\(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^5} \, dx\) [63]

Optimal result

Integrand size = 40, antiderivative size = 445 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {B^2 d^2 i (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B^2 d i (c+d x)^3}{27 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B^2 i (c+d x)^4}{32 (b c-a d)^3 g^5 (a+b x)^4}-\frac {B d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^3 g^5 (a+b x)^4} \] Output:

-1/4*B^2*d^2*i*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+4/27*b*B^2*d*i*(d*x+c) 
^3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/32*b^2*B^2*i*(d*x+c)^4/(-a*d+b*c)^3/g^5/(b 
*x+a)^4-1/2*B*d^2*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5 
/(b*x+a)^2+4/9*b*B*d*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/ 
g^5/(b*x+a)^3-1/8*b^2*B*i*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c) 
^3/g^5/(b*x+a)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b 
*c)^3/g^5/(b*x+a)^2+2/3*b*d*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a* 
d+b*c)^3/g^5/(b*x+a)^3-1/4*b^2*i*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/( 
-a*d+b*c)^3/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 = 1.21 (sec) , antiderivative size = 1255, normalized size of antiderivative = 2.82 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/864*(i*(216*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 288* 
d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 16*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 - 144*A*d^ 
4*(a + b*x)^4*Log[a + b*x] - 300*B*d^4*(a + b*x)^4*Log[a + b*x] + 72*B*...
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 334, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, 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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(i*(-1/4*(B^2*d^2*(c + d*x)^2)/(a + b*x)^2 + (4*b*B^2*d*(c + d*x)^3)/(27*( 
a + b*x)^3) - (b^2*B^2*(c + d*x)^4)/(32*(a + b*x)^4) - (B*d^2*(c + d*x)^2* 
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(a + b*x)^2) + (4*b*B*d*(c + d*x) 
^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(a + b*x)^3) - (b^2*B*(c + d*x 
)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(a + b*x)^4) - (d^2*(c + d*x) 
^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(a + b*x)^2) + (2*b*d*(c + d 
*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(a + b*x)^3) - (b^2*(c + 
d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(a + b*x)^4)))/((b*c - a 
*d)^3*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. \(925\) vs. \(2(427)=854\).

Time = 2.36 (sec) , antiderivative size = 926, normalized size of antiderivative = 2.08

method	result	size
parts	\(\frac {i \,A^{2} \left (-\frac {-d a +b c}{4 b^{2} \left (b x +a \right )^{4}}-\frac {d}{3 b^{2} \left (b x +a \right )^{3}}\right )}{g^{5}}-\frac {i \,B^{2} \left (d a -b c \right )^{2} e^{2} \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}}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{5}}-\frac {2 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}}{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^{3} b^{2} e^{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}}{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^{3}}-\frac {2 i A B \left (d a -b c \right )^{2} e^{2} \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 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (d a -b c \right )^{5}}-\frac {2 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 )}{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^{3} b^{2} e^{2} \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^{3}}\)	\(926\)
derivativedivides	\(\text {Expression too large to display}\)	\(1058\)
default	\(\text {Expression too large to display}\)	\(1058\)
orering	\(\text {Expression too large to display}\)	\(1478\)
norman	\(\text {Expression too large to display}\)	\(1577\)
parallelrisch	\(\text {Expression too large to display}\)	\(1949\)
risch	\(\text {Expression too large to display}\)	\(3657\)

Input:

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

Output:

i*A^2/g^5*(-1/4*(-a*d+b*c)/b^2/(b*x+a)^4-1/3*d/b^2/(b*x+a)^3)-i*B^2/g^5/d^ 
3*(a*d-b*c)^2*e^2*(d^5/(a*d-b*c)^5*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*l 
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b* 
e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)-2*d^4/(a*d 
-b*c)^5*b*e*(-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^3/(a*d-b*c)^5*b^2*e^2*(-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*A*B/g^5/d^3*(a*d-b*c)^2*e^2*(d^5/(a*d-b*c)^5*(- 
1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b 
*e/d+(a*d-b*c)*e/d/(d*x+c))^2)-2*d^4/(a*d-b*c)^5*b*e*(-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^3/(a*d-b*c)^5*b^2*e^2*(-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. 985 vs. \(2 (427) = 854\).

Time = 0.10 (sec) , antiderivative size = 985, normalized size of antiderivative = 2.21 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/864*(12*((12*A*B + 13*B^2)*b^4*c*d^3 - (12*A*B + 13*B^2)*a*b^3*d^4)*i*x 
^3 - 6*((12*A*B + B^2)*b^4*c^2*d^2 - 16*(6*A*B + 5*B^2)*a*b^3*c*d^3 + (84* 
A*B + 79*B^2)*a^2*b^2*d^4)*i*x^2 + 4*((72*A^2 + 12*A*B - 5*B^2)*b^4*c^3*d 
- 12*(18*A^2 + 6*A*B - B^2)*a*b^3*c^2*d^2 + 108*(2*A^2 + 2*A*B + B^2)*a^2* 
b^2*c*d^3 - (72*A^2 + 156*A*B + 115*B^2)*a^3*b*d^4)*i*x + 72*(B^2*b^4*d^4* 
i*x^4 + 4*B^2*a*b^3*d^4*i*x^3 + 6*B^2*a^2*b^2*d^4*i*x^2 + 4*(B^2*b^4*c^3*d 
 - 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2*c*d^3)*i*x + (3*B^2*b^4*c^4 - 8*B^2 
*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2)*i)*log((b*e*x + a*e)/(d*x + c))^2 + 
(27*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 64*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^3*d 
 + 216*(2*A^2 + 2*A*B + B^2)*a^2*b^2*c^2*d^2 - (72*A^2 + 156*A*B + 115*B^2 
)*a^4*d^4)*i + 12*((12*A*B + 13*B^2)*b^4*d^4*i*x^4 + 4*(3*B^2*b^4*c*d^3 + 
2*(6*A*B + 5*B^2)*a*b^3*d^4)*i*x^3 - 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^ 
3 - 6*(2*A*B + B^2)*a^2*b^2*d^4)*i*x^2 + 4*((12*A*B + B^2)*b^4*c^3*d - 6*( 
6*A*B + B^2)*a*b^3*c^2*d^2 + 18*(2*A*B + B^2)*a^2*b^2*c*d^3)*i*x + (9*(4*A 
*B + B^2)*b^4*c^4 - 32*(3*A*B + B^2)*a*b^3*c^3*d + 36*(2*A*B + B^2)*a^2*b^ 
2*c^2*d^2)*i)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^3 - 3*a*b^8*c^2*d + 3* 
a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3* 
a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 
3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d 
+ 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d...
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2251 vs. \(2 (416) = 832\).

Time = 90.49 (sec) , antiderivative size = 2251, normalized size of antiderivative = 5.06 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-B*d**4*i*(12*A + 13*B)*log(x + (12*A*B*a*d**5*i + 12*A*B*b*c*d**4*i + 13* 
B**2*a*d**5*i + 13*B**2*b*c*d**4*i - B*a**4*d**8*i*(12*A + 13*B)/(a*d - b* 
c)**3 + 4*B*a**3*b*c*d**7*i*(12*A + 13*B)/(a*d - b*c)**3 - 6*B*a**2*b**2*c 
**2*d**6*i*(12*A + 13*B)/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**5*i*(12*A + 1 
3*B)/(a*d - b*c)**3 - B*b**4*c**4*d**4*i*(12*A + 13*B)/(a*d - b*c)**3)/(24 
*A*B*b*d**5*i + 26*B**2*b*d**5*i))/(72*b**2*g**5*(a*d - b*c)**3) + B*d**4* 
i*(12*A + 13*B)*log(x + (12*A*B*a*d**5*i + 12*A*B*b*c*d**4*i + 13*B**2*a*d 
**5*i + 13*B**2*b*c*d**4*i + B*a**4*d**8*i*(12*A + 13*B)/(a*d - b*c)**3 - 
4*B*a**3*b*c*d**7*i*(12*A + 13*B)/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**6 
*i*(12*A + 13*B)/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**5*i*(12*A + 13*B)/(a* 
d - b*c)**3 + B*b**4*c**4*d**4*i*(12*A + 13*B)/(a*d - b*c)**3)/(24*A*B*b*d 
**5*i + 26*B**2*b*d**5*i))/(72*b**2*g**5*(a*d - b*c)**3) + (6*B**2*a**2*c* 
*2*d**2*i + 12*B**2*a**2*c*d**3*i*x + 6*B**2*a**2*d**4*i*x**2 - 8*B**2*a*b 
*c**3*d*i - 12*B**2*a*b*c**2*d**2*i*x + 4*B**2*a*b*d**4*i*x**3 + 3*B**2*b* 
*2*c**4*i + 4*B**2*b**2*c**3*d*i*x + B**2*b**2*d**4*i*x**4)*log(e*(a + b*x 
)/(c + d*x))**2/(12*a**7*d**3*g**5 - 36*a**6*b*c*d**2*g**5 + 48*a**6*b*d** 
3*g**5*x + 36*a**5*b**2*c**2*d*g**5 - 144*a**5*b**2*c*d**2*g**5*x + 72*a** 
5*b**2*d**3*g**5*x**2 - 12*a**4*b**3*c**3*g**5 + 144*a**4*b**3*c**2*d*g**5 
*x - 216*a**4*b**3*c*d**2*g**5*x**2 + 48*a**4*b**3*d**3*g**5*x**3 - 48*a** 
3*b**4*c**3*g**5*x + 216*a**3*b**4*c**2*d*g**5*x**2 - 144*a**3*b**4*c*d...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4808 vs. \(2 (427) = 854\).

Time = 0.41 (sec) , antiderivative size = 4808, normalized size of antiderivative = 10.80 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/12*(4*b*x + a)*B^2*d*i*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) 
 + 1/288*(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 - 4*(7*b^4*c^3*d - 60*a*b^3*c^ 
2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(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) + 1...
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.67 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/864*(72*(3*B^2*b^2*e^5*i - 8*(b*e*x + a*e)*B^2*b*d*e^4*i/(d*x + c) + 6* 
(b*e*x + a*e)^2*B^2*d^2*e^3*i/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))^2/ 
((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b*c*d*g^5/( 
d*x + c)^4 + (b*e*x + a*e)^4*a^2*d^2*g^5/(d*x + c)^4) + 12*(36*A*B*b^2*e^5 
*i + 9*B^2*b^2*e^5*i - 96*(b*e*x + a*e)*A*B*b*d*e^4*i/(d*x + c) - 32*(b*e* 
x + a*e)*B^2*b*d*e^4*i/(d*x + c) + 72*(b*e*x + a*e)^2*A*B*d^2*e^3*i/(d*x + 
 c)^2 + 36*(b*e*x + a*e)^2*B^2*d^2*e^3*i/(d*x + c)^2)*log((b*e*x + a*e)/(d 
*x + c))/((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b* 
c*d*g^5/(d*x + c)^4 + (b*e*x + a*e)^4*a^2*d^2*g^5/(d*x + c)^4) + (216*A^2* 
b^2*e^5*i + 108*A*B*b^2*e^5*i + 27*B^2*b^2*e^5*i - 576*(b*e*x + a*e)*A^2*b 
*d*e^4*i/(d*x + c) - 384*(b*e*x + a*e)*A*B*b*d*e^4*i/(d*x + c) - 128*(b*e* 
x + a*e)*B^2*b*d*e^4*i/(d*x + c) + 432*(b*e*x + a*e)^2*A^2*d^2*e^3*i/(d*x 
+ c)^2 + 432*(b*e*x + a*e)^2*A*B*d^2*e^3*i/(d*x + c)^2 + 216*(b*e*x + a*e) 
^2*B^2*d^2*e^3*i/(d*x + c)^2)/((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2 
*(b*e*x + a*e)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*e*x + a*e)^4*a^2*d^2*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 = 33.25 (sec) , antiderivative size = 1870, normalized size of antiderivative = 4.20 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

((72*A^2*a^3*d^3*i + 216*A^2*b^3*c^3*i + 115*B^2*a^3*d^3*i + 27*B^2*b^3*c^ 
3*i + 156*A*B*a^3*d^3*i + 108*A*B*b^3*c^3*i - 360*A^2*a*b^2*c^2*d*i + 72*A 
^2*a^2*b*c*d^2*i - 101*B^2*a*b^2*c^2*d*i + 115*B^2*a^2*b*c*d^2*i - 276*A*B 
*a*b^2*c^2*d*i + 156*A*B*a^2*b*c*d^2*i)/(12*(a*d - b*c)) + (x^2*(79*B^2*a* 
b^2*d^3*i - B^2*b^3*c*d^2*i + 84*A*B*a*b^2*d^3*i - 12*A*B*b^3*c*d^2*i))/(2 
*(a*d - b*c)) + (x*(72*A^2*a^2*b*d^3*i + 115*B^2*a^2*b*d^3*i + 72*A^2*b^3* 
c^2*d*i - 5*B^2*b^3*c^2*d*i + 156*A*B*a^2*b*d^3*i + 12*A*B*b^3*c^2*d*i - 1 
44*A^2*a*b^2*c*d^2*i + 7*B^2*a*b^2*c*d^2*i - 60*A*B*a*b^2*c*d^2*i))/(3*(a* 
d - b*c)) + (d*x^3*(13*B^2*b^3*d^2*i + 12*A*B*b^3*d^2*i))/(a*d - b*c))/(x* 
(288*a^3*b^4*c*g^5 - 288*a^4*b^3*d*g^5) - x^3*(288*a^2*b^5*d*g^5 - 288*a*b 
^6*c*g^5) + x^4*(72*b^7*c*g^5 - 72*a*b^6*d*g^5) + x^2*(432*a^2*b^5*c*g^5 - 
 432*a^3*b^4*d*g^5) + 72*a^4*b^3*c*g^5 - 72*a^5*b^2*d*g^5) - log((e*(a + b 
*x))/(c + d*x))^2*(((B^2*c*i)/(4*b^2*g^5) + (B^2*a*d*i)/(12*b^3*g^5) + (B^ 
2*d*i*x)/(3*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)/(12*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b 
*c*d^2))) - (log((e*(a + b*x))/(c + d*x))*(x*((2*A*B*i)/(3*b^2*g^5) + (B^2 
*d^4*i*(b*(a*((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)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(12 
*b*d^4)) + a*(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 ...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 2224, normalized size of antiderivative = 5.00 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*(144*log(a + b*x)*a**6*b*d**4 + 576*log(a + b*x)*a**5*b**2*d**4*x + 120 
*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 + 480*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 + 720* 
log(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 + 480*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 + 120*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 - 57 
6*log(c + d*x)*a**5*b**2*d**4*x - 120*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 - 480*lo 
g(c + d*x)*a**4*b**3*d**4*x - 144*log(c + d*x)*a**3*b**4*c*d**3*x - 576*lo 
g(c + d*x)*a**3*b**4*d**4*x**3 - 720*log(c + d*x)*a**3*b**4*d**4*x**2 - 21 
6*log(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 - 480*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 - 120*log(c + d*x)*a*b**6*d**4*x**4 - 36*log(c + d*x)*b**7*c*d**3*x** 
4 + 432*log((a*e + b*e*x)/(c + d*x))**2*a**3*b**4*c**2*d**2 + 864*log((a*e 
 + b*e*x)/(c + d*x))**2*a**3*b**4*c*d**3*x + 432*log((a*e + b*e*x)/(c + d* 
x))**2*a**3*b**4*d**4*x**2 - 576*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 + 288* 
log((a*e + b*e*x)/(c + d*x))**2*a**2*b**5*d**4*x**3 + 216*log((a*e + b*...