\(\int \frac {(c i+d i x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^6} \, dx\) [82]

Optimal result

Integrand size = 42, antiderivative size = 299 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\frac {B^2 d i^3 (c+d x)^4}{32 (b c-a d)^2 g^6 (a+b x)^4}-\frac {2 b B^2 i^3 (c+d x)^5}{125 (b c-a d)^2 g^6 (a+b x)^5}+\frac {B d i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^2 g^6 (a+b x)^4}-\frac {2 b B i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 (b c-a d)^2 g^6 (a+b x)^5}+\frac {d i^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^2 g^6 (a+b x)^4}-\frac {b i^3 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 (b c-a d)^2 g^6 (a+b x)^5} \] Output:

1/32*B^2*d*i^3*(d*x+c)^4/(-a*d+b*c)^2/g^6/(b*x+a)^4-2/125*b*B^2*i^3*(d*x+c 
)^5/(-a*d+b*c)^2/g^6/(b*x+a)^5+1/8*B*d*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d* 
x+c)))/(-a*d+b*c)^2/g^6/(b*x+a)^4-2/25*b*B*i^3*(d*x+c)^5*(A+B*ln(e*(b*x+a) 
/(d*x+c)))/(-a*d+b*c)^2/g^6/(b*x+a)^5+1/4*d*i^3*(d*x+c)^4*(A+B*ln(e*(b*x+a 
)/(d*x+c)))^2/(-a*d+b*c)^2/g^6/(b*x+a)^4-1/5*b*i^3*(d*x+c)^5*(A+B*ln(e*(b* 
x+a)/(d*x+c)))^2/(-a*d+b*c)^2/g^6/(b*x+a)^5
 

Mathematica [C] (verified)

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

Time = 5.02 (sec) , antiderivative size = 2456, normalized size of antiderivative = 8.21 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Result too large to show} \] Input:

Integrate[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + 
b*g*x)^6,x]
 

Output:

-1/36000*(i^3*(7200*(b*c - a*d)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 
 27000*d*(b*c - a*d)^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 
36000*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)] 
)^2 + 18000*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d* 
x)])^2 + 2000*B*d^2*(a + b*x)^2*(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*L 
og[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*Poly 
Log[2, (b*(c + d*x))/(b*c - a*d)]) + 375*B*d*(a + b*x)*(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...
 

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.

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

\(\Big \downarrow \) 2962

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

\(\Big \downarrow \) 2795

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {i^3 \left (-\frac {b (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 (a+b x)^5}-\frac {2 b B (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{25 (a+b x)^5}+\frac {d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}+\frac {B d (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 (a+b x)^4}-\frac {2 b B^2 (c+d x)^5}{125 (a+b x)^5}+\frac {B^2 d (c+d x)^4}{32 (a+b x)^4}\right )}{g^6 (b c-a d)^2}\)

Input:

Int[((c*i + d*i*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x) 
^6,x]
 

Output:

(i^3*((B^2*d*(c + d*x)^4)/(32*(a + b*x)^4) - (2*b*B^2*(c + d*x)^5)/(125*(a 
 + b*x)^5) + (B*d*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8*(a 
+ b*x)^4) - (2*b*B*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(25*( 
a + b*x)^5) + (d*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(a 
 + b*x)^4) - (b*(c + d*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*(a 
+ b*x)^5)))/((b*c - a*d)^2*g^6)
 

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. \(721\) vs. \(2(287)=574\).

Time = 2.50 (sec) , antiderivative size = 722, normalized size of antiderivative = 2.41

method	result	size
derivativedivides	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{3} d^{2} e^{4} A^{2} b}{5 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {i^{3} d^{3} e^{3} A^{2}}{4 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {2 i^{3} d^{2} e^{4} 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 )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{3} g^{6}}+\frac {2 i^{3} d^{3} e^{3} 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 )}{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^{6}}-\frac {i^{3} d^{2} e^{4} 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}}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {2}{125 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{3} g^{6}}+\frac {i^{3} d^{3} e^{3} 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}}{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^{6}}\right )}{d^{2}}\)	\(722\)
default	\(-\frac {e \left (d a -b c \right ) \left (\frac {i^{3} d^{2} e^{4} A^{2} b}{5 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {i^{3} d^{3} e^{3} A^{2}}{4 \left (d a -b c \right )^{3} g^{6} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {2 i^{3} d^{2} e^{4} 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 )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{3} g^{6}}+\frac {2 i^{3} d^{3} e^{3} 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 )}{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^{6}}-\frac {i^{3} d^{2} e^{4} 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}}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {2}{125 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{3} g^{6}}+\frac {i^{3} d^{3} e^{3} 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}}{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^{6}}\right )}{d^{2}}\)	\(722\)
parts	\(\frac {i^{3} A^{2} \left (-\frac {-a^{3} d^{3}+3 a^{2} b c \,d^{2}-3 a \,b^{2} c^{2} d +b^{3} c^{3}}{5 b^{4} \left (b x +a \right )^{5}}-\frac {d^{3}}{2 b^{4} \left (b x +a \right )^{2}}-\frac {3 d \left (a^{2} d^{2}-2 a c d b +c^{2} b^{2}\right )}{4 b^{4} \left (b x +a \right )^{4}}+\frac {d^{2} \left (d a -b c \right )}{b^{4} \left (b x +a \right )^{3}}\right )}{g^{6}}-\frac {i^{3} B^{2} \left (d a -b c \right )^{4} e^{4} \left (\frac {d^{6} \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 )^{6}}-\frac {d^{5} 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}}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {2}{125 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{6}}\right )}{g^{6} d^{5}}-\frac {2 i^{3} A B \left (d a -b c \right )^{4} e^{4} \left (\frac {d^{6} \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 )^{6}}-\frac {d^{5} 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 )}{5 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}-\frac {1}{25 \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{5}}\right )}{\left (d a -b c \right )^{6}}\right )}{g^{6} d^{5}}\)	\(735\)
norman	\(\text {Expression too large to display}\)	\(1608\)
orering	\(\text {Expression too large to display}\)	\(1795\)
parallelrisch	\(\text {Expression too large to display}\)	\(1824\)
risch	\(\text {Expression too large to display}\)	\(4428\)

Input:

int((d*i*x+c*i)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x,method=_RE 
TURNVERBOSE)
 

Output:

-1/d^2*e*(a*d-b*c)*(1/5*i^3*d^2*e^4/(a*d-b*c)^3/g^6*A^2*b/(b*e/d+(a*d-b*c) 
*e/d/(d*x+c))^5-1/4*i^3*d^3*e^3/(a*d-b*c)^3/g^6*A^2/(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))^4-2*i^3*d^2*e^4/(a*d-b*c)^3/g^6*A*B*b*(-1/5/(b*e/d+(a*d-b*c)*e/d/( 
d*x+c))^5*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c 
))^5)+2*i^3*d^3*e^3/(a*d-b*c)^3/g^6*A*B*(-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)- 
i^3*d^2*e^4/(a*d-b*c)^3/g^6*B^2*b*(-1/5/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5*ln 
(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/25/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5*ln(b* 
e/d+(a*d-b*c)*e/d/(d*x+c))-2/125/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^5)+i^3*d^3* 
e^3/(a*d-b*c)^3/g^6*B^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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (287) = 574\).

Time = 0.10 (sec) , antiderivative size = 1045, normalized size of antiderivative = 3.49 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x, al 
gorithm="fricas")
 

Output:

1/4000*(20*((20*A*B + 9*B^2)*b^5*c*d^4 - (20*A*B + 9*B^2)*a*b^4*d^5)*i^3*x 
^4 - 10*((200*A^2 + 20*A*B - 11*B^2)*b^5*c^2*d^3 - 50*(8*A^2 + 4*A*B + B^2 
)*a*b^4*c*d^4 + (200*A^2 + 180*A*B + 61*B^2)*a^2*b^3*d^5)*i^3*x^3 - 10*(2* 
(200*A^2 + 60*A*B + 7*B^2)*b^5*c^3*d^2 - 75*(8*A^2 + 4*A*B + B^2)*a*b^4*c^ 
2*d^3 + (200*A^2 + 180*A*B + 61*B^2)*a^3*b^2*d^5)*i^3*x^2 - 5*((600*A^2 + 
220*A*B + 39*B^2)*b^5*c^4*d - 100*(8*A^2 + 4*A*B + B^2)*a*b^4*c^3*d^2 + (2 
00*A^2 + 180*A*B + 61*B^2)*a^4*b*d^5)*i^3*x - (32*(25*A^2 + 10*A*B + 2*B^2 
)*b^5*c^5 - 125*(8*A^2 + 4*A*B + B^2)*a*b^4*c^4*d + (200*A^2 + 180*A*B + 6 
1*B^2)*a^5*d^5)*i^3 + 200*(B^2*b^5*d^5*i^3*x^5 + 5*B^2*a*b^4*d^5*i^3*x^4 - 
 10*(B^2*b^5*c^2*d^3 - 2*B^2*a*b^4*c*d^4)*i^3*x^3 - 10*(2*B^2*b^5*c^3*d^2 
- 3*B^2*a*b^4*c^2*d^3)*i^3*x^2 - 5*(3*B^2*b^5*c^4*d - 4*B^2*a*b^4*c^3*d^2) 
*i^3*x - (4*B^2*b^5*c^5 - 5*B^2*a*b^4*c^4*d)*i^3)*log((b*e*x + a*e)/(d*x + 
 c))^2 + 20*((20*A*B + 9*B^2)*b^5*d^5*i^3*x^5 + 5*(4*B^2*b^5*c*d^4 + 5*(4* 
A*B + B^2)*a*b^4*d^5)*i^3*x^4 - 10*((20*A*B + B^2)*b^5*c^2*d^3 - 10*(4*A*B 
 + B^2)*a*b^4*c*d^4)*i^3*x^3 - 10*(2*(20*A*B + 3*B^2)*b^5*c^3*d^2 - 15*(4* 
A*B + B^2)*a*b^4*c^2*d^3)*i^3*x^2 - 5*((60*A*B + 11*B^2)*b^5*c^4*d - 20*(4 
*A*B + B^2)*a*b^4*c^3*d^2)*i^3*x - (16*(5*A*B + B^2)*b^5*c^5 - 25*(4*A*B + 
 B^2)*a*b^4*c^4*d)*i^3)*log((b*e*x + a*e)/(d*x + c)))/((b^11*c^2 - 2*a*b^1 
0*c*d + a^2*b^9*d^2)*g^6*x^5 + 5*(a*b^10*c^2 - 2*a^2*b^9*c*d + a^3*b^8*d^2 
)*g^6*x^4 + 10*(a^2*b^9*c^2 - 2*a^3*b^8*c*d + a^4*b^7*d^2)*g^6*x^3 + 10...
                                                                                    
                                                                                    
 

Sympy [F(-1)]

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

integrate((d*i*x+c*i)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**6,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15765 vs. \(2 (287) = 574\).

Time = 1.26 (sec) , antiderivative size = 15765, normalized size of antiderivative = 52.73 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x, al 
gorithm="maxima")
 

Output:

-3/20*(5*b*x + a)*B^2*c^2*d*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^ 
7*g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5* 
a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/10*(10*b^2*x^2 + 5*a*b*x + a^2)*B^2*c*d^2 
*i^3*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 
 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6 
) - 1/20*(10*b^3*x^3 + 10*a*b^2*x^2 + 5*a^2*b*x + a^3)*B^2*d^3*i^3*log(b*e 
*x/(d*x + c) + a*e/(d*x + c))^2/(b^9*g^6*x^5 + 5*a*b^8*g^6*x^4 + 10*a^2*b^ 
7*g^6*x^3 + 10*a^3*b^6*g^6*x^2 + 5*a^4*b^5*g^6*x + a^5*b^4*g^6) - 1/9000*( 
60*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 
163*a^3*b*c*d^3 + 137*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b 
^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3*b^4*c^3*d - 17*a* 
b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3 
*d + 6*a^2*b^8*c^2*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9 
*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6*c*d^3 + a^5*b^5*d^4 
)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5* 
b^5*c*d^3 + a^6*b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a 
^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*x^2 + 5*(a^4*b^6*c^4 - 
 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6* 
x + (a^5*b^5*c^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + 
 a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a...
 

Giac [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.60 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=-\frac {1}{4000} \, {\left (\frac {200 \, {\left (4 \, B^{2} b e^{6} i^{3} - \frac {5 \, {\left (b e x + a e\right )} B^{2} d e^{5} i^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b e x + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {20 \, {\left (80 \, A B b e^{6} i^{3} + 16 \, B^{2} b e^{6} i^{3} - \frac {100 \, {\left (b e x + a e\right )} A B d e^{5} i^{3}}{d x + c} - \frac {25 \, {\left (b e x + a e\right )} B^{2} d e^{5} i^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b e x + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {800 \, A^{2} b e^{6} i^{3} + 320 \, A B b e^{6} i^{3} + 64 \, B^{2} b e^{6} i^{3} - \frac {1000 \, {\left (b e x + a e\right )} A^{2} d e^{5} i^{3}}{d x + c} - \frac {500 \, {\left (b e x + a e\right )} A B d e^{5} i^{3}}{d x + c} - \frac {125 \, {\left (b e x + a e\right )} B^{2} d e^{5} i^{3}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{5} b c g^{6}}{{\left (d x + c\right )}^{5}} - \frac {{\left (b e x + a e\right )}^{5} a d g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \] Input:

integrate((d*i*x+c*i)^3*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^6,x, al 
gorithm="giac")
 

Output:

-1/4000*(200*(4*B^2*b*e^6*i^3 - 5*(b*e*x + a*e)*B^2*d*e^5*i^3/(d*x + c))*l 
og((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^5*b*c*g^6/(d*x + c)^5 - (b*e* 
x + a*e)^5*a*d*g^6/(d*x + c)^5) + 20*(80*A*B*b*e^6*i^3 + 16*B^2*b*e^6*i^3 
- 100*(b*e*x + a*e)*A*B*d*e^5*i^3/(d*x + c) - 25*(b*e*x + a*e)*B^2*d*e^5*i 
^3/(d*x + c))*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^5*b*c*g^6/(d*x + 
 c)^5 - (b*e*x + a*e)^5*a*d*g^6/(d*x + c)^5) + (800*A^2*b*e^6*i^3 + 320*A* 
B*b*e^6*i^3 + 64*B^2*b*e^6*i^3 - 1000*(b*e*x + a*e)*A^2*d*e^5*i^3/(d*x + c 
) - 500*(b*e*x + a*e)*A*B*d*e^5*i^3/(d*x + c) - 125*(b*e*x + a*e)*B^2*d*e^ 
5*i^3/(d*x + c))/((b*e*x + a*e)^5*b*c*g^6/(d*x + c)^5 - (b*e*x + a*e)^5*a* 
d*g^6/(d*x + c)^5))*(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 = 35.59 (sec) , antiderivative size = 3720, normalized size of antiderivative = 12.44 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx=\text {Too large to display} \] Input:

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

Output:

- log((e*(a + b*x))/(c + d*x))^2*((x*(a*(b*((B^2*a*d^3*i^3)/(20*b^5*g^6) + 
 (B^2*c*d^2*i^3)/(10*b^4*g^6)) + (3*B^2*a*d^3*i^3)/(20*b^4*g^6) + (3*B^2*c 
*d^2*i^3)/(10*b^3*g^6)) + b*(a*((B^2*a*d^3*i^3)/(20*b^5*g^6) + (B^2*c*d^2* 
i^3)/(10*b^4*g^6)) + (3*B^2*c^2*d*i^3)/(20*b^3*g^6)) + (3*B^2*c^2*d*i^3)/( 
5*b^2*g^6)) + x^2*(b*(b*((B^2*a*d^3*i^3)/(20*b^5*g^6) + (B^2*c*d^2*i^3)/(1 
0*b^4*g^6)) + (3*B^2*a*d^3*i^3)/(20*b^4*g^6) + (3*B^2*c*d^2*i^3)/(10*b^3*g 
^6)) + (3*B^2*a*d^3*i^3)/(10*b^3*g^6) + (3*B^2*c*d^2*i^3)/(5*b^2*g^6)) + a 
*(a*((B^2*a*d^3*i^3)/(20*b^5*g^6) + (B^2*c*d^2*i^3)/(10*b^4*g^6)) + (3*B^2 
*c^2*d*i^3)/(20*b^3*g^6)) + (B^2*c^3*i^3)/(5*b^2*g^6) + (B^2*d^3*i^3*x^3)/ 
(2*b^2*g^6))/(5*a^4*x + a^5/b + b^4*x^5 + 10*a^3*b*x^2 + 5*a*b^3*x^4 + 10* 
a^2*b^2*x^3) - (B^2*d^5*i^3)/(20*b^4*g^6*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) 
 - ((200*A^2*a^4*d^4*i^3 - 800*A^2*b^4*c^4*i^3 + 61*B^2*a^4*d^4*i^3 - 64*B 
^2*b^4*c^4*i^3 + 180*A*B*a^4*d^4*i^3 - 320*A*B*b^4*c^4*i^3 + 200*A^2*a*b^3 
*c^3*d*i^3 + 200*A^2*a^3*b*c*d^3*i^3 + 61*B^2*a*b^3*c^3*d*i^3 + 61*B^2*a^3 
*b*c*d^3*i^3 + 200*A^2*a^2*b^2*c^2*d^2*i^3 + 61*B^2*a^2*b^2*c^2*d^2*i^3 + 
180*A*B*a^2*b^2*c^2*d^2*i^3 + 180*A*B*a*b^3*c^3*d*i^3 + 180*A*B*a^3*b*c*d^ 
3*i^3)/(20*(a*d - b*c)) + (x^4*(9*B^2*b^4*d^4*i^3 + 20*A*B*b^4*d^4*i^3))/( 
a*d - b*c) + (x^3*(200*A^2*a*b^3*d^4*i^3 + 61*B^2*a*b^3*d^4*i^3 - 200*A^2* 
b^4*c*d^3*i^3 + 11*B^2*b^4*c*d^3*i^3 + 180*A*B*a*b^3*d^4*i^3 - 20*A*B*b^4* 
c*d^3*i^3))/(2*(a*d - b*c)) + (x*(200*A^2*a^3*b*d^4*i^3 + 61*B^2*a^3*b*...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 2741, normalized size of antiderivative = 9.17 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx =\text {Too large to display} \] Input:

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

Output:

(i*( - 400*log(a + b*x)*a**7*b*d**5 - 2000*log(a + b*x)*a**6*b**2*d**5*x - 
 100*log(a + b*x)*a**6*b**2*d**5 - 80*log(a + b*x)*a**5*b**3*c*d**4 - 4000 
*log(a + b*x)*a**5*b**3*d**5*x**2 - 500*log(a + b*x)*a**5*b**3*d**5*x - 40 
0*log(a + b*x)*a**4*b**4*c*d**4*x - 4000*log(a + b*x)*a**4*b**4*d**5*x**3 
- 1000*log(a + b*x)*a**4*b**4*d**5*x**2 - 800*log(a + b*x)*a**3*b**5*c*d** 
4*x**2 - 2000*log(a + b*x)*a**3*b**5*d**5*x**4 - 1000*log(a + b*x)*a**3*b* 
*5*d**5*x**3 - 800*log(a + b*x)*a**2*b**6*c*d**4*x**3 - 400*log(a + b*x)*a 
**2*b**6*d**5*x**5 - 500*log(a + b*x)*a**2*b**6*d**5*x**4 - 400*log(a + b* 
x)*a*b**7*c*d**4*x**4 - 100*log(a + b*x)*a*b**7*d**5*x**5 - 80*log(a + b*x 
)*b**8*c*d**4*x**5 + 400*log(c + d*x)*a**7*b*d**5 + 2000*log(c + d*x)*a**6 
*b**2*d**5*x + 100*log(c + d*x)*a**6*b**2*d**5 + 80*log(c + d*x)*a**5*b**3 
*c*d**4 + 4000*log(c + d*x)*a**5*b**3*d**5*x**2 + 500*log(c + d*x)*a**5*b* 
*3*d**5*x + 400*log(c + d*x)*a**4*b**4*c*d**4*x + 4000*log(c + d*x)*a**4*b 
**4*d**5*x**3 + 1000*log(c + d*x)*a**4*b**4*d**5*x**2 + 800*log(c + d*x)*a 
**3*b**5*c*d**4*x**2 + 2000*log(c + d*x)*a**3*b**5*d**5*x**4 + 1000*log(c 
+ d*x)*a**3*b**5*d**5*x**3 + 800*log(c + d*x)*a**2*b**6*c*d**4*x**3 + 400* 
log(c + d*x)*a**2*b**6*d**5*x**5 + 500*log(c + d*x)*a**2*b**6*d**5*x**4 + 
400*log(c + d*x)*a*b**7*c*d**4*x**4 + 100*log(c + d*x)*a*b**7*d**5*x**5 + 
80*log(c + d*x)*b**8*c*d**4*x**5 - 1000*log((a*e + b*e*x)/(c + d*x))**2*a* 
*2*b**6*c**4*d - 4000*log((a*e + b*e*x)/(c + d*x))**2*a**2*b**6*c**3*d*...