\(\int (a g+b g x)^3 (c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [10]

Optimal result

Integrand size = 40, antiderivative size = 423 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^5 g^3 i^2 x}{60 b^2 d^3}+\frac {B (b c-a d)^4 g^3 i^2 (c+d x)^2}{120 b d^4}-\frac {19 B (b c-a d)^3 g^3 i^2 (c+d x)^3}{180 d^4}+\frac {13 b B (b c-a d)^2 g^3 i^2 (c+d x)^4}{120 d^4}-\frac {b^2 B (b c-a d) g^3 i^2 (c+d x)^5}{30 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log \left (\frac {a+b x}{c+d x}\right )}{60 b^3 d^4}-\frac {(b c-a d)^3 g^3 i^2 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 d^4}+\frac {3 b (b c-a d)^2 g^3 i^2 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{4 d^4}-\frac {3 b^2 (b c-a d) g^3 i^2 (c+d x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{5 d^4}+\frac {b^3 g^3 i^2 (c+d x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 d^4}+\frac {B (b c-a d)^6 g^3 i^2 \log (c+d x)}{60 b^3 d^4} \] Output:

1/60*B*(-a*d+b*c)^5*g^3*i^2*x/b^2/d^3+1/120*B*(-a*d+b*c)^4*g^3*i^2*(d*x+c) 
^2/b/d^4-19/180*B*(-a*d+b*c)^3*g^3*i^2*(d*x+c)^3/d^4+13/120*b*B*(-a*d+b*c) 
^2*g^3*i^2*(d*x+c)^4/d^4-1/30*b^2*B*(-a*d+b*c)*g^3*i^2*(d*x+c)^5/d^4+1/60* 
B*(-a*d+b*c)^6*g^3*i^2*ln((b*x+a)/(d*x+c))/b^3/d^4-1/3*(-a*d+b*c)^3*g^3*i^ 
2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+3/4*b*(-a*d+b*c)^2*g^3*i^2*(d* 
x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4-3/5*b^2*(-a*d+b*c)*g^3*i^2*(d*x+c)^ 
5*(A+B*ln(e*(b*x+a)/(d*x+c)))/d^4+1/6*b^3*g^3*i^2*(d*x+c)^6*(A+B*ln(e*(b*x 
+a)/(d*x+c)))/d^4+1/60*B*(-a*d+b*c)^6*g^3*i^2*ln(d*x+c)/b^3/d^4
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.01 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^3 i^2 \left (90 d^4 (b c-a d)^2 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+144 d^5 (b c-a d) (a+b x)^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+60 d^6 (a+b x)^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-15 B (b c-a d)^3 \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )+12 B (b c-a d)^2 \left (12 b d (b c-a d)^3 x-6 d^2 (b c-a d)^2 (a+b x)^2+4 d^3 (b c-a d) (a+b x)^3-3 d^4 (a+b x)^4-12 (b c-a d)^4 \log (c+d x)\right )-B (b c-a d) \left (60 b d (b c-a d)^4 x+30 d^2 (-b c+a d)^3 (a+b x)^2+20 d^3 (b c-a d)^2 (a+b x)^3+15 d^4 (-b c+a d) (a+b x)^4+12 d^5 (a+b x)^5-60 (b c-a d)^5 \log (c+d x)\right )\right )}{360 b^3 d^4} \] Input:

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

Output:

(g^3*i^2*(90*d^4*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d 
*x)]) + 144*d^5*(b*c - a*d)*(a + b*x)^5*(A + B*Log[(e*(a + b*x))/(c + d*x) 
]) + 60*d^6*(a + b*x)^6*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 15*B*(b*c - 
 a*d)^3*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3* 
(a + b*x)^3 - 6*(b*c - a*d)^3*Log[c + d*x]) + 12*B*(b*c - a*d)^2*(12*b*d*( 
b*c - a*d)^3*x - 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 4*d^3*(b*c - a*d)*(a + 
b*x)^3 - 3*d^4*(a + b*x)^4 - 12*(b*c - a*d)^4*Log[c + d*x]) - B*(b*c - a*d 
)*(60*b*d*(b*c - a*d)^4*x + 30*d^2*(-(b*c) + a*d)^3*(a + b*x)^2 + 20*d^3*( 
b*c - a*d)^2*(a + b*x)^3 + 15*d^4*(-(b*c) + a*d)*(a + b*x)^4 + 12*d^5*(a + 
 b*x)^5 - 60*(b*c - a*d)^5*Log[c + d*x])))/(360*b^3*d^4)
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2962, 2782, 27, 2123, 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.

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

\(\Big \downarrow \) 2962

\(\displaystyle g^3 i^2 (b c-a d)^6 \int \frac {(a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}\)

\(\Big \downarrow \) 2782

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (-B \int -\frac {(c+d x) \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {15 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {20 d^3 (a+b x)^3}{(c+d x)^3}\right )}{60 d^4 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {B \int \frac {(c+d x) \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {15 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {20 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{60 d^4}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\)

\(\Big \downarrow \) 2123

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {B \int \left (-\frac {10 d b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {26 d b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {19 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3 b}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2 b^2}+\frac {c+d x}{(a+b x) b^3}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^3}\right )d\frac {a+b x}{c+d x}}{60 d^4}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {b^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{60 d^4}\right )\)

Input:

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

Output:

(b*c - a*d)^6*g^3*i^2*((b^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*d^4*( 
b - (d*(a + b*x))/(c + d*x))^6) - (3*b^2*(A + B*Log[(e*(a + b*x))/(c + d*x 
)]))/(5*d^4*(b - (d*(a + b*x))/(c + d*x))^5) + (3*b*(A + B*Log[(e*(a + b*x 
))/(c + d*x)]))/(4*d^4*(b - (d*(a + b*x))/(c + d*x))^4) - (A + B*Log[(e*(a 
 + b*x))/(c + d*x)])/(3*d^4*(b - (d*(a + b*x))/(c + d*x))^3) + (B*((-2*b^2 
)/(b - (d*(a + b*x))/(c + d*x))^5 + (13*b)/(2*(b - (d*(a + b*x))/(c + d*x) 
)^4) - 19/(3*(b - (d*(a + b*x))/(c + d*x))^3) + 1/(2*b*(b - (d*(a + b*x))/ 
(c + d*x))^2) + 1/(b^2*(b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + 
 d*x)]/b^3 - Log[b - (d*(a + b*x))/(c + d*x)]/b^3))/(60*d^4))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2123

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] 
:> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c 
, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
 

rule 2782

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q 
_), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* 
x^n])   u, x] - Simp[b*n   Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ 
{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 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. \(924\) vs. \(2(401)=802\).

Time = 1.76 (sec) , antiderivative size = 925, normalized size of antiderivative = 2.19

method	result	size
risch	\(i^{2} g^{3} A \,a^{3} c^{2} x +\frac {3 i^{2} g^{3} b^{2} d A a c \,x^{4}}{2}-\frac {i^{2} g^{3} b^{2} d B a c \,x^{4}}{20}+2 i^{2} g^{3} b d A \,a^{2} c \,x^{3}+i^{2} g^{3} b^{2} A a \,c^{2} x^{3}+\frac {7 i^{2} g^{3} b d B \,a^{2} c \,x^{3}}{60}-\frac {13 i^{2} g^{3} b^{2} B a \,c^{2} x^{3}}{60}+i^{2} g^{3} d A \,a^{3} c \,x^{2}+\frac {3 i^{2} g^{3} b A \,a^{2} c^{2} x^{2}}{2}+\frac {17 i^{2} g^{3} d B \,a^{3} c \,x^{2}}{60}-\frac {i^{2} g^{3} b B \,a^{2} c^{2} x^{2}}{4}-\frac {i^{2} g^{3} b^{2} B a \,c^{3} x^{2}}{20 d}+\frac {i^{2} g^{3} d B \,a^{4} c x}{10 b}+\frac {i^{2} g^{3} B \,a^{3} c^{2} x}{12}-\frac {i^{2} g^{3} b B \,a^{2} c^{3} x}{4 d}+\frac {i^{2} g^{3} b^{2} B a \,c^{4} x}{10 d^{2}}-\frac {i^{2} g^{3} d B \ln \left (-b x -a \right ) a^{5} c}{10 b^{2}}+\frac {i^{2} g^{3} b B \ln \left (d x +c \right ) a^{2} c^{4}}{4 d^{2}}-\frac {i^{2} g^{3} b^{2} B \ln \left (d x +c \right ) a \,c^{5}}{10 d^{3}}+\frac {3 i^{2} g^{3} b^{2} d^{2} A a \,x^{5}}{5}+\frac {2 i^{2} g^{3} b^{3} d A c \,x^{5}}{5}+\frac {i^{2} g^{3} b^{2} d^{2} B a \,x^{5}}{30}-\frac {i^{2} g^{3} b^{3} d B c \,x^{5}}{30}+\frac {3 i^{2} g^{3} b \,d^{2} A \,a^{2} x^{4}}{4}+\frac {i^{2} g^{3} b^{3} A \,c^{2} x^{4}}{4}+\frac {13 i^{2} g^{3} b \,d^{2} B \,a^{2} x^{4}}{120}-\frac {7 i^{2} g^{3} b^{3} B \,c^{2} x^{4}}{120}+\frac {i^{2} g^{3} d^{2} A \,a^{3} x^{3}}{3}+\frac {19 i^{2} g^{3} d^{2} B \,a^{3} x^{3}}{180}-\frac {i^{2} g^{3} b^{3} B \,c^{3} x^{3}}{180 d}+\frac {i^{2} g^{3} d^{2} B \,a^{4} x^{2}}{120 b}+\frac {i^{2} g^{3} b^{3} B \,c^{4} x^{2}}{120 d^{2}}-\frac {i^{2} g^{3} d^{2} B \,a^{5} x}{60 b^{2}}-\frac {i^{2} g^{3} b^{3} B \,c^{5} x}{60 d^{3}}+\frac {i^{2} g^{3} B \ln \left (-b x -a \right ) a^{4} c^{2}}{4 b}-\frac {i^{2} g^{3} B \ln \left (d x +c \right ) a^{3} c^{3}}{3 d}+\frac {i^{2} g^{3} d^{2} B \ln \left (-b x -a \right ) a^{6}}{60 b^{3}}+\frac {i^{2} g^{3} b^{3} B \ln \left (d x +c \right ) c^{6}}{60 d^{4}}+\frac {i^{2} g^{3} b^{3} d^{2} A \,x^{6}}{6}+\frac {i^{2} g^{3} B x \left (10 d^{2} b^{3} x^{5}+36 a \,b^{2} d^{2} x^{4}+24 b^{3} c d \,x^{4}+45 a^{2} b \,d^{2} x^{3}+90 a \,b^{2} c d \,x^{3}+15 b^{3} c^{2} x^{3}+20 a^{3} d^{2} x^{2}+120 a^{2} b c d \,x^{2}+60 a \,b^{2} c^{2} x^{2}+60 a^{3} c d x +90 a^{2} b \,c^{2} x +60 a^{3} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{60}\)	\(925\)
parallelrisch	\(\text {Expression too large to display}\)	\(1578\)
parts	\(\text {Expression too large to display}\)	\(2106\)
derivativedivides	\(\text {Expression too large to display}\)	\(2168\)
default	\(\text {Expression too large to display}\)	\(2168\)

Input:

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

Output:

i^2*g^3*A*a^3*c^2*x+3/2*i^2*g^3*b^2*d*A*a*c*x^4-1/20*i^2*g^3*b^2*d*B*a*c*x 
^4+2*i^2*g^3*b*d*A*a^2*c*x^3+i^2*g^3*b^2*A*a*c^2*x^3+7/60*i^2*g^3*b*d*B*a^ 
2*c*x^3-13/60*i^2*g^3*b^2*B*a*c^2*x^3+i^2*g^3*d*A*a^3*c*x^2+3/2*i^2*g^3*b* 
A*a^2*c^2*x^2+17/60*i^2*g^3*d*B*a^3*c*x^2-1/4*i^2*g^3*b*B*a^2*c^2*x^2-1/20 
*i^2*g^3*b^2/d*B*a*c^3*x^2+1/10*i^2*g^3/b*d*B*a^4*c*x+1/12*i^2*g^3*B*a^3*c 
^2*x-1/4*i^2*g^3*b/d*B*a^2*c^3*x+1/10*i^2*g^3*b^2/d^2*B*a*c^4*x-1/10*i^2*g 
^3/b^2*d*B*ln(-b*x-a)*a^5*c+1/4*i^2*g^3*b/d^2*B*ln(d*x+c)*a^2*c^4-1/10*i^2 
*g^3*b^2/d^3*B*ln(d*x+c)*a*c^5+3/5*i^2*g^3*b^2*d^2*A*a*x^5+2/5*i^2*g^3*b^3 
*d*A*c*x^5+1/30*i^2*g^3*b^2*d^2*B*a*x^5-1/30*i^2*g^3*b^3*d*B*c*x^5+3/4*i^2 
*g^3*b*d^2*A*a^2*x^4+1/4*i^2*g^3*b^3*A*c^2*x^4+13/120*i^2*g^3*b*d^2*B*a^2* 
x^4-7/120*i^2*g^3*b^3*B*c^2*x^4+1/3*i^2*g^3*d^2*A*a^3*x^3+19/180*i^2*g^3*d 
^2*B*a^3*x^3-1/180*i^2*g^3*b^3/d*B*c^3*x^3+1/120*i^2*g^3/b*d^2*B*a^4*x^2+1 
/120*i^2*g^3*b^3/d^2*B*c^4*x^2-1/60*i^2*g^3/b^2*d^2*B*a^5*x-1/60*i^2*g^3*b 
^3/d^3*B*c^5*x+1/4*i^2*g^3/b*B*ln(-b*x-a)*a^4*c^2-1/3*i^2*g^3/d*B*ln(d*x+c 
)*a^3*c^3+1/60*i^2*g^3/b^3*d^2*B*ln(-b*x-a)*a^6+1/60*i^2*g^3*b^3/d^4*B*ln( 
d*x+c)*c^6+1/6*i^2*g^3*b^3*d^2*A*x^6+1/60*i^2*g^3*B*x*(10*b^3*d^2*x^5+36*a 
*b^2*d^2*x^4+24*b^3*c*d*x^4+45*a^2*b*d^2*x^3+90*a*b^2*c*d*x^3+15*b^3*c^2*x 
^3+20*a^3*d^2*x^2+120*a^2*b*c*d*x^2+60*a*b^2*c^2*x^2+60*a^3*c*d*x+90*a^2*b 
*c^2*x+60*a^3*c^2)*ln(e*(b*x+a)/(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.71 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {60 \, A b^{6} d^{6} g^{3} i^{2} x^{6} + 12 \, {\left ({\left (12 \, A - B\right )} b^{6} c d^{5} + {\left (18 \, A + B\right )} a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 3 \, {\left ({\left (30 \, A - 7 \, B\right )} b^{6} c^{2} d^{4} + 6 \, {\left (30 \, A - B\right )} a b^{5} c d^{5} + {\left (90 \, A + 13 \, B\right )} a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} - 2 \, {\left (B b^{6} c^{3} d^{3} - 3 \, {\left (60 \, A - 13 \, B\right )} a b^{5} c^{2} d^{4} - 3 \, {\left (120 \, A + 7 \, B\right )} a^{2} b^{4} c d^{5} - {\left (60 \, A + 19 \, B\right )} a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 3 \, {\left (B b^{6} c^{4} d^{2} - 6 \, B a b^{5} c^{3} d^{3} + 30 \, {\left (6 \, A - B\right )} a^{2} b^{4} c^{2} d^{4} + 2 \, {\left (60 \, A + 17 \, B\right )} a^{3} b^{3} c d^{5} + B a^{4} b^{2} d^{6}\right )} g^{3} i^{2} x^{2} - 6 \, {\left (B b^{6} c^{5} d - 6 \, B a b^{5} c^{4} d^{2} + 15 \, B a^{2} b^{4} c^{3} d^{3} - 5 \, {\left (12 \, A + B\right )} a^{3} b^{3} c^{2} d^{4} - 6 \, B a^{4} b^{2} c d^{5} + B a^{5} b d^{6}\right )} g^{3} i^{2} x + 6 \, {\left (15 \, B a^{4} b^{2} c^{2} d^{4} - 6 \, B a^{5} b c d^{5} + B a^{6} d^{6}\right )} g^{3} i^{2} \log \left (b x + a\right ) + 6 \, {\left (B b^{6} c^{6} - 6 \, B a b^{5} c^{5} d + 15 \, B a^{2} b^{4} c^{4} d^{2} - 20 \, B a^{3} b^{3} c^{3} d^{3}\right )} g^{3} i^{2} \log \left (d x + c\right ) + 6 \, {\left (10 \, B b^{6} d^{6} g^{3} i^{2} x^{6} + 60 \, B a^{3} b^{3} c^{2} d^{4} g^{3} i^{2} x + 12 \, {\left (2 \, B b^{6} c d^{5} + 3 \, B a b^{5} d^{6}\right )} g^{3} i^{2} x^{5} + 15 \, {\left (B b^{6} c^{2} d^{4} + 6 \, B a b^{5} c d^{5} + 3 \, B a^{2} b^{4} d^{6}\right )} g^{3} i^{2} x^{4} + 20 \, {\left (3 \, B a b^{5} c^{2} d^{4} + 6 \, B a^{2} b^{4} c d^{5} + B a^{3} b^{3} d^{6}\right )} g^{3} i^{2} x^{3} + 30 \, {\left (3 \, B a^{2} b^{4} c^{2} d^{4} + 2 \, B a^{3} b^{3} c d^{5}\right )} g^{3} i^{2} x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{360 \, b^{3} d^{4}} \] Input:

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

Output:

1/360*(60*A*b^6*d^6*g^3*i^2*x^6 + 12*((12*A - B)*b^6*c*d^5 + (18*A + B)*a* 
b^5*d^6)*g^3*i^2*x^5 + 3*((30*A - 7*B)*b^6*c^2*d^4 + 6*(30*A - B)*a*b^5*c* 
d^5 + (90*A + 13*B)*a^2*b^4*d^6)*g^3*i^2*x^4 - 2*(B*b^6*c^3*d^3 - 3*(60*A 
- 13*B)*a*b^5*c^2*d^4 - 3*(120*A + 7*B)*a^2*b^4*c*d^5 - (60*A + 19*B)*a^3* 
b^3*d^6)*g^3*i^2*x^3 + 3*(B*b^6*c^4*d^2 - 6*B*a*b^5*c^3*d^3 + 30*(6*A - B) 
*a^2*b^4*c^2*d^4 + 2*(60*A + 17*B)*a^3*b^3*c*d^5 + B*a^4*b^2*d^6)*g^3*i^2* 
x^2 - 6*(B*b^6*c^5*d - 6*B*a*b^5*c^4*d^2 + 15*B*a^2*b^4*c^3*d^3 - 5*(12*A 
+ B)*a^3*b^3*c^2*d^4 - 6*B*a^4*b^2*c*d^5 + B*a^5*b*d^6)*g^3*i^2*x + 6*(15* 
B*a^4*b^2*c^2*d^4 - 6*B*a^5*b*c*d^5 + B*a^6*d^6)*g^3*i^2*log(b*x + a) + 6* 
(B*b^6*c^6 - 6*B*a*b^5*c^5*d + 15*B*a^2*b^4*c^4*d^2 - 20*B*a^3*b^3*c^3*d^3 
)*g^3*i^2*log(d*x + c) + 6*(10*B*b^6*d^6*g^3*i^2*x^6 + 60*B*a^3*b^3*c^2*d^ 
4*g^3*i^2*x + 12*(2*B*b^6*c*d^5 + 3*B*a*b^5*d^6)*g^3*i^2*x^5 + 15*(B*b^6*c 
^2*d^4 + 6*B*a*b^5*c*d^5 + 3*B*a^2*b^4*d^6)*g^3*i^2*x^4 + 20*(3*B*a*b^5*c^ 
2*d^4 + 6*B*a^2*b^4*c*d^5 + B*a^3*b^3*d^6)*g^3*i^2*x^3 + 30*(3*B*a^2*b^4*c 
^2*d^4 + 2*B*a^3*b^3*c*d^5)*g^3*i^2*x^2)*log((b*e*x + a*e)/(d*x + c)))/(b^ 
3*d^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1727 vs. \(2 (398) = 796\).

Time = 6.07 (sec) , antiderivative size = 1727, normalized size of antiderivative = 4.08 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

A*b**3*d**2*g**3*i**2*x**6/6 + B*a**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 1 
5*b**2*c**2)*log(x + (B*a**6*c*d**5*g**3*i**2 - 6*B*a**5*b*c**2*d**4*g**3* 
i**2 + B*a**5*d**4*g**3*i**2*(a**2*d**2 - 6*a*b*c*d + 15*b**2*c**2)/b + 35 
*B*a**4*b**2*c**3*d**3*g**3*i**2 - B*a**4*c*d**3*g**3*i**2*(a**2*d**2 - 6* 
a*b*c*d + 15*b**2*c**2) - 15*B*a**3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2*b* 
*4*c**5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2)/(B*a**6*d**6*g**3*i**2 - 6* 
B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2*d**4*g**3*i**2 + 20*B*a**3 
*b**3*c**3*d**3*g**3*i**2 - 15*B*a**2*b**4*c**4*d**2*g**3*i**2 + 6*B*a*b** 
5*c**5*d*g**3*i**2 - B*b**6*c**6*g**3*i**2))/(60*b**3) - B*c**3*g**3*i**2* 
(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3)*log(x + (B 
*a**6*c*d**5*g**3*i**2 - 6*B*a**5*b*c**2*d**4*g**3*i**2 + 35*B*a**4*b**2*c 
**3*d**3*g**3*i**2 - 15*B*a**3*b**3*c**4*d**2*g**3*i**2 + 6*B*a**2*b**4*c* 
*5*d*g**3*i**2 - B*a*b**5*c**6*g**3*i**2 - B*a*b**2*c**3*g**3*i**2*(20*a** 
3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3) + B*b**3*c**4*g** 
3*i**2*(20*a**3*d**3 - 15*a**2*b*c*d**2 + 6*a*b**2*c**2*d - b**3*c**3)/d)/ 
(B*a**6*d**6*g**3*i**2 - 6*B*a**5*b*c*d**5*g**3*i**2 + 15*B*a**4*b**2*c**2 
*d**4*g**3*i**2 + 20*B*a**3*b**3*c**3*d**3*g**3*i**2 - 15*B*a**2*b**4*c**4 
*d**2*g**3*i**2 + 6*B*a*b**5*c**5*d*g**3*i**2 - B*b**6*c**6*g**3*i**2))/(6 
0*d**4) + x**5*(3*A*a*b**2*d**2*g**3*i**2/5 + 2*A*b**3*c*d*g**3*i**2/5 + B 
*a*b**2*d**2*g**3*i**2/30 - B*b**3*c*d*g**3*i**2/30) + x**4*(3*A*a**2*b...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1789 vs. \(2 (401) = 802\).

Time = 0.08 (sec) , antiderivative size = 1789, normalized size of antiderivative = 4.23 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

1/6*A*b^3*d^2*g^3*i^2*x^6 + 2/5*A*b^3*c*d*g^3*i^2*x^5 + 3/5*A*a*b^2*d^2*g^ 
3*i^2*x^5 + 1/4*A*b^3*c^2*g^3*i^2*x^4 + 3/2*A*a*b^2*c*d*g^3*i^2*x^4 + 3/4* 
A*a^2*b*d^2*g^3*i^2*x^4 + A*a*b^2*c^2*g^3*i^2*x^3 + 2*A*a^2*b*c*d*g^3*i^2* 
x^3 + 1/3*A*a^3*d^2*g^3*i^2*x^3 + 3/2*A*a^2*b*c^2*g^3*i^2*x^2 + A*a^3*c*d* 
g^3*i^2*x^2 + (x*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + a*log(b*x + a)/b - 
 c*log(d*x + c)/d)*B*a^3*c^2*g^3*i^2 + 3/2*(x^2*log(b*e*x/(d*x + c) + a*e/ 
(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c)/d^2 - (b*c - a*d)*x/( 
b*d))*B*a^2*b*c^2*g^3*i^2 + 1/2*(2*x^3*log(b*e*x/(d*x + c) + a*e/(d*x + c) 
) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2) 
*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a*b^2*c^2*g^3*i^2 + 1/24*(6*x 
^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a)/b^4 + 6*c^4*l 
og(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^ 
3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*b^3*c^2*g^3*i^2 + (x^2*log( 
b*e*x/(d*x + c) + a*e/(d*x + c)) - a^2*log(b*x + a)/b^2 + c^2*log(d*x + c) 
/d^2 - (b*c - a*d)*x/(b*d))*B*a^3*c*d*g^3*i^2 + (2*x^3*log(b*e*x/(d*x + c) 
 + a*e/(d*x + c)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^ 
2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*a^2*b*c*d*g^3 
*i^2 + 1/4*(6*x^4*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 6*a^4*log(b*x + a 
)/b^4 + 6*c^4*log(d*x + c)/d^4 - (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c 
^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3))*B*a*b^2*c*d...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4746 vs. \(2 (401) = 802\).

Time = 0.36 (sec) , antiderivative size = 4746, normalized size of antiderivative = 11.22 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/360*(6*(B*b^10*c^7*e^7*g^3*i^2 - 7*B*a*b^9*c^6*d*e^7*g^3*i^2 + 21*B*a^2 
*b^8*c^5*d^2*e^7*g^3*i^2 - 35*B*a^3*b^7*c^4*d^3*e^7*g^3*i^2 + 35*B*a^4*b^6 
*c^3*d^4*e^7*g^3*i^2 - 21*B*a^5*b^5*c^2*d^5*e^7*g^3*i^2 + 7*B*a^6*b^4*c*d^ 
6*e^7*g^3*i^2 - B*a^7*b^3*d^7*e^7*g^3*i^2 - 6*(b*e*x + a*e)*B*b^9*c^7*d*e^ 
6*g^3*i^2/(d*x + c) + 42*(b*e*x + a*e)*B*a*b^8*c^6*d^2*e^6*g^3*i^2/(d*x + 
c) - 126*(b*e*x + a*e)*B*a^2*b^7*c^5*d^3*e^6*g^3*i^2/(d*x + c) + 210*(b*e* 
x + a*e)*B*a^3*b^6*c^4*d^4*e^6*g^3*i^2/(d*x + c) - 210*(b*e*x + a*e)*B*a^4 
*b^5*c^3*d^5*e^6*g^3*i^2/(d*x + c) + 126*(b*e*x + a*e)*B*a^5*b^4*c^2*d^6*e 
^6*g^3*i^2/(d*x + c) - 42*(b*e*x + a*e)*B*a^6*b^3*c*d^7*e^6*g^3*i^2/(d*x + 
 c) + 6*(b*e*x + a*e)*B*a^7*b^2*d^8*e^6*g^3*i^2/(d*x + c) + 15*(b*e*x + a* 
e)^2*B*b^8*c^7*d^2*e^5*g^3*i^2/(d*x + c)^2 - 105*(b*e*x + a*e)^2*B*a*b^7*c 
^6*d^3*e^5*g^3*i^2/(d*x + c)^2 + 315*(b*e*x + a*e)^2*B*a^2*b^6*c^5*d^4*e^5 
*g^3*i^2/(d*x + c)^2 - 525*(b*e*x + a*e)^2*B*a^3*b^5*c^4*d^5*e^5*g^3*i^2/( 
d*x + c)^2 + 525*(b*e*x + a*e)^2*B*a^4*b^4*c^3*d^6*e^5*g^3*i^2/(d*x + c)^2 
 - 315*(b*e*x + a*e)^2*B*a^5*b^3*c^2*d^7*e^5*g^3*i^2/(d*x + c)^2 + 105*(b* 
e*x + a*e)^2*B*a^6*b^2*c*d^8*e^5*g^3*i^2/(d*x + c)^2 - 15*(b*e*x + a*e)^2* 
B*a^7*b*d^9*e^5*g^3*i^2/(d*x + c)^2 - 20*(b*e*x + a*e)^3*B*b^7*c^7*d^3*e^4 
*g^3*i^2/(d*x + c)^3 + 140*(b*e*x + a*e)^3*B*a*b^6*c^6*d^4*e^4*g^3*i^2/(d* 
x + c)^3 - 420*(b*e*x + a*e)^3*B*a^2*b^5*c^5*d^5*e^4*g^3*i^2/(d*x + c)^3 + 
 700*(b*e*x + a*e)^3*B*a^3*b^4*c^4*d^6*e^4*g^3*i^2/(d*x + c)^3 - 700*(b...
 

Mupad [B] (verification not implemented)

Time = 27.93 (sec) , antiderivative size = 2473, normalized size of antiderivative = 5.85 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

x^3*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A 
*a*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(12* 
d) + ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B 
*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60* 
b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 
 + 60*A*a*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(180*b*d) - (a*c*( 
(b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2 
*(60*a*d + 60*b*c))/60))/(3*b*d)) - x^4*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A 
*b*c + B*a*d - B*b*c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d 
 + 60*b*c))/(240*b*d) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2* 
d^2 - 2*B*b^2*c^2 + 60*A*a*b*c*d - B*a*b*c*d))/20 + (A*a*b^2*c*d*g^3*i^2)/ 
4) + x^2*((a*c*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b*c))/6 
- (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d) - (b 
*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2 + 60*A*a 
*b*c*d - B*a*b*c*d))/5 + A*a*b^2*c*d*g^3*i^2))/(2*b*d) - ((60*a*d + 60*b*c 
)*((g^3*i^2*(16*A*a^3*d^3 + 4*A*b^3*c^3 + 3*B*a^3*d^3 - B*b^3*c^3 + 48*A*a 
*b^2*c^2*d + 72*A*a^2*b*c*d^2 - 5*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2))/(4*d) 
+ ((60*a*d + 60*b*c)*((((b^2*d*g^3*i^2*(24*A*a*d + 18*A*b*c + B*a*d - B*b* 
c))/6 - (A*b^2*d*g^3*i^2*(60*a*d + 60*b*c))/60)*(60*a*d + 60*b*c))/(60*b*d 
) - (b*g^3*i^2*(30*A*a^2*d^2 + 15*A*b^2*c^2 + 3*B*a^2*d^2 - 2*B*b^2*c^2...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 1036, normalized size of antiderivative = 2.45 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

(g**3*( - 6*log(c + d*x)*a**6*d**6 + 36*log(c + d*x)*a**5*b*c*d**5 - 90*lo 
g(c + d*x)*a**4*b**2*c**2*d**4 + 120*log(c + d*x)*a**3*b**3*c**3*d**3 - 90 
*log(c + d*x)*a**2*b**4*c**4*d**2 + 36*log(c + d*x)*a*b**5*c**5*d - 6*log( 
c + d*x)*b**6*c**6 - 6*log((a*e + b*e*x)/(c + d*x))*a**6*d**6 + 36*log((a* 
e + b*e*x)/(c + d*x))*a**5*b*c*d**5 - 90*log((a*e + b*e*x)/(c + d*x))*a**4 
*b**2*c**2*d**4 - 360*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*c**2*d**4*x - 
 360*log((a*e + b*e*x)/(c + d*x))*a**3*b**3*c*d**5*x**2 - 120*log((a*e + b 
*e*x)/(c + d*x))*a**3*b**3*d**6*x**3 - 540*log((a*e + b*e*x)/(c + d*x))*a* 
*2*b**4*c**2*d**4*x**2 - 720*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*c*d**5 
*x**3 - 270*log((a*e + b*e*x)/(c + d*x))*a**2*b**4*d**6*x**4 - 360*log((a* 
e + b*e*x)/(c + d*x))*a*b**5*c**2*d**4*x**3 - 540*log((a*e + b*e*x)/(c + d 
*x))*a*b**5*c*d**5*x**4 - 216*log((a*e + b*e*x)/(c + d*x))*a*b**5*d**6*x** 
5 - 90*log((a*e + b*e*x)/(c + d*x))*b**6*c**2*d**4*x**4 - 144*log((a*e + b 
*e*x)/(c + d*x))*b**6*c*d**5*x**5 - 60*log((a*e + b*e*x)/(c + d*x))*b**6*d 
**6*x**6 + 6*a**5*b*d**6*x - 360*a**4*b**2*c**2*d**4*x - 360*a**4*b**2*c*d 
**5*x**2 - 36*a**4*b**2*c*d**5*x - 120*a**4*b**2*d**6*x**3 - 3*a**4*b**2*d 
**6*x**2 - 540*a**3*b**3*c**2*d**4*x**2 - 30*a**3*b**3*c**2*d**4*x - 720*a 
**3*b**3*c*d**5*x**3 - 102*a**3*b**3*c*d**5*x**2 - 270*a**3*b**3*d**6*x**4 
 - 38*a**3*b**3*d**6*x**3 + 90*a**2*b**4*c**3*d**3*x - 360*a**2*b**4*c**2* 
d**4*x**3 + 90*a**2*b**4*c**2*d**4*x**2 - 540*a**2*b**4*c*d**5*x**4 - 4...