\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\) [71]

Optimal result

Integrand size = 42, antiderivative size = 147 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {2 B^2 i^2 (c+d x)^3}{27 (b c-a d) g^4 (a+b x)^3}-\frac {2 B i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d) g^4 (a+b x)^3}-\frac {i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d) g^4 (a+b x)^3} \] Output:

-2/27*B^2*i^2*(d*x+c)^3/(-a*d+b*c)/g^4/(b*x+a)^3-2/9*B*i^2*(d*x+c)^3*(A+B* 
ln(e*(b*x+a)/(d*x+c)))/(-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)))^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 = 2.03 (sec) , antiderivative size = 1352, normalized size of antiderivative = 9.20 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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)])^2 + 54*d 
*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 54*d^2*( 
-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + B*(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*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 
54*B*d^2*(a + b*x)^2*(2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 
 2*d*(a + b*x)*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*d*(a 
+ b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 2*B*(b*c - a*d 
+ d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*d*(a + b*x)*...
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2962, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2742

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

\(\Big \downarrow \) 2741

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

Input:

Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^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)])^2)/(a + b*x)^ 
3 + (2*B*(-1/9*(B*(c + d*x)^3)/(a + b*x)^3 - ((c + d*x)^3*(A + B*Log[(e*(a 
 + b*x))/(c + d*x)]))/(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 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. \(366\) vs. \(2(141)=282\).

Time = 1.72 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.50

method	result	size
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{2} d^{2} e^{2} A^{2}}{3 \left (d a -b c \right )^{2} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {2 i^{2} d^{2} e^{2} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{2} g^{4}}+\frac {i^{2} d^{2} e^{2} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{2} g^{4}}\right )}{d^{2}}\)	\(367\)
default	\(-\frac {e \left (d a -b c \right ) \left (-\frac {i^{2} d^{2} e^{2} A^{2}}{3 \left (d a -b c \right )^{2} g^{4} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {2 i^{2} d^{2} e^{2} A B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{2} g^{4}}+\frac {i^{2} d^{2} e^{2} B^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (d a -b c \right )^{2} g^{4}}\right )}{d^{2}}\)	\(367\)
parts	\(\frac {i^{2} A^{2} \left (\frac {d \left (d a -b c \right )}{b^{3} \left (b x +a \right )^{2}}-\frac {d^{2}}{b^{3} \left (b x +a \right )}-\frac {a^{2} d^{2}-2 a c d b +c^{2} b^{2}}{3 b^{3} \left (b x +a \right )^{3}}\right )}{g^{4}}-\frac {i^{2} B^{2} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{g^{4} \left (d a -b c \right )}-\frac {2 i^{2} A B \,e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{g^{4} \left (d a -b c \right )}\)	\(371\)
risch	\(\frac {i^{2} A^{2} d^{2} a}{g^{4} b^{3} \left (b x +a \right )^{2}}-\frac {i^{2} A^{2} d c}{g^{4} b^{2} \left (b x +a \right )^{2}}-\frac {i^{2} A^{2} d^{2}}{g^{4} b^{3} \left (b x +a \right )}-\frac {i^{2} A^{2} a^{2} d^{2}}{3 g^{4} b^{3} \left (b x +a \right )^{3}}+\frac {2 i^{2} A^{2} a c d}{3 g^{4} b^{2} \left (b x +a \right )^{3}}-\frac {i^{2} A^{2} c^{2}}{3 g^{4} b \left (b x +a \right )^{3}}+\frac {i^{2} B^{2} e^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 g^{4} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}+\frac {2 i^{2} B^{2} e^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{9 g^{4} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}+\frac {2 i^{2} B^{2} e^{3}}{27 g^{4} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}+\frac {2 i^{2} A B \,e^{3} \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{3 g^{4} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}+\frac {2 i^{2} A B \,e^{3}}{9 g^{4} \left (d a -b c \right ) \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e b c}{d \left (d x +c \right )}\right )^{3}}\)	\(522\)
norman	\(\frac {\frac {B^{2} c \,d^{2} i^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (d a -b c \right )}+\frac {B^{2} c^{2} d \,i^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g \left (d a -b c \right )}-\frac {9 A^{2} a^{2} d^{2} i^{2}+9 A^{2} a b c d \,i^{2}+9 A^{2} b^{2} c^{2} i^{2}+6 A B \,a^{2} d^{2} i^{2}+6 A B a b c d \,i^{2}+6 A B \,b^{2} c^{2} i^{2}+2 B^{2} a^{2} d^{2} i^{2}+2 B^{2} a b c \,i^{2} d +2 B^{2} b^{2} c^{2} i^{2}}{27 g \,b^{3}}-\frac {\left (9 A^{2} d^{2} i^{2}+6 d^{2} B \,i^{2} A +2 d^{2} B^{2} i^{2}\right ) x^{2}}{9 b g}-\frac {\left (9 A^{2} a \,i^{2} d^{2}+9 A^{2} b c d \,i^{2}+6 A B a \,d^{2} i^{2}+6 A B b c d \,i^{2}+2 B^{2} a \,d^{2} i^{2}+2 B^{2} b c \,i^{2} d \right ) x}{9 g \,b^{2}}+\frac {B^{2} c^{3} i^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{3 \left (d a -b c \right ) g}+\frac {B^{2} d^{3} i^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{3 \left (d a -b c \right ) g}+\frac {2 c^{3} B \,i^{2} \left (3 A +B \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{9 g \left (d a -b c \right )}+\frac {2 B \,d^{3} i^{2} \left (3 A +B \right ) x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{9 g \left (d a -b c \right )}+\frac {2 B c \,d^{2} i^{2} \left (3 A +B \right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (d a -b c \right )}+\frac {2 c^{2} B \,i^{2} d \left (3 A +B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 g \left (d a -b c \right )}}{g^{3} \left (b x +a \right )^{3}}\)	\(608\)
parallelrisch	\(-\frac {27 A^{2} x^{2} a \,b^{4} d^{4} i^{2}-27 A^{2} x^{2} b^{5} c \,d^{3} i^{2}+6 B^{2} x^{2} a \,b^{4} d^{4} i^{2}-6 B^{2} x^{2} b^{5} c \,d^{3} i^{2}-9 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} c^{3} d \,i^{2}+27 A^{2} x \,a^{2} b^{3} d^{4} i^{2}-27 A^{2} x \,b^{5} c^{2} d^{2} i^{2}+6 B^{2} x \,a^{2} b^{3} d^{4} i^{2}-6 B^{2} x \,b^{5} c^{2} d^{2} i^{2}-6 B^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{3} d \,i^{2}-18 A B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{4} i^{2}+2 B^{2} a^{3} b^{2} d^{4} i^{2}-2 B^{2} b^{5} c^{3} d \,i^{2}+9 A^{2} a^{3} b^{2} d^{4} i^{2}-9 A^{2} b^{5} c^{3} d \,i^{2}+6 A B \,a^{3} b^{2} d^{4} i^{2}-6 A B \,b^{5} c^{3} d \,i^{2}-9 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} d^{4} i^{2}-6 B^{2} x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} d^{4} i^{2}-54 A B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c \,d^{3} i^{2}-54 A B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d^{2} i^{2}-27 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} c \,d^{3} i^{2}-18 B^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c \,d^{3} i^{2}-27 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{5} c^{2} d^{2} i^{2}+18 A B \,x^{2} a \,b^{4} d^{4} i^{2}-18 A B \,x^{2} b^{5} c \,d^{3} i^{2}-18 B^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{2} d^{2} i^{2}+18 A B x \,a^{2} b^{3} d^{4} i^{2}-18 A B x \,b^{5} c^{2} d^{2} i^{2}-18 A B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{5} c^{3} d \,i^{2}}{27 g^{4} \left (b x +a \right )^{3} b^{5} d \left (d a -b c \right )}\)	\(724\)
orering	\(\text {Expression too large to display}\)	\(1433\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (141) = 282\).

Time = 0.09 (sec) , antiderivative size = 444, normalized size of antiderivative = 3.02 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {3 \, {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c d^{2} - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{2} d - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} i^{2} x + {\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{3} d^{3}\right )} i^{2} + 9 \, {\left (B^{2} b^{3} d^{3} i^{2} x^{3} + 3 \, B^{2} b^{3} c d^{2} i^{2} x^{2} + 3 \, B^{2} b^{3} c^{2} d i^{2} x + B^{2} b^{3} c^{3} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 6 \, {\left ({\left (3 \, A B + B^{2}\right )} b^{3} d^{3} i^{2} x^{3} + 3 \, {\left (3 \, A B + B^{2}\right )} b^{3} c d^{2} i^{2} x^{2} + 3 \, {\left (3 \, A B + B^{2}\right )} b^{3} c^{2} d i^{2} x + {\left (3 \, A B + B^{2}\right )} b^{3} c^{3} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{27 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \] Input:

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

Output:

-1/27*(3*((9*A^2 + 6*A*B + 2*B^2)*b^3*c*d^2 - (9*A^2 + 6*A*B + 2*B^2)*a*b^ 
2*d^3)*i^2*x^2 + 3*((9*A^2 + 6*A*B + 2*B^2)*b^3*c^2*d - (9*A^2 + 6*A*B + 2 
*B^2)*a^2*b*d^3)*i^2*x + ((9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - (9*A^2 + 6*A*B 
 + 2*B^2)*a^3*d^3)*i^2 + 9*(B^2*b^3*d^3*i^2*x^3 + 3*B^2*b^3*c*d^2*i^2*x^2 
+ 3*B^2*b^3*c^2*d*i^2*x + B^2*b^3*c^3*i^2)*log((b*e*x + a*e)/(d*x + c))^2 
+ 6*((3*A*B + B^2)*b^3*d^3*i^2*x^3 + 3*(3*A*B + B^2)*b^3*c*d^2*i^2*x^2 + 3 
*(3*A*B + B^2)*b^3*c^2*d*i^2*x + (3*A*B + B^2)*b^3*c^3*i^2)*log((b*e*x + a 
*e)/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1182 vs. \(2 (131) = 262\).

Time = 27.70 (sec) , antiderivative size = 1182, normalized size of antiderivative = 8.04 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-2*B*d**3*i**2*(3*A + B)*log(x + (6*A*B*a*d**4*i**2 + 6*A*B*b*c*d**3*i**2 
+ 2*B**2*a*d**4*i**2 + 2*B**2*b*c*d**3*i**2 - 2*B*a**2*d**5*i**2*(3*A + B) 
/(a*d - b*c) + 4*B*a*b*c*d**4*i**2*(3*A + B)/(a*d - b*c) - 2*B*b**2*c**2*d 
**3*i**2*(3*A + B)/(a*d - b*c))/(12*A*B*b*d**4*i**2 + 4*B**2*b*d**4*i**2)) 
/(9*b**3*g**4*(a*d - b*c)) + 2*B*d**3*i**2*(3*A + B)*log(x + (6*A*B*a*d**4 
*i**2 + 6*A*B*b*c*d**3*i**2 + 2*B**2*a*d**4*i**2 + 2*B**2*b*c*d**3*i**2 + 
2*B*a**2*d**5*i**2*(3*A + B)/(a*d - b*c) - 4*B*a*b*c*d**4*i**2*(3*A + B)/( 
a*d - b*c) + 2*B*b**2*c**2*d**3*i**2*(3*A + B)/(a*d - b*c))/(12*A*B*b*d**4 
*i**2 + 4*B**2*b*d**4*i**2))/(9*b**3*g**4*(a*d - b*c)) + (B**2*c**3*i**2 + 
 3*B**2*c**2*d*i**2*x + 3*B**2*c*d**2*i**2*x**2 + B**2*d**3*i**2*x**3)*log 
(e*(a + b*x)/(c + d*x))**2/(3*a**4*d*g**4 - 3*a**3*b*c*g**4 + 9*a**3*b*d*g 
**4*x - 9*a**2*b**2*c*g**4*x + 9*a**2*b**2*d*g**4*x**2 - 9*a*b**3*c*g**4*x 
**2 + 3*a*b**3*d*g**4*x**3 - 3*b**4*c*g**4*x**3) + (-9*A**2*a**2*d**2*i**2 
 - 9*A**2*a*b*c*d*i**2 - 9*A**2*b**2*c**2*i**2 - 6*A*B*a**2*d**2*i**2 - 6* 
A*B*a*b*c*d*i**2 - 6*A*B*b**2*c**2*i**2 - 2*B**2*a**2*d**2*i**2 - 2*B**2*a 
*b*c*d*i**2 - 2*B**2*b**2*c**2*i**2 + x**2*(-27*A**2*b**2*d**2*i**2 - 18*A 
*B*b**2*d**2*i**2 - 6*B**2*b**2*d**2*i**2) + x*(-27*A**2*a*b*d**2*i**2 - 2 
7*A**2*b**2*c*d*i**2 - 18*A*B*a*b*d**2*i**2 - 18*A*B*b**2*c*d*i**2 - 6*B** 
2*a*b*d**2*i**2 - 6*B**2*b**2*c*d*i**2))/(27*a**3*b**3*g**4 + 81*a**2*b**4 
*g**4*x + 81*a*b**5*g**4*x**2 + 27*b**6*g**4*x**3) + (-6*A*B*a**2*d**2*...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5532 vs. \(2 (141) = 282\).

Time = 0.41 (sec) , antiderivative size = 5532, normalized size of antiderivative = 37.63 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*(3*b*x + a)*B^2*c*d*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^5*g 
^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*(3*b^2*x^2 
 + 3*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^6* 
g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/54*(6*((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))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (4*b^3*c^3 - 27*a*b^2*c^2*d + 1 
08*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3 
*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3 
*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*( 
5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2 
*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33* 
a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3 
*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))/(a^3*b^4*c^3*g 
^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3* 
g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + ...
 

Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.49 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx=-\frac {1}{27} \, {\left (\frac {9 \, {\left (d x + c\right )}^{3} B^{2} e^{4} i^{2} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{{\left (b e x + a e\right )}^{3} g^{4}} + \frac {6 \, {\left (3 \, A B e^{4} i^{2} + B^{2} e^{4} i^{2}\right )} {\left (d x + c\right )}^{3} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{3} g^{4}} + \frac {{\left (9 \, A^{2} e^{4} i^{2} + 6 \, A B e^{4} i^{2} + 2 \, B^{2} e^{4} i^{2}\right )} {\left (d x + c\right )}^{3}}{{\left (b e x + a e\right )}^{3} g^{4}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

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

Output:

-1/27*(9*(d*x + c)^3*B^2*e^4*i^2*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + 
a*e)^3*g^4) + 6*(3*A*B*e^4*i^2 + B^2*e^4*i^2)*(d*x + c)^3*log((b*e*x + a*e 
)/(d*x + c))/((b*e*x + a*e)^3*g^4) + (9*A^2*e^4*i^2 + 6*A*B*e^4*i^2 + 2*B^ 
2*e^4*i^2)*(d*x + c)^3/((b*e*x + a*e)^3*g^4))*(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 = 29.44 (sec) , antiderivative size = 1153, normalized size of antiderivative = 7.84 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1067, normalized size of antiderivative = 7.26 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 18*log(a + b*x)*a**4*b*c*d**2 - 54*log(a + b*x)*a**3*b**2*c*d**2*x - 6 
*log(a + b*x)*a**3*b**2*c*d**2 - 54*log(a + b*x)*a**2*b**3*c*d**2*x**2 - 1 
8*log(a + b*x)*a**2*b**3*c*d**2*x - 18*log(a + b*x)*a*b**4*c*d**2*x**3 - 1 
8*log(a + b*x)*a*b**4*c*d**2*x**2 - 6*log(a + b*x)*b**5*c*d**2*x**3 + 18*l 
og(c + d*x)*a**4*b*c*d**2 + 54*log(c + d*x)*a**3*b**2*c*d**2*x + 6*log(c + 
 d*x)*a**3*b**2*c*d**2 + 54*log(c + d*x)*a**2*b**3*c*d**2*x**2 + 18*log(c 
+ d*x)*a**2*b**3*c*d**2*x + 18*log(c + d*x)*a*b**4*c*d**2*x**3 + 18*log(c 
+ d*x)*a*b**4*c*d**2*x**2 + 6*log(c + d*x)*b**5*c*d**2*x**3 - 9*log((a*e + 
 b*e*x)/(c + d*x))**2*a*b**4*c**3 - 27*log((a*e + b*e*x)/(c + d*x))**2*a*b 
**4*c**2*d*x - 27*log((a*e + b*e*x)/(c + d*x))**2*a*b**4*c*d**2*x**2 - 9*l 
og((a*e + b*e*x)/(c + d*x))**2*a*b**4*d**3*x**3 + 18*log((a*e + b*e*x)/(c 
+ d*x))*a**4*b*c*d**2 + 54*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*c*d**2*x 
 + 6*log((a*e + b*e*x)/(c + d*x))*a**3*b**2*c*d**2 - 18*log((a*e + b*e*x)/ 
(c + d*x))*a**2*b**3*c**3 - 54*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c**2 
*d*x + 18*log((a*e + b*e*x)/(c + d*x))*a**2*b**3*c*d**2*x - 18*log((a*e + 
b*e*x)/(c + d*x))*a**2*b**3*d**3*x**3 - 6*log((a*e + b*e*x)/(c + d*x))*a*b 
**4*c**3 - 18*log((a*e + b*e*x)/(c + d*x))*a*b**4*c**2*d*x + 18*log((a*e + 
 b*e*x)/(c + d*x))*a*b**4*c*d**2*x**3 - 6*log((a*e + b*e*x)/(c + d*x))*a*b 
**4*d**3*x**3 + 6*log((a*e + b*e*x)/(c + d*x))*b**5*c*d**2*x**3 + 9*a**5*c 
*d**2 + 27*a**4*b*c*d**2*x + 6*a**4*b*c*d**2 - 9*a**3*b**2*c**3 - 27*a*...