\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^6} \, dx\) [126]

Optimal result

Integrand size = 43, antiderivative size = 293 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=-\frac {B d^2 i^2 n (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 n (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 n (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \] Output:

-1/9*B*d^2*i^2*n*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/8*b*B*d*i^2*n*(d*x 
+c)^4/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/25*b^2*B*i^2*n*(d*x+c)^5/(-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))^n))/(-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))^n))/ 
(-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 
))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^5
 

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.22 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\frac {i^2 \left (-\frac {360 A b^2 c^2}{(a+b x)^5}+\frac {720 a A b c d}{(a+b x)^5}-\frac {360 a^2 A d^2}{(a+b x)^5}-\frac {72 b^2 B c^2 n}{(a+b x)^5}+\frac {144 a b B c d n}{(a+b x)^5}-\frac {72 a^2 B d^2 n}{(a+b x)^5}-\frac {900 A b c d}{(a+b x)^4}+\frac {900 a A d^2}{(a+b x)^4}-\frac {135 b B c d n}{(a+b x)^4}+\frac {135 a B d^2 n}{(a+b x)^4}-\frac {600 A d^2}{(a+b x)^3}-\frac {20 B d^2 n}{(a+b x)^3}+\frac {30 B d^3 n}{(b c-a d) (a+b x)^2}-\frac {60 B d^4 n}{(b c-a d)^2 (a+b x)}-\frac {60 B d^5 n \log (a+b x)}{(b c-a d)^3}-\frac {60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5}+\frac {60 B d^5 n \log (c+d x)}{(b c-a d)^3}\right )}{1800 b^3 g^6} \] Input:

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

Output:

(i^2*((-360*A*b^2*c^2)/(a + b*x)^5 + (720*a*A*b*c*d)/(a + b*x)^5 - (360*a^ 
2*A*d^2)/(a + b*x)^5 - (72*b^2*B*c^2*n)/(a + b*x)^5 + (144*a*b*B*c*d*n)/(a 
 + b*x)^5 - (72*a^2*B*d^2*n)/(a + b*x)^5 - (900*A*b*c*d)/(a + b*x)^4 + (90 
0*a*A*d^2)/(a + b*x)^4 - (135*b*B*c*d*n)/(a + b*x)^4 + (135*a*B*d^2*n)/(a 
+ b*x)^4 - (600*A*d^2)/(a + b*x)^3 - (20*B*d^2*n)/(a + b*x)^3 + (30*B*d^3* 
n)/((b*c - a*d)*(a + b*x)^2) - (60*B*d^4*n)/((b*c - a*d)^2*(a + b*x)) - (6 
0*B*d^5*n*Log[a + b*x])/(b*c - a*d)^3 - (60*B*(a^2*d^2 + a*b*d*(3*c + 5*d* 
x) + b^2*(6*c^2 + 15*c*d*x + 10*d^2*x^2))*Log[e*((a + b*x)/(c + d*x))^n])/ 
(a + b*x)^5 + (60*B*d^5*n*Log[c + d*x])/(b*c - a*d)^3))/(1800*b^3*g^6)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.73, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2961, 2772, 27, 1140, 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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2772

\(\displaystyle \frac {i^2 \left (-B n \int -\frac {(c+d x)^6 \left (6 b^2-\frac {15 d (a+b x) b}{c+d x}+\frac {10 d^2 (a+b x)^2}{(c+d x)^2}\right )}{30 (a+b x)^6}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 (a+b x)^5}-\frac {d^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {b d (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^4}\right )}{g^6 (b c-a d)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {i^2 \left (\frac {1}{30} B n \int \frac {(c+d x)^6 \left (6 b^2-\frac {15 d (a+b x) b}{c+d x}+\frac {10 d^2 (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^6}d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 (a+b x)^5}-\frac {d^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {b d (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^4}\right )}{g^6 (b c-a d)^3}\)

\(\Big \downarrow \) 1140

\(\displaystyle \frac {i^2 \left (\frac {1}{30} B n \int \left (\frac {6 b^2 (c+d x)^6}{(a+b x)^6}-\frac {15 b d (c+d x)^5}{(a+b x)^5}+\frac {10 d^2 (c+d x)^4}{(a+b x)^4}\right )d\frac {a+b x}{c+d x}-\frac {b^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 (a+b x)^5}-\frac {d^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {b d (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^4}\right )}{g^6 (b c-a d)^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^2 \left (-\frac {b^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 (a+b x)^5}-\frac {d^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 (a+b x)^3}+\frac {b d (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 (a+b x)^4}+\frac {1}{30} B n \left (-\frac {6 b^2 (c+d x)^5}{5 (a+b x)^5}-\frac {10 d^2 (c+d x)^3}{3 (a+b x)^3}+\frac {15 b d (c+d x)^4}{4 (a+b x)^4}\right )\right )}{g^6 (b c-a d)^3}\)

Input:

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

Output:

(i^2*((B*n*((-10*d^2*(c + d*x)^3)/(3*(a + b*x)^3) + (15*b*d*(c + d*x)^4)/( 
4*(a + b*x)^4) - (6*b^2*(c + d*x)^5)/(5*(a + b*x)^5)))/30 - (d^2*(c + d*x) 
^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(a + b*x)^3) + (b*d*(c + d*x 
)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(a + b*x)^4) - (b^2*(c + d* 
x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*(a + b*x)^5)))/((b*c - a*d 
)^3*g^6)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 2009

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

rule 2772

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ 
.))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + 
 b*Log[c*x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] 
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q 
, 1] && EqQ[m, -1])
 

rule 2961

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q   Subst[Int[x^m*((A + B*L 
og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ 
b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1048\) vs. \(2(281)=562\).

Time = 61.25 (sec) , antiderivative size = 1049, normalized size of antiderivative = 3.58

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1049\)

Input:

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

Output:

-1/1800*(1800*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c^2*d^4*i^2*n-1800*B*x 
*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c^2*d^4*i^2*n+2400*B*x*ln(e*((b*x+a)/(d 
*x+c))^n)*a*b^8*c^3*d^3*i^2*n-300*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*d^ 
6*i^2*n-600*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*d^6*i^2*n-300*B*x^3*a* 
b^8*c*d^5*i^2*n^2-600*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^3*d^3*i^2*n-60 
0*B*x^2*a^2*b^7*c*d^5*i^2*n^2-60*B*x^4*b^9*c*d^5*i^2*n^2+270*B*x^3*a^2*b^7 
*d^6*i^2*n^2+30*B*x^3*b^9*c^2*d^4*i^2*n^2+470*B*x^2*a^3*b^6*d^6*i^2*n^2-20 
*B*x^2*b^9*c^3*d^3*i^2*n^2+600*A*x^2*a^3*b^6*d^6*i^2*n-600*A*x^2*b^9*c^3*d 
^3*i^2*n+235*B*x*a^4*b^5*d^6*i^2*n^2-135*B*x*b^9*c^4*d^2*i^2*n^2+300*A*x*a 
^4*b^5*d^6*i^2*n-900*A*x*b^9*c^4*d^2*i^2*n-360*B*ln(e*((b*x+a)/(d*x+c))^n) 
*b^9*c^5*d*i^2*n-1800*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c*d^5*i^2*n+ 
150*B*x^2*a*b^8*c^2*d^4*i^2*n^2-1800*A*x^2*a^2*b^7*c*d^5*i^2*n+1800*A*x^2* 
a*b^8*c^2*d^4*i^2*n-900*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^4*d^2*i^2*n-60 
0*B*x*a^2*b^7*c^2*d^4*i^2*n^2+500*B*x*a*b^8*c^3*d^3*i^2*n^2-1800*A*x*a^2*b 
^7*c^2*d^4*i^2*n+2400*A*x*a*b^8*c^3*d^3*i^2*n-600*B*ln(e*((b*x+a)/(d*x+c)) 
^n)*a^2*b^7*c^3*d^3*i^2*n+900*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c^4*d^2*i^ 
2*n+47*B*a^5*b^4*d^6*i^2*n^2-72*B*b^9*c^5*d*i^2*n^2+60*A*a^5*b^4*d^6*i^2*n 
-360*A*b^9*c^5*d*i^2*n-200*B*a^2*b^7*c^3*d^3*i^2*n^2+225*B*a*b^8*c^4*d^2*i 
^2*n^2-600*A*a^2*b^7*c^3*d^3*i^2*n+900*A*a*b^8*c^4*d^2*i^2*n-60*B*x^5*ln(e 
*((b*x+a)/(d*x+c))^n)*b^9*d^6*i^2*n+60*B*x^4*a*b^8*d^6*i^2*n^2)/g^6/(b*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1087 vs. \(2 (281) = 562\).

Time = 0.13 (sec) , antiderivative size = 1087, normalized size of antiderivative = 3.71 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/1800*(60*(B*b^5*c*d^4 - B*a*b^4*d^5)*i^2*n*x^4 - 30*(B*b^5*c^2*d^3 - 10 
*B*a*b^4*c*d^4 + 9*B*a^2*b^3*d^5)*i^2*n*x^3 + (72*B*b^5*c^5 - 225*B*a*b^4* 
c^4*d + 200*B*a^2*b^3*c^3*d^2 - 47*B*a^5*d^5)*i^2*n + 60*(6*A*b^5*c^5 - 15 
*A*a*b^4*c^4*d + 10*A*a^2*b^3*c^3*d^2 - A*a^5*d^5)*i^2 + 10*((2*B*b^5*c^3* 
d^2 - 15*B*a*b^4*c^2*d^3 + 60*B*a^2*b^3*c*d^4 - 47*B*a^3*b^2*d^5)*i^2*n + 
60*(A*b^5*c^3*d^2 - 3*A*a*b^4*c^2*d^3 + 3*A*a^2*b^3*c*d^4 - A*a^3*b^2*d^5) 
*i^2)*x^2 + 5*((27*B*b^5*c^4*d - 100*B*a*b^4*c^3*d^2 + 120*B*a^2*b^3*c^2*d 
^3 - 47*B*a^4*b*d^5)*i^2*n + 60*(3*A*b^5*c^4*d - 8*A*a*b^4*c^3*d^2 + 6*A*a 
^2*b^3*c^2*d^3 - A*a^4*b*d^5)*i^2)*x + 60*(10*(B*b^5*c^3*d^2 - 3*B*a*b^4*c 
^2*d^3 + 3*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*i^2*x^2 + 5*(3*B*b^5*c^4*d - 8 
*B*a*b^4*c^3*d^2 + 6*B*a^2*b^3*c^2*d^3 - B*a^4*b*d^5)*i^2*x + (6*B*b^5*c^5 
 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - B*a^5*d^5)*i^2)*log(e) + 60*( 
B*b^5*d^5*i^2*n*x^5 + 5*B*a*b^4*d^5*i^2*n*x^4 + 10*B*a^2*b^3*d^5*i^2*n*x^3 
 + 10*(B*b^5*c^3*d^2 - 3*B*a*b^4*c^2*d^3 + 3*B*a^2*b^3*c*d^4)*i^2*n*x^2 + 
5*(3*B*b^5*c^4*d - 8*B*a*b^4*c^3*d^2 + 6*B*a^2*b^3*c^2*d^3)*i^2*n*x + (6*B 
*b^5*c^5 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2)*i^2*n)*log((b*x + a)/( 
d*x + c)))/((b^11*c^3 - 3*a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^ 
6*x^5 + 5*(a*b^10*c^3 - 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^4*b^7*d^3)*g 
^6*x^4 + 10*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c*d^2 - a^5*b^6*d^3 
)*g^6*x^3 + 10*(a^3*b^8*c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b...
 

Sympy [F(-1)]

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

integrate((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(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. 3058 vs. \(2 (281) = 562\).

Time = 0.18 (sec) , antiderivative size = 3058, normalized size of antiderivative = 10.44 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/300*B*c^2*i^2*n*((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)) - 1/1800*B 
*d^2*i^2*n*((47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 27 
8*a^5*b*c*d^3 + 47*a^6*d^4 + 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4* 
d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*d^3 - 9*a^3* 
b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 24 
8*a^3*b^3*c*d^3 + 47*a^4*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c...
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 2.42 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.43 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=-\frac {1}{1800} \, {\left (\frac {60 \, {\left (6 \, B b^{2} i^{2} n - \frac {15 \, {\left (b x + a\right )} B b d i^{2} n}{d x + c} + \frac {10 \, {\left (b x + a\right )}^{2} B d^{2} i^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {72 \, B b^{2} i^{2} n - \frac {225 \, {\left (b x + a\right )} B b d i^{2} n}{d x + c} + \frac {200 \, {\left (b x + a\right )}^{2} B d^{2} i^{2} n}{{\left (d x + c\right )}^{2}} + 360 \, B b^{2} i^{2} \log \left (e\right ) - \frac {900 \, {\left (b x + a\right )} B b d i^{2} \log \left (e\right )}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} B d^{2} i^{2} \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 360 \, A b^{2} i^{2} - \frac {900 \, {\left (b x + a\right )} A b d i^{2}}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} A d^{2} i^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

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

Output:

-1/1800*(60*(6*B*b^2*i^2*n - 15*(b*x + a)*B*b*d*i^2*n/(d*x + c) + 10*(b*x 
+ a)^2*B*d^2*i^2*n/(d*x + c)^2)*log((b*x + a)/(d*x + c))/((b*x + a)^5*b^2* 
c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x + a)^5* 
a^2*d^2*g^6/(d*x + c)^5) + (72*B*b^2*i^2*n - 225*(b*x + a)*B*b*d*i^2*n/(d* 
x + c) + 200*(b*x + a)^2*B*d^2*i^2*n/(d*x + c)^2 + 360*B*b^2*i^2*log(e) - 
900*(b*x + a)*B*b*d*i^2*log(e)/(d*x + c) + 600*(b*x + a)^2*B*d^2*i^2*log(e 
)/(d*x + c)^2 + 360*A*b^2*i^2 - 900*(b*x + a)*A*b*d*i^2/(d*x + c) + 600*(b 
*x + a)^2*A*d^2*i^2/(d*x + c)^2)/((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2* 
(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x + a)^5*a^2*d^2*g^6/(d*x + c)^5) 
)*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 27.94 (sec) , antiderivative size = 954, normalized size of antiderivative = 3.26 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\frac {B\,d^5\,i^2\,n\,\mathrm {atanh}\left (\frac {30\,a^3\,b^3\,d^3\,g^6-30\,a^2\,b^4\,c\,d^2\,g^6-30\,a\,b^5\,c^2\,d\,g^6+30\,b^6\,c^3\,g^6}{30\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{15\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^3}+\frac {B\,c\,d\,i^2}{10\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^3}+\frac {B\,c\,d\,i^2}{10\,b^2}\right )+\frac {2\,B\,a\,d^2\,i^2}{15\,b^2}+\frac {2\,B\,c\,d\,i^2}{5\,b}\right )+\frac {B\,c^2\,i^2}{5\,b}+\frac {B\,d^2\,i^2\,x^2}{3\,b}\right )}{a^5\,g^6+5\,a^4\,b\,g^6\,x+10\,a^3\,b^2\,g^6\,x^2+10\,a^2\,b^3\,g^6\,x^3+5\,a\,b^4\,g^6\,x^4+b^5\,g^6\,x^5}-\frac {\frac {60\,A\,a^4\,d^4\,i^2+360\,A\,b^4\,c^4\,i^2+47\,B\,a^4\,d^4\,i^2\,n+72\,B\,b^4\,c^4\,i^2\,n+60\,A\,a^2\,b^2\,c^2\,d^2\,i^2-540\,A\,a\,b^3\,c^3\,d\,i^2+60\,A\,a^3\,b\,c\,d^3\,i^2-153\,B\,a\,b^3\,c^3\,d\,i^2\,n+47\,B\,a^3\,b\,c\,d^3\,i^2\,n+47\,B\,a^2\,b^2\,c^2\,d^2\,i^2\,n}{60\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (60\,A\,a^2\,b^2\,d^4\,i^2+60\,A\,b^4\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,d^4\,i^2\,n+2\,B\,b^4\,c^2\,d^2\,i^2\,n-120\,A\,a\,b^3\,c\,d^3\,i^2-13\,B\,a\,b^3\,c\,d^3\,i^2\,n\right )}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (60\,A\,a^3\,b\,d^4\,i^2+180\,A\,b^4\,c^3\,d\,i^2-300\,A\,a\,b^3\,c^2\,d^2\,i^2+60\,A\,a^2\,b^2\,c\,d^3\,i^2+47\,B\,a^3\,b\,d^4\,i^2\,n+27\,B\,b^4\,c^3\,d\,i^2\,n-73\,B\,a\,b^3\,c^2\,d^2\,i^2\,n+47\,B\,a^2\,b^2\,c\,d^3\,i^2\,n\right )}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^3\,\left (9\,B\,a\,b^3\,d^3\,i^2\,n-B\,b^4\,c\,d^2\,i^2\,n\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^4\,d^4\,i^2\,n\,x^4}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{30\,a^5\,b^3\,g^6+150\,a^4\,b^4\,g^6\,x+300\,a^3\,b^5\,g^6\,x^2+300\,a^2\,b^6\,g^6\,x^3+150\,a\,b^7\,g^6\,x^4+30\,b^8\,g^6\,x^5} \] Input:

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

Output:

(B*d^5*i^2*n*atanh((30*b^6*c^3*g^6 + 30*a^3*b^3*d^3*g^6 - 30*a*b^5*c^2*d*g 
^6 - 30*a^2*b^4*c*d^2*g^6)/(30*b^3*g^6*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 
+ b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(15*b^3*g^6*(a*d - b*c)^3) - (log( 
e*((a + b*x)/(c + d*x))^n)*(a*((B*a*d^2*i^2)/(30*b^3) + (B*c*d*i^2)/(10*b^ 
2)) + x*(b*((B*a*d^2*i^2)/(30*b^3) + (B*c*d*i^2)/(10*b^2)) + (2*B*a*d^2*i^ 
2)/(15*b^2) + (2*B*c*d*i^2)/(5*b)) + (B*c^2*i^2)/(5*b) + (B*d^2*i^2*x^2)/( 
3*b)))/(a^5*g^6 + b^5*g^6*x^5 + 5*a*b^4*g^6*x^4 + 10*a^3*b^2*g^6*x^2 + 10* 
a^2*b^3*g^6*x^3 + 5*a^4*b*g^6*x) - ((60*A*a^4*d^4*i^2 + 360*A*b^4*c^4*i^2 
+ 47*B*a^4*d^4*i^2*n + 72*B*b^4*c^4*i^2*n + 60*A*a^2*b^2*c^2*d^2*i^2 - 540 
*A*a*b^3*c^3*d*i^2 + 60*A*a^3*b*c*d^3*i^2 - 153*B*a*b^3*c^3*d*i^2*n + 47*B 
*a^3*b*c*d^3*i^2*n + 47*B*a^2*b^2*c^2*d^2*i^2*n)/(60*(a^2*d^2 + b^2*c^2 - 
2*a*b*c*d)) + (x^2*(60*A*a^2*b^2*d^4*i^2 + 60*A*b^4*c^2*d^2*i^2 + 47*B*a^2 
*b^2*d^4*i^2*n + 2*B*b^4*c^2*d^2*i^2*n - 120*A*a*b^3*c*d^3*i^2 - 13*B*a*b^ 
3*c*d^3*i^2*n))/(6*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x*(60*A*a^3*b*d^4*i 
^2 + 180*A*b^4*c^3*d*i^2 - 300*A*a*b^3*c^2*d^2*i^2 + 60*A*a^2*b^2*c*d^3*i^ 
2 + 47*B*a^3*b*d^4*i^2*n + 27*B*b^4*c^3*d*i^2*n - 73*B*a*b^3*c^2*d^2*i^2*n 
 + 47*B*a^2*b^2*c*d^3*i^2*n))/(12*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*x^ 
3*(9*B*a*b^3*d^3*i^2*n - B*b^4*c*d^2*i^2*n))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b 
*c*d)) + (B*b^4*d^4*i^2*n*x^4)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(30*a^5*b^ 
3*g^6 + 30*b^8*g^6*x^5 + 150*a^4*b^4*g^6*x + 150*a*b^7*g^6*x^4 + 300*a^...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1378, normalized size of antiderivative = 4.70 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 60*log(a + b*x)*a**6*b*d**5*n - 300*log(a + b*x)*a**5*b**2*d**5*n*x - 
600*log(a + b*x)*a**4*b**3*d**5*n*x**2 - 600*log(a + b*x)*a**3*b**4*d**5*n 
*x**3 - 300*log(a + b*x)*a**2*b**5*d**5*n*x**4 - 60*log(a + b*x)*a*b**6*d* 
*5*n*x**5 + 60*log(c + d*x)*a**6*b*d**5*n + 300*log(c + d*x)*a**5*b**2*d** 
5*n*x + 600*log(c + d*x)*a**4*b**3*d**5*n*x**2 + 600*log(c + d*x)*a**3*b** 
4*d**5*n*x**3 + 300*log(c + d*x)*a**2*b**5*d**5*n*x**4 + 60*log(c + d*x)*a 
*b**6*d**5*n*x**5 + 60*log(((a + b*x)**n*e)/(c + d*x)**n)*a**6*b*d**5 + 30 
0*log(((a + b*x)**n*e)/(c + d*x)**n)*a**5*b**2*d**5*x + 600*log(((a + b*x) 
**n*e)/(c + d*x)**n)*a**4*b**3*d**5*x**2 - 600*log(((a + b*x)**n*e)/(c + d 
*x)**n)*a**3*b**4*c**3*d**2 - 1800*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3 
*b**4*c**2*d**3*x - 1800*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**4*c*d* 
*4*x**2 + 900*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**5*c**4*d + 2400*l 
og(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**5*c**3*d**2*x + 1800*log(((a + b 
*x)**n*e)/(c + d*x)**n)*a**2*b**5*c**2*d**3*x**2 - 360*log(((a + b*x)**n*e 
)/(c + d*x)**n)*a*b**6*c**5 - 900*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b** 
6*c**4*d*x - 600*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**6*c**3*d**2*x**2 
+ 60*a**7*d**5 + 35*a**6*b*d**5*n + 300*a**6*b*d**5*x + 12*a**5*b**2*c*d** 
4*n + 175*a**5*b**2*d**5*n*x + 600*a**5*b**2*d**5*x**2 - 600*a**4*b**3*c** 
3*d**2 - 1800*a**4*b**3*c**2*d**3*x + 60*a**4*b**3*c*d**4*n*x - 1800*a**4* 
b**3*c*d**4*x**2 + 350*a**4*b**3*d**5*n*x**2 + 900*a**3*b**4*c**4*d - 2...