^2*b^3*c^3*d^2*i^3 + 919*B^2*a^3*b^2*c^2*d^3*i^3 + 
2220*A*B*a^2*b^3*c^3*d^2*i^3 + 2220*A*B*a^3*b^2*c^2*d^3*i^3 - 11280*A*B*a* 
b^4*c^4*d*i^3 + 2220*A*B*a^4*b*c*d^4*i^3)/(60*(a*d - b*c)) + (x^4*(347*B^2 
*a*b^4*d^5*i^3 + 23*B^2*b^5*c*d^4*i^3 + 660*A*B*a*b^4*d^5*i^3 - 60*A*B*b^5 
*c*d^4*i^3))/(2*(a*d - b*c)) + (x^2*(1800*A^2*a^3*b^2*d^5*i^3 + 919*B^2*a^ 
3*b^2*d^5*i^3 + 5400*A^2*b^5*c^3*d^2*i^3 + 73*B^2*b^5*c^3*d^2*i^3 - 9000*A 
^2*a*b^4*c^2*d^3*i^3 + 1800*A^2*a^2*b^3*c*d^4*i^3 - 431*B^2*a*b^4*c^2*d^3* 
i^3 + 919*B^2*a^2*b^3*c*d^4*i^3 + 2220*A*B*a^3*b^2*d^5*i^3 + 1140*A*B*b^5* 
c^3*d^2*i^3 - 3180*A*B*a*b^4*c^2*d^3*i^3 + 2220*A*B*a^2*b^3*c*d^4*i^3))/(4 
*(a*d - b*c)) + (x^3*(1800*A^2*a^2*b^3*d^5*i^3 + 919*B^2*a^2*b^3*d^5*i^3 + 
 1800*A^2*b^5*c^2*d^3*i^3 - 53*B^2*b^5*c^2*d^3*i^3 - 3600*A^2*a*b^4*c*d^4* 
i^3 + 244*B^2*a*b^4*c*d^4*i^3 + 2220*A*B*a^2*b^3*d^5*i^3 + 60*A*B*b^5*c^2* 
d^3*i^3 - 480*A*B*a*b^4*c*d^4*i^3))/(3*(a*d - b*c)) + (x*(1800*A^2*a^4*b*d 
^5*i^3 + 919*B^2*a^4*b*d^5*i^3 + 10800*A^2*b^5*c^4*d*i^3 + 424*B^2*b^5*c^4 
*d*i^3 - 16200*A^2*a*b^4*c^3*d^2*i^3 + 1800*A^2*a^3*b^2*c*d^4*i^3 - 1331*B 
^2*a*b^4*c^3*d^2*i^3 + 919*B^2*a^3*b^2*c*d^4*i^3 + 2220*A*B*a^4*b*d^5*i...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 3828, normalized size of antiderivative = 8.27 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^7} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^7,x)
 

Output:

(i*( - 3600*log(a + b*x)*a**8*b*d**6 - 21600*log(a + b*x)*a**7*b**2*d**6*x 
 - 1620*log(a + b*x)*a**7*b**2*d**6 - 600*log(a + b*x)*a**6*b**3*c*d**5 - 
54000*log(a + b*x)*a**6*b**3*d**6*x**2 - 9720*log(a + b*x)*a**6*b**3*d**6* 
x - 3600*log(a + b*x)*a**5*b**4*c*d**5*x - 72000*log(a + b*x)*a**5*b**4*d* 
*6*x**3 - 24300*log(a + b*x)*a**5*b**4*d**6*x**2 - 9000*log(a + b*x)*a**4* 
b**5*c*d**5*x**2 - 54000*log(a + b*x)*a**4*b**5*d**6*x**4 - 32400*log(a + 
b*x)*a**4*b**5*d**6*x**3 - 12000*log(a + b*x)*a**3*b**6*c*d**5*x**3 - 2160 
0*log(a + b*x)*a**3*b**6*d**6*x**5 - 24300*log(a + b*x)*a**3*b**6*d**6*x** 
4 - 9000*log(a + b*x)*a**2*b**7*c*d**5*x**4 - 3600*log(a + b*x)*a**2*b**7* 
d**6*x**6 - 9720*log(a + b*x)*a**2*b**7*d**6*x**5 - 3600*log(a + b*x)*a*b* 
*8*c*d**5*x**5 - 1620*log(a + b*x)*a*b**8*d**6*x**6 - 600*log(a + b*x)*b** 
9*c*d**5*x**6 + 3600*log(c + d*x)*a**8*b*d**6 + 21600*log(c + d*x)*a**7*b* 
*2*d**6*x + 1620*log(c + d*x)*a**7*b**2*d**6 + 600*log(c + d*x)*a**6*b**3* 
c*d**5 + 54000*log(c + d*x)*a**6*b**3*d**6*x**2 + 9720*log(c + d*x)*a**6*b 
**3*d**6*x + 3600*log(c + d*x)*a**5*b**4*c*d**5*x + 72000*log(c + d*x)*a** 
5*b**4*d**6*x**3 + 24300*log(c + d*x)*a**5*b**4*d**6*x**2 + 9000*log(c + d 
*x)*a**4*b**5*c*d**5*x**2 + 54000*log(c + d*x)*a**4*b**5*d**6*x**4 + 32400 
*log(c + d*x)*a**4*b**5*d**6*x**3 + 12000*log(c + d*x)*a**3*b**6*c*d**5*x* 
*3 + 21600*log(c + d*x)*a**3*b**6*d**6*x**5 + 24300*log(c + d*x)*a**3*b**6 
*d**6*x**4 + 9000*log(c + d*x)*a**2*b**7*c*d**5*x**4 + 3600*log(c + d*x...