\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^5} \, dx\) [72]

Optimal result

Integrand size = 42, antiderivative size = 299 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {2 B^2 d i^2 (c+d x)^3}{27 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B^2 i^2 (c+d x)^4}{32 (b c-a d)^2 g^5 (a+b x)^4}+\frac {2 B d i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^2 g^5 (a+b x)^4}+\frac {d i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^2 g^5 (a+b x)^4} \] Output:

2/27*B^2*d*i^2*(d*x+c)^3/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/32*b*B^2*i^2*(d*x+c) 
^4/(-a*d+b*c)^2/g^5/(b*x+a)^4+2/9*B*d*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x 
+c)))/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/8*b*B*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/( 
d*x+c)))/(-a*d+b*c)^2/g^5/(b*x+a)^4+1/3*d*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/ 
(d*x+c)))^2/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/4*b*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+ 
a)/(d*x+c)))^2/(-a*d+b*c)^2/g^5/(b*x+a)^4
 

Mathematica [C] (verified)

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

Time = 3.01 (sec) , antiderivative size = 1703, normalized size of antiderivative = 5.70 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + 
b*g*x)^5,x]
 

Output:

-1/864*(i^2*(216*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 57 
6*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 43 
2*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 3 
2*B*d*(a + b*x)*(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)]) + 3*B*(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*(-(b*c) + a*d)*(a + b*x)^3 - 14...
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.74, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^2 \left (-\frac {b (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}-\frac {b B (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 (a+b x)^4}+\frac {d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 (a+b x)^3}+\frac {2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 (a+b x)^3}-\frac {b B^2 (c+d x)^4}{32 (a+b x)^4}+\frac {2 B^2 d (c+d x)^3}{27 (a+b x)^3}\right )}{g^5 (b c-a d)^2}\)

Input:

Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x) 
^5,x]
 

Output:

(i^2*((2*B^2*d*(c + d*x)^3)/(27*(a + b*x)^3) - (b*B^2*(c + d*x)^4)/(32*(a 
+ b*x)^4) + (2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(a 
 + b*x)^3) - (b*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(a 
+ b*x)^4) + (d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(a + 
 b*x)^3) - (b*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(a + 
b*x)^4)))/((b*c - a*d)^2*g^5)
 

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. \(684\) vs. \(2(287)=574\).

Time = 2.43 (sec) , antiderivative size = 685, normalized size of antiderivative = 2.29

method	result	size
parts	\(\frac {i^{2} A^{2} \left (-\frac {d^{2}}{2 b^{3} \left (b x +a \right )^{2}}-\frac {a^{2} d^{2}-2 a c d b +c^{2} b^{2}}{4 b^{3} \left (b x +a \right )^{4}}+\frac {2 d \left (d a -b c \right )}{3 b^{3} \left (b x +a \right )^{3}}\right )}{g^{5}}-\frac {i^{2} B^{2} \left (d a -b c \right )^{3} e^{3} \left (\frac {d^{5} \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 )^{5}}-\frac {d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{5}}\right )}{g^{5} d^{4}}-\frac {2 i^{2} A B \left (d a -b c \right )^{3} e^{3} \left (\frac {d^{5} \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 )^{5}}-\frac {d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{5}}\right )}{g^{5} d^{4}}\)	\(685\)
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{2} d^{2} e^{3} A^{2} b}{4 \left (d a -b c \right )^{3} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{3} e^{2} A^{2}}{3 \left (d a -b c \right )^{3} g^{5} \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^{3} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{5}}+\frac {2 i^{2} d^{3} 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 )^{3} g^{5}}-\frac {i^{2} d^{2} e^{3} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{5}}+\frac {i^{2} d^{3} 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 )^{3} g^{5}}\right )}{d^{2}}\)	\(722\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{2} d^{2} e^{3} A^{2} b}{4 \left (d a -b c \right )^{3} g^{5} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {i^{2} d^{3} e^{2} A^{2}}{3 \left (d a -b c \right )^{3} g^{5} \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^{3} A B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{5}}+\frac {2 i^{2} d^{3} 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 )^{3} g^{5}}-\frac {i^{2} d^{2} e^{3} B^{2} b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (d a -b c \right )^{3} g^{5}}+\frac {i^{2} d^{3} 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 )^{3} g^{5}}\right )}{d^{2}}\)	\(722\)
norman	\(\text {Expression too large to display}\)	\(1311\)
orering	\(\text {Expression too large to display}\)	\(1513\)
parallelrisch	\(\text {Expression too large to display}\)	\(1666\)
risch	\(\text {Expression too large to display}\)	\(3450\)

Input:

int((d*i*x+c*i)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x,method=_RE 
TURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 837 vs. \(2 (287) = 574\).

Time = 0.10 (sec) , antiderivative size = 837, normalized size of antiderivative = 2.80 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, al 
gorithm="fricas")
 

Output:

1/864*(12*((12*A*B + 7*B^2)*b^4*c*d^3 - (12*A*B + 7*B^2)*a*b^3*d^4)*i^2*x^ 
3 - 6*((72*A^2 + 12*A*B - 5*B^2)*b^4*c^2*d^2 - 16*(9*A^2 + 6*A*B + 2*B^2)* 
a*b^3*c*d^3 + (72*A^2 + 84*A*B + 37*B^2)*a^2*b^2*d^4)*i^2*x^2 - 4*((144*A^ 
2 + 60*A*B + 11*B^2)*b^4*c^3*d - 24*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^2*d^2 
+ (72*A^2 + 84*A*B + 37*B^2)*a^3*b*d^4)*i^2*x - (27*(8*A^2 + 4*A*B + B^2)* 
b^4*c^4 - 32*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^3*d + (72*A^2 + 84*A*B + 37*B 
^2)*a^4*d^4)*i^2 + 72*(B^2*b^4*d^4*i^2*x^4 + 4*B^2*a*b^3*d^4*i^2*x^3 - 6*( 
B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3)*i^2*x^2 - 4*(2*B^2*b^4*c^3*d - 3*B^2* 
a*b^3*c^2*d^2)*i^2*x - (3*B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d)*i^2)*log((b*e*x 
 + a*e)/(d*x + c))^2 + 12*((12*A*B + 7*B^2)*b^4*d^4*i^2*x^4 + 4*(3*B^2*b^4 
*c*d^3 + 4*(3*A*B + B^2)*a*b^3*d^4)*i^2*x^3 - 6*((12*A*B + B^2)*b^4*c^2*d^ 
2 - 8*(3*A*B + B^2)*a*b^3*c*d^3)*i^2*x^2 - 4*((24*A*B + 5*B^2)*b^4*c^3*d - 
 12*(3*A*B + B^2)*a*b^3*c^2*d^2)*i^2*x - (9*(4*A*B + B^2)*b^4*c^4 - 16*(3* 
A*B + B^2)*a*b^3*c^3*d)*i^2)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^2 - 2*a 
*b^8*c*d + a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d + a^3*b^6*d 
^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)*g^5*x^2 + 4*(a 
^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 2*a^5*b^4 
*c*d + a^6*b^3*d^2)*g^5)
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2055 vs. \(2 (277) = 554\).

Time = 54.50 (sec) , antiderivative size = 2055, normalized size of antiderivative = 6.87 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**5,x)
 

Output:

-B*d**4*i**2*(12*A + 7*B)*log(x + (12*A*B*a*d**5*i**2 + 12*A*B*b*c*d**4*i* 
*2 + 7*B**2*a*d**5*i**2 + 7*B**2*b*c*d**4*i**2 - B*a**3*d**7*i**2*(12*A + 
7*B)/(a*d - b*c)**2 + 3*B*a**2*b*c*d**6*i**2*(12*A + 7*B)/(a*d - b*c)**2 - 
 3*B*a*b**2*c**2*d**5*i**2*(12*A + 7*B)/(a*d - b*c)**2 + B*b**3*c**3*d**4* 
i**2*(12*A + 7*B)/(a*d - b*c)**2)/(24*A*B*b*d**5*i**2 + 14*B**2*b*d**5*i** 
2))/(72*b**3*g**5*(a*d - b*c)**2) + B*d**4*i**2*(12*A + 7*B)*log(x + (12*A 
*B*a*d**5*i**2 + 12*A*B*b*c*d**4*i**2 + 7*B**2*a*d**5*i**2 + 7*B**2*b*c*d* 
*4*i**2 + B*a**3*d**7*i**2*(12*A + 7*B)/(a*d - b*c)**2 - 3*B*a**2*b*c*d**6 
*i**2*(12*A + 7*B)/(a*d - b*c)**2 + 3*B*a*b**2*c**2*d**5*i**2*(12*A + 7*B) 
/(a*d - b*c)**2 - B*b**3*c**3*d**4*i**2*(12*A + 7*B)/(a*d - b*c)**2)/(24*A 
*B*b*d**5*i**2 + 14*B**2*b*d**5*i**2))/(72*b**3*g**5*(a*d - b*c)**2) + (4* 
B**2*a*c**3*d*i**2 + 12*B**2*a*c**2*d**2*i**2*x + 12*B**2*a*c*d**3*i**2*x* 
*2 + 4*B**2*a*d**4*i**2*x**3 - 3*B**2*b*c**4*i**2 - 8*B**2*b*c**3*d*i**2*x 
 - 6*B**2*b*c**2*d**2*i**2*x**2 + B**2*b*d**4*i**2*x**4)*log(e*(a + b*x)/( 
c + d*x))**2/(12*a**6*d**2*g**5 - 24*a**5*b*c*d*g**5 + 48*a**5*b*d**2*g**5 
*x + 12*a**4*b**2*c**2*g**5 - 96*a**4*b**2*c*d*g**5*x + 72*a**4*b**2*d**2* 
g**5*x**2 + 48*a**3*b**3*c**2*g**5*x - 144*a**3*b**3*c*d*g**5*x**2 + 48*a* 
*3*b**3*d**2*g**5*x**3 + 72*a**2*b**4*c**2*g**5*x**2 - 96*a**2*b**4*c*d*g* 
*5*x**3 + 12*a**2*b**4*d**2*g**5*x**4 + 48*a*b**5*c**2*g**5*x**3 - 24*a*b* 
*5*c*d*g**5*x**4 + 12*b**6*c**2*g**5*x**4) + (-12*A*B*a**3*d**3*i**2 - ...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8031 vs. \(2 (287) = 574\).

Time = 0.61 (sec) , antiderivative size = 8031, normalized size of antiderivative = 26.86 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, al 
gorithm="maxima")
 

Output:

-1/6*(4*b*x + a)*B^2*c*d*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^6*g 
^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g 
^5) - 1/12*(6*b^2*x^2 + 4*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^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3 
*b^4*g^5*x + a^4*b^3*g^5) + 1/288*(12*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a* 
b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 
+ 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2* 
d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2* 
d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^ 
2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4* 
c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^ 
2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 
4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12 
*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^ 
2*c*d^3 + a^4*b*d^4)*g^5))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (9*b^4*c 
^4 - 64*a*b^3*c^3*d + 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 
- 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 
163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x 
^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d 
^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c)^2 - ...
 

Giac [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.60 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b e^{5} i^{2} - \frac {4 \, {\left (b e x + a e\right )} B^{2} d e^{4} i^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (36 \, A B b e^{5} i^{2} + 9 \, B^{2} b e^{5} i^{2} - \frac {48 \, {\left (b e x + a e\right )} A B d e^{4} i^{2}}{d x + c} - \frac {16 \, {\left (b e x + a e\right )} B^{2} d e^{4} i^{2}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {216 \, A^{2} b e^{5} i^{2} + 108 \, A B b e^{5} i^{2} + 27 \, B^{2} b e^{5} i^{2} - \frac {288 \, {\left (b e x + a e\right )} A^{2} d e^{4} i^{2}}{d x + c} - \frac {192 \, {\left (b e x + a e\right )} A B d e^{4} i^{2}}{d x + c} - \frac {64 \, {\left (b e x + a e\right )} B^{2} d e^{4} i^{2}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, al 
gorithm="giac")
 

Output:

-1/864*(72*(3*B^2*b*e^5*i^2 - 4*(b*e*x + a*e)*B^2*d*e^4*i^2/(d*x + c))*log 
((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^4*b*c*g^5/(d*x + c)^4 - (b*e*x 
+ a*e)^4*a*d*g^5/(d*x + c)^4) + 12*(36*A*B*b*e^5*i^2 + 9*B^2*b*e^5*i^2 - 4 
8*(b*e*x + a*e)*A*B*d*e^4*i^2/(d*x + c) - 16*(b*e*x + a*e)*B^2*d*e^4*i^2/( 
d*x + c))*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b*c*g^5/(d*x + c)^ 
4 - (b*e*x + a*e)^4*a*d*g^5/(d*x + c)^4) + (216*A^2*b*e^5*i^2 + 108*A*B*b* 
e^5*i^2 + 27*B^2*b*e^5*i^2 - 288*(b*e*x + a*e)*A^2*d*e^4*i^2/(d*x + c) - 1 
92*(b*e*x + a*e)*A*B*d*e^4*i^2/(d*x + c) - 64*(b*e*x + a*e)*B^2*d*e^4*i^2/ 
(d*x + c))/((b*e*x + a*e)^4*b*c*g^5/(d*x + c)^4 - (b*e*x + a*e)^4*a*d*g^5/ 
(d*x + c)^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 = 32.74 (sec) , antiderivative size = 1940, normalized size of antiderivative = 6.49 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x) 
^5,x)
 

Output:

- log((e*(a + b*x))/(c + d*x))^2*((x*(b*((B^2*c*d*i^2)/(6*b^3*g^5) + (B^2* 
a*d^2*i^2)/(12*b^4*g^5)) + (B^2*c*d*i^2)/(2*b^2*g^5) + (B^2*a*d^2*i^2)/(4* 
b^3*g^5)) + a*((B^2*c*d*i^2)/(6*b^3*g^5) + (B^2*a*d^2*i^2)/(12*b^4*g^5)) + 
 (B^2*c^2*i^2)/(4*b^2*g^5) + (B^2*d^2*i^2*x^2)/(2*b^2*g^5))/(4*a^3*x + a^4 
/b + b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3) - (B^2*d^4*i^2)/(12*b^3*g^5*(a^2 
*d^2 + b^2*c^2 - 2*a*b*c*d))) - ((72*A^2*a^3*d^3*i^2 - 216*A^2*b^3*c^3*i^2 
 + 37*B^2*a^3*d^3*i^2 - 27*B^2*b^3*c^3*i^2 + 84*A*B*a^3*d^3*i^2 - 108*A*B* 
b^3*c^3*i^2 + 72*A^2*a*b^2*c^2*d*i^2 + 72*A^2*a^2*b*c*d^2*i^2 + 37*B^2*a*b 
^2*c^2*d*i^2 + 37*B^2*a^2*b*c*d^2*i^2 + 84*A*B*a*b^2*c^2*d*i^2 + 84*A*B*a^ 
2*b*c*d^2*i^2)/(12*(a*d - b*c)) + (x^3*(7*B^2*b^3*d^3*i^2 + 12*A*B*b^3*d^3 
*i^2))/(a*d - b*c) + (x*(72*A^2*a^2*b*d^3*i^2 + 37*B^2*a^2*b*d^3*i^2 - 144 
*A^2*b^3*c^2*d*i^2 - 11*B^2*b^3*c^2*d*i^2 + 72*A^2*a*b^2*c*d^2*i^2 + 37*B^ 
2*a*b^2*c*d^2*i^2 + 84*A*B*a^2*b*d^3*i^2 - 60*A*B*b^3*c^2*d*i^2 + 84*A*B*a 
*b^2*c*d^2*i^2))/(3*(a*d - b*c)) + (x^2*(72*A^2*a*b^2*d^3*i^2 + 37*B^2*a*b 
^2*d^3*i^2 - 72*A^2*b^3*c*d^2*i^2 + 5*B^2*b^3*c*d^2*i^2 + 84*A*B*a*b^2*d^3 
*i^2 - 12*A*B*b^3*c*d^2*i^2))/(2*(a*d - b*c)))/(72*a^4*b^3*g^5 + 72*b^7*g^ 
5*x^4 + 288*a^3*b^4*g^5*x + 288*a*b^6*g^5*x^3 + 432*a^2*b^5*g^5*x^2) - (lo 
g((e*(a + b*x))/(c + d*x))*(x^2*((A*B*d*i^2)/(b^2*g^5) + (B^2*d^4*i^2*(b*( 
b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(12*b*d^3) + (a*(a*d - b*c))/(4*b*d^2 
)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*d^3) + (a*(a*d - b*c))/(2*d^2...
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 2107, normalized size of antiderivative = 7.05 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

int((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x)
 

Output:

( - 144*log(a + b*x)*a**6*b*d**4 - 576*log(a + b*x)*a**5*b**2*d**4*x - 48* 
log(a + b*x)*a**5*b**2*d**4 - 36*log(a + b*x)*a**4*b**3*c*d**3 - 864*log(a 
 + b*x)*a**4*b**3*d**4*x**2 - 192*log(a + b*x)*a**4*b**3*d**4*x - 144*log( 
a + b*x)*a**3*b**4*c*d**3*x - 576*log(a + b*x)*a**3*b**4*d**4*x**3 - 288*l 
og(a + b*x)*a**3*b**4*d**4*x**2 - 216*log(a + b*x)*a**2*b**5*c*d**3*x**2 - 
 144*log(a + b*x)*a**2*b**5*d**4*x**4 - 192*log(a + b*x)*a**2*b**5*d**4*x* 
*3 - 144*log(a + b*x)*a*b**6*c*d**3*x**3 - 48*log(a + b*x)*a*b**6*d**4*x** 
4 - 36*log(a + b*x)*b**7*c*d**3*x**4 + 144*log(c + d*x)*a**6*b*d**4 + 576* 
log(c + d*x)*a**5*b**2*d**4*x + 48*log(c + d*x)*a**5*b**2*d**4 + 36*log(c 
+ d*x)*a**4*b**3*c*d**3 + 864*log(c + d*x)*a**4*b**3*d**4*x**2 + 192*log(c 
 + d*x)*a**4*b**3*d**4*x + 144*log(c + d*x)*a**3*b**4*c*d**3*x + 576*log(c 
 + d*x)*a**3*b**4*d**4*x**3 + 288*log(c + d*x)*a**3*b**4*d**4*x**2 + 216*l 
og(c + d*x)*a**2*b**5*c*d**3*x**2 + 144*log(c + d*x)*a**2*b**5*d**4*x**4 + 
 192*log(c + d*x)*a**2*b**5*d**4*x**3 + 144*log(c + d*x)*a*b**6*c*d**3*x** 
3 + 48*log(c + d*x)*a*b**6*d**4*x**4 + 36*log(c + d*x)*b**7*c*d**3*x**4 - 
288*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**5*c**3*d - 864*log((a*e + b*e* 
x)/(c + d*x))**2*a**2*b**5*c**2*d**2*x - 864*log((a*e + b*e*x)/(c + d*x))* 
*2*a**2*b**5*c*d**3*x**2 - 288*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**5*d 
**4*x**3 + 216*log((a*e + b*e*x)/(c + d*x))**2*a*b**6*c**4 + 576*log((a*e 
+ b*e*x)/(c + d*x))**2*a*b**6*c**3*d*x + 432*log((a*e + b*e*x)/(c + d*x...