\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4} \, dx\) [175]

Optimal result

Integrand size = 45, antiderivative size = 157 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 B^2 i^2 n^2 (c+d x)^3}{27 (b c-a d) g^4 (a+b x)^3}-\frac {2 B i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d) g^4 (a+b x)^3} \] Output:

-2/27*B^2*i^2*n^2*(d*x+c)^3/(-a*d+b*c)/g^4/(b*x+a)^3-2/9*B*i^2*n*(d*x+c)^3 
*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)/g^4/(b*x+a)^3-1/3*i^2*(d*x+c)^ 
3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)/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 = 2.43 (sec) , antiderivative size = 1418, normalized size of antiderivative = 9.03 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/54*(i^2*(18*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 54 
*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 54*d 
^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + B 
*n*(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3*n - 18*A*d*(b*c - a*d)^2*(a + b 
*x) - 15*B*d*(b*c - a*d)^2*n*(a + b*x) + 36*A*d^2*(b*c - a*d)*(a + b*x)^2 
+ 66*B*d^2*(b*c - a*d)*n*(a + b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 
 66*B*d^3*n*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*n*(a + b*x)^3*Log[a + b*x] 
^2 + 12*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n] - 18*B*d*(b*c - a*d 
)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^2*(b*c - a*d)*(a + b 
*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*L 
og[e*((a + b*x)/(c + d*x))^n] - 36*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d 
^3*n*(a + b*x)^3*Log[c + d*x] + 36*B*d^3*n*(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))^n]*Log[c + d*x] - 18*B*d^3*n*(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*n* 
(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] + 36*B*d^3*n*(a + 
b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36*B*d^3*n*(a + b*x)^3*P 
olyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 54*B*d^2*n*(a + b*x)^2*(2*(b*c - a 
*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d*(a + b*x)*Log[a + b*x]*(A 
 + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*d*(a + b*x)*(A + B*Log[e*((a + b* 
x)/(c + d*x))^n])*Log[c + d*x] + 2*B*n*(b*c - a*d + d*(a + b*x)*Log[a +...
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2742, 2741}

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.

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

\(\Big \downarrow \) 2961

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

\(\Big \downarrow \) 2742

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

\(\Big \downarrow \) 2741

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2742

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* 
(p/(m + 1))   Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b 
, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]
 

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. \(824\) vs. \(2(151)=302\).

Time = 10.30 (sec) , antiderivative size = 825, normalized size of antiderivative = 5.25

method	result	size
parallelrisch	\(-\frac {18 A B \,x^{2} a \,b^{4} d^{4} i^{2} n^{2}-54 A B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c \,d^{3} i^{2} n -54 A B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{2} d^{2} i^{2} n -9 B^{2} x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{5} d^{4} i^{2} n -6 B^{2} x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} d^{4} i^{2} n^{2}+6 B^{2} x^{2} a \,b^{4} d^{4} i^{2} n^{3}-6 B^{2} x^{2} b^{5} c \,d^{3} i^{2} n^{3}+6 B^{2} x \,a^{2} b^{3} d^{4} i^{2} n^{3}-6 B^{2} x \,b^{5} c^{2} d^{2} i^{2} n^{3}+27 A^{2} x^{2} a \,b^{4} d^{4} i^{2} n -27 A^{2} x^{2} b^{5} c \,d^{3} i^{2} n +6 A B \,a^{3} b^{2} d^{4} i^{2} n^{2}-6 A B \,b^{5} c^{3} d \,i^{2} n^{2}-18 A B \,x^{2} b^{5} c \,d^{3} i^{2} n^{2}-27 B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{5} c^{2} d^{2} i^{2} n -18 B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{2} d^{2} i^{2} n^{2}+18 A B x \,a^{2} b^{3} d^{4} i^{2} n^{2}-18 A B x \,b^{5} c^{2} d^{2} i^{2} n^{2}-18 A B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{3} d \,i^{2} n -18 A B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} d^{4} i^{2} n -27 B^{2} x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{5} c \,d^{3} i^{2} n -18 B^{2} x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c \,d^{3} i^{2} n^{2}+2 B^{2} a^{3} b^{2} d^{4} i^{2} n^{3}-2 B^{2} b^{5} c^{3} d \,i^{2} n^{3}+9 A^{2} a^{3} b^{2} d^{4} i^{2} n -9 A^{2} b^{5} c^{3} d \,i^{2} n -9 B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{5} c^{3} d \,i^{2} n -6 B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{5} c^{3} d \,i^{2} n^{2}+27 A^{2} x \,a^{2} b^{3} d^{4} i^{2} n -27 A^{2} x \,b^{5} c^{2} d^{2} i^{2} n}{27 g^{4} \left (b x +a \right )^{3} b^{5} d n \left (d a -b c \right )}\)	\(825\)

Input:

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

Output:

-1/27*(18*A*B*x^2*a*b^4*d^4*i^2*n^2-54*A*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b 
^5*c*d^3*i^2*n-54*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^2*d^2*i^2*n-9*B^2* 
x^3*ln(e*((b*x+a)/(d*x+c))^n)^2*b^5*d^4*i^2*n-6*B^2*x^3*ln(e*((b*x+a)/(d*x 
+c))^n)*b^5*d^4*i^2*n^2+6*B^2*x^2*a*b^4*d^4*i^2*n^3-6*B^2*x^2*b^5*c*d^3*i^ 
2*n^3+6*B^2*x*a^2*b^3*d^4*i^2*n^3-6*B^2*x*b^5*c^2*d^2*i^2*n^3+27*A^2*x^2*a 
*b^4*d^4*i^2*n-27*A^2*x^2*b^5*c*d^3*i^2*n+6*A*B*a^3*b^2*d^4*i^2*n^2-6*A*B* 
b^5*c^3*d*i^2*n^2-18*A*B*x^2*b^5*c*d^3*i^2*n^2-27*B^2*x*ln(e*((b*x+a)/(d*x 
+c))^n)^2*b^5*c^2*d^2*i^2*n-18*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^2*d^2 
*i^2*n^2+18*A*B*x*a^2*b^3*d^4*i^2*n^2-18*A*B*x*b^5*c^2*d^2*i^2*n^2-18*A*B* 
ln(e*((b*x+a)/(d*x+c))^n)*b^5*c^3*d*i^2*n-18*A*B*x^3*ln(e*((b*x+a)/(d*x+c) 
)^n)*b^5*d^4*i^2*n-27*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^5*c*d^3*i^2*n- 
18*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^5*c*d^3*i^2*n^2+2*B^2*a^3*b^2*d^4*i 
^2*n^3-2*B^2*b^5*c^3*d*i^2*n^3+9*A^2*a^3*b^2*d^4*i^2*n-9*A^2*b^5*c^3*d*i^2 
*n-9*B^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^5*c^3*d*i^2*n-6*B^2*ln(e*((b*x+a)/( 
d*x+c))^n)*b^5*c^3*d*i^2*n^2+27*A^2*x*a^2*b^3*d^4*i^2*n-27*A^2*x*b^5*c^2*d 
^2*i^2*n)/g^4/(b*x+a)^3/b^5/d/n/(a*d-b*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 975 vs. \(2 (151) = 302\).

Time = 0.10 (sec) , antiderivative size = 975, normalized size of antiderivative = 6.21 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/27*(2*(B^2*b^3*c^3 - B^2*a^3*d^3)*i^2*n^2 + 6*(A*B*b^3*c^3 - A*B*a^3*d^ 
3)*i^2*n + 9*(A^2*b^3*c^3 - A^2*a^3*d^3)*i^2 + 3*(2*(B^2*b^3*c*d^2 - B^2*a 
*b^2*d^3)*i^2*n^2 + 6*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*i^2*n + 9*(A^2*b^3*c 
*d^2 - A^2*a*b^2*d^3)*i^2)*x^2 + 9*(3*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*i^2* 
x^2 + 3*(B^2*b^3*c^2*d - B^2*a^2*b*d^3)*i^2*x + (B^2*b^3*c^3 - B^2*a^3*d^3 
)*i^2)*log(e)^2 + 9*(B^2*b^3*d^3*i^2*n^2*x^3 + 3*B^2*b^3*c*d^2*i^2*n^2*x^2 
 + 3*B^2*b^3*c^2*d*i^2*n^2*x + B^2*b^3*c^3*i^2*n^2)*log((b*x + a)/(d*x + c 
))^2 + 3*(2*(B^2*b^3*c^2*d - B^2*a^2*b*d^3)*i^2*n^2 + 6*(A*B*b^3*c^2*d - A 
*B*a^2*b*d^3)*i^2*n + 9*(A^2*b^3*c^2*d - A^2*a^2*b*d^3)*i^2)*x + 6*((B^2*b 
^3*c^3 - B^2*a^3*d^3)*i^2*n + 3*(A*B*b^3*c^3 - A*B*a^3*d^3)*i^2 + 3*((B^2* 
b^3*c*d^2 - B^2*a*b^2*d^3)*i^2*n + 3*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*i^2)* 
x^2 + 3*((B^2*b^3*c^2*d - B^2*a^2*b*d^3)*i^2*n + 3*(A*B*b^3*c^2*d - A*B*a^ 
2*b*d^3)*i^2)*x + 3*(B^2*b^3*d^3*i^2*n*x^3 + 3*B^2*b^3*c*d^2*i^2*n*x^2 + 3 
*B^2*b^3*c^2*d*i^2*n*x + B^2*b^3*c^3*i^2*n)*log((b*x + a)/(d*x + c)))*log( 
e) + 6*(B^2*b^3*c^3*i^2*n^2 + 3*A*B*b^3*c^3*i^2*n + (B^2*b^3*d^3*i^2*n^2 + 
 3*A*B*b^3*d^3*i^2*n)*x^3 + 3*(B^2*b^3*c*d^2*i^2*n^2 + 3*A*B*b^3*c*d^2*i^2 
*n)*x^2 + 3*(B^2*b^3*c^2*d*i^2*n^2 + 3*A*B*b^3*c^2*d*i^2*n)*x)*log((b*x + 
a)/(d*x + c)))/((b^7*c - a*b^6*d)*g^4*x^3 + 3*(a*b^6*c - a^2*b^5*d)*g^4*x^ 
2 + 3*(a^2*b^5*c - a^3*b^4*d)*g^4*x + (a^3*b^4*c - a^4*b^3*d)*g^4)
 

Sympy [F]

\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\frac {i^{2} \left (\int \frac {A^{2} c^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {A^{2} d^{2} x^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {B^{2} c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {2 A B c^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {2 A^{2} c d x}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {B^{2} d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {2 A B d^{2} x^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {2 B^{2} c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx + \int \frac {4 A B c d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{4} + 4 a^{3} b x + 6 a^{2} b^{2} x^{2} + 4 a b^{3} x^{3} + b^{4} x^{4}}\, dx\right )}{g^{4}} \] Input:

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

Output:

i**2*(Integral(A**2*c**2/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3* 
x**3 + b**4*x**4), x) + Integral(A**2*d**2*x**2/(a**4 + 4*a**3*b*x + 6*a** 
2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(B**2*c**2*log(e*(a 
/(c + d*x) + b*x/(c + d*x))**n)**2/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 
 4*a*b**3*x**3 + b**4*x**4), x) + Integral(2*A*B*c**2*log(e*(a/(c + d*x) + 
 b*x/(c + d*x))**n)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 
+ b**4*x**4), x) + Integral(2*A**2*c*d*x/(a**4 + 4*a**3*b*x + 6*a**2*b**2* 
x**2 + 4*a*b**3*x**3 + b**4*x**4), x) + Integral(B**2*d**2*x**2*log(e*(a/( 
c + d*x) + b*x/(c + d*x))**n)**2/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4 
*a*b**3*x**3 + b**4*x**4), x) + Integral(2*A*B*d**2*x**2*log(e*(a/(c + d*x 
) + b*x/(c + d*x))**n)/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x* 
*3 + b**4*x**4), x) + Integral(2*B**2*c*d*x*log(e*(a/(c + d*x) + b*x/(c + 
d*x))**n)**2/(a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4* 
x**4), x) + Integral(4*A*B*c*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/( 
a**4 + 4*a**3*b*x + 6*a**2*b**2*x**2 + 4*a*b**3*x**3 + b**4*x**4), x))/g** 
4
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5588 vs. \(2 (151) = 302\).

Time = 0.36 (sec) , antiderivative size = 5588, normalized size of antiderivative = 35.59 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/9*A*B*d^2*i^2*n*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c 
^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a 
^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7*c^2 
 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + 
 a^4*b^4*d^2)*g^4*x + (a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6 
*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 - 3*a*b^5*c^ 
2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + 
 a^2*d^3)*log(d*x + c)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b 
^3*d^3)*g^4)) - 1/9*A*B*c^2*i^2*n*((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)) - 1/9*A*B*c*d*i^2*n*((5*a*b^2*c 
^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c 
^2 - 16*a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2) 
*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^ 
5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d 
+ a^5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*...
 

Giac [A] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.38 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{27} \, {\left (\frac {9 \, {\left (d x + c\right )}^{3} B^{2} i^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )}^{3} g^{4}} + \frac {6 \, {\left (B^{2} i^{2} n^{2} + 3 \, B^{2} i^{2} n \log \left (e\right ) + 3 \, A B i^{2} n\right )} {\left (d x + c\right )}^{3} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{3} g^{4}} + \frac {{\left (2 \, B^{2} i^{2} n^{2} + 6 \, B^{2} i^{2} n \log \left (e\right ) + 9 \, B^{2} i^{2} \log \left (e\right )^{2} + 6 \, A B i^{2} n + 18 \, A B i^{2} \log \left (e\right ) + 9 \, A^{2} i^{2}\right )} {\left (d x + c\right )}^{3}}{{\left (b x + a\right )}^{3} g^{4}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \] Input:

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

Output:

-1/27*(9*(d*x + c)^3*B^2*i^2*n^2*log((b*x + a)/(d*x + c))^2/((b*x + a)^3*g 
^4) + 6*(B^2*i^2*n^2 + 3*B^2*i^2*n*log(e) + 3*A*B*i^2*n)*(d*x + c)^3*log(( 
b*x + a)/(d*x + c))/((b*x + a)^3*g^4) + (2*B^2*i^2*n^2 + 6*B^2*i^2*n*log(e 
) + 9*B^2*i^2*log(e)^2 + 6*A*B*i^2*n + 18*A*B*i^2*log(e) + 9*A^2*i^2)*(d*x 
 + c)^3/((b*x + a)^3*g^4))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
 

Mupad [B] (verification not implemented)

Time = 29.03 (sec) , antiderivative size = 1195, normalized size of antiderivative = 7.61 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1161, normalized size of antiderivative = 7.39 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 18*log(a + b*x)*a**4*b*c*d**2*n - 6*log(a + b*x)*a**3*b**2*c*d**2*n**2 
 - 54*log(a + b*x)*a**3*b**2*c*d**2*n*x - 18*log(a + b*x)*a**2*b**3*c*d**2 
*n**2*x - 54*log(a + b*x)*a**2*b**3*c*d**2*n*x**2 - 18*log(a + b*x)*a*b**4 
*c*d**2*n**2*x**2 - 18*log(a + b*x)*a*b**4*c*d**2*n*x**3 - 6*log(a + b*x)* 
b**5*c*d**2*n**2*x**3 + 18*log(c + d*x)*a**4*b*c*d**2*n + 6*log(c + d*x)*a 
**3*b**2*c*d**2*n**2 + 54*log(c + d*x)*a**3*b**2*c*d**2*n*x + 18*log(c + d 
*x)*a**2*b**3*c*d**2*n**2*x + 54*log(c + d*x)*a**2*b**3*c*d**2*n*x**2 + 18 
*log(c + d*x)*a*b**4*c*d**2*n**2*x**2 + 18*log(c + d*x)*a*b**4*c*d**2*n*x* 
*3 + 6*log(c + d*x)*b**5*c*d**2*n**2*x**3 - 9*log(((a + b*x)**n*e)/(c + d* 
x)**n)**2*a*b**4*c**3 - 27*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c* 
*2*d*x - 27*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*c*d**2*x**2 - 9*l 
og(((a + b*x)**n*e)/(c + d*x)**n)**2*a*b**4*d**3*x**3 + 18*log(((a + b*x)* 
*n*e)/(c + d*x)**n)*a**4*b*c*d**2 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a 
**3*b**2*c*d**2*n + 54*log(((a + b*x)**n*e)/(c + d*x)**n)*a**3*b**2*c*d**2 
*x - 18*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2*b**3*c**3 - 54*log(((a + b 
*x)**n*e)/(c + d*x)**n)*a**2*b**3*c**2*d*x + 18*log(((a + b*x)**n*e)/(c + 
d*x)**n)*a**2*b**3*c*d**2*n*x - 18*log(((a + b*x)**n*e)/(c + d*x)**n)*a**2 
*b**3*d**3*x**3 - 6*log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**4*c**3*n - 18* 
log(((a + b*x)**n*e)/(c + d*x)**n)*a*b**4*c**2*d*n*x + 18*log(((a + b*x)** 
n*e)/(c + d*x)**n)*a*b**4*c*d**2*x**3 - 6*log(((a + b*x)**n*e)/(c + d*x...