Integrand size = 30, antiderivative size = 283 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx=-\frac {B (b c-a d) n}{6 (b f-a g) (d f-c g) (f+g x)^2}-\frac {B (b c-a d) (2 b d f-b c g-a d g) n}{3 (b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {b^3 B n \log (a+b x)}{3 g (b f-a g)^3}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 g (f+g x)^3}-\frac {B d^3 n \log (c+d x)}{3 g (d f-c g)^3}+\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n \log (f+g x)}{3 (b f-a g)^3 (d f-c g)^3} \] Output:
-1/6*B*(-a*d+b*c)*n/(-a*g+b*f)/(-c*g+d*f)/(g*x+f)^2-1/3*B*(-a*d+b*c)*(-a*d *g-b*c*g+2*b*d*f)*n/(-a*g+b*f)^2/(-c*g+d*f)^2/(g*x+f)+1/3*b^3*B*n*ln(b*x+a )/g/(-a*g+b*f)^3-1/3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/g/(g*x+f)^3-1/3*B*d^3 *n*ln(d*x+c)/g/(-c*g+d*f)^3+1/3*B*(-a*d+b*c)*(a^2*d^2*g^2-a*b*d*g*(-c*g+3* d*f)+b^2*(c^2*g^2-3*c*d*f*g+3*d^2*f^2))*n*ln(g*x+f)/(-a*g+b*f)^3/(-c*g+d*f )^3
Time = 0.71 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.93 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx=\frac {-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^3}+B (b c-a d) n \left (-\frac {g}{2 (b f-a g) (d f-c g) (f+g x)^2}+\frac {g (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}+\frac {d^3 \log (c+d x)}{(b c-a d) (-d f+c g)^3}+\frac {g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{(b f-a g)^3 (d f-c g)^3}\right )}{3 g} \] Input:
Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^4,x]
Output:
(-((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^3) + B*(b*c - a*d)*n*( -1/2*g/((b*f - a*g)*(d*f - c*g)*(f + g*x)^2) + (g*(-2*b*d*f + b*c*g + a*d* g))/((b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)) + (b^3*Log[a + b*x])/((b*c - a *d)*(b*f - a*g)^3) + (d^3*Log[c + d*x])/((b*c - a*d)*(-(d*f) + c*g)^3) + ( g*(a^2*d^2*g^2 + a*b*d*g*(-3*d*f + c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2 *g^2))*Log[f + g*x])/((b*f - a*g)^3*(d*f - c*g)^3)))/(3*g)
Time = 0.65 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2947, 93, 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.
\(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(f+g x)^4} \, dx\) |
\(\Big \downarrow \) 2947 |
\(\displaystyle \frac {B n (b c-a d) \int \frac {1}{(a+b x) (c+d x) (f+g x)^3}dx}{3 g}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 93 |
\(\displaystyle \frac {B n (b c-a d) \int \left (\frac {b^4}{(b c-a d) (b f-a g)^3 (a+b x)}+\frac {d^4}{(b c-a d) (c g-d f)^3 (c+d x)}+\frac {g^2 \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)^2}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^3}\right )dx}{3 g}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 g (f+g x)^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {B n (b c-a d) \left (\frac {g \log (f+g x) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{(b f-a g)^3 (d f-c g)^3}+\frac {b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}-\frac {d^3 \log (c+d x)}{(b c-a d) (d f-c g)^3}-\frac {g (-a d g-b c g+2 b d f)}{(f+g x) (b f-a g)^2 (d f-c g)^2}-\frac {g}{2 (f+g x)^2 (b f-a g) (d f-c g)}\right )}{3 g}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 g (f+g x)^3}\) |
Input:
Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(f + g*x)^4,x]
Output:
-1/3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(f + g*x)^3) + (B*(b*c - a* d)*n*(-1/2*g/((b*f - a*g)*(d*f - c*g)*(f + g*x)^2) - (g*(2*b*d*f - b*c*g - a*d*g))/((b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)) + (b^3*Log[a + b*x])/((b* c - a*d)*(b*f - a*g)^3) - (d^3*Log[c + d*x])/((b*c - a*d)*(d*f - c*g)^3) + (g*(a^2*d^2*g^2 - a*b*d*g*(3*d*f - c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^ 2*g^2))*Log[f + g*x])/((b*f - a*g)^3*(d*f - c*g)^3)))/(3*g)
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/(g*(m + 1))), x] - Simp[B*n*((b*c - a*d) /(g*(m + 1))) Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; Free Q[{a, b, c, d, e, f, g, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(3261\) vs. \(2(271)=542\).
Time = 27.26 (sec) , antiderivative size = 3262, normalized size of antiderivative = 11.53
Input:
int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^4,x,method=_RETURNVERBOSE)
Output:
-1/6*(-6*B*ln(b*x+a)*x^3*b^4*c^2*d^2*f*g^7*n^2+6*B*ln(b*x+a)*x^3*b^4*c*d^3 *f^2*g^6*n^2-6*B*ln(g*x+f)*x^3*a^2*b^2*d^4*f*g^7*n^2+6*B*ln(g*x+f)*x^3*a*b ^3*d^4*f^2*g^6*n^2+6*B*ln(g*x+f)*x^3*b^4*c^2*d^2*f*g^7*n^2-6*B*ln(g*x+f)*x ^3*b^4*c*d^3*f^2*g^6*n^2-6*B*ln(b*x+a)*x^2*a^3*b*d^4*f*g^7*n^2+18*B*ln(b*x +a)*x^2*a^2*b^2*d^4*f^2*g^6*n^2-18*B*ln(b*x+a)*x^2*a*b^3*d^4*f^3*g^5*n^2+6 *B*ln(b*x+a)*x^2*b^4*c^3*d*f*g^7*n^2-18*B*ln(b*x+a)*x^2*b^4*c^2*d^2*f^2*g^ 6*n^2+18*B*ln(b*x+a)*x^2*b^4*c*d^3*f^3*g^5*n^2+6*B*ln(g*x+f)*x^2*a^3*b*d^4 *f*g^7*n^2-18*B*ln(g*x+f)*x^2*a^2*b^2*d^4*f^2*g^6*n^2+18*B*ln(g*x+f)*x^2*a *b^3*d^4*f^3*g^5*n^2-6*B*ln(g*x+f)*x^2*b^4*c^3*d*f*g^7*n^2+2*B*x^2*a^3*b*c *d^3*g^8*n^2-2*B*x^2*a^3*b*d^4*f*g^7*n^2+6*B*x^2*a^2*b^2*d^4*f^2*g^6*n^2-2 *B*x^2*a*b^3*c^3*d*g^8*n^2-4*B*x^2*a*b^3*d^4*f^3*g^5*n^2+2*B*x^2*b^4*c^3*d *f*g^7*n^2-6*B*x^2*b^4*c^2*d^2*f^2*g^6*n^2+4*B*x^2*b^4*c*d^3*f^3*g^5*n^2-6 *B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^4*d^4*f^5*g^3*n-B*x*a^3*b*c^2*d^2*g^8*n^2 -5*B*x*a^3*b*d^4*f^2*g^6*n^2+B*x*a^2*b^2*c^3*d*g^8*n^2+14*B*x*a^2*b^2*d^4* f^3*g^5*n^2-9*B*x*a*b^3*d^4*f^4*g^4*n^2+5*B*x*b^4*c^3*d*f^2*g^6*n^2-14*B*x *b^4*c^2*d^2*f^3*g^5*n^2+9*B*x*b^4*c*d^3*f^4*g^4*n^2+2*B*ln(e*((b*x+a)/(d* x+c))^n)*a^3*b*c^3*d*g^8*n-2*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^3*d*f^3*g^5 *n+6*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d^2*f^4*g^4*n-6*B*ln(e*((b*x+a)/( d*x+c))^n)*b^4*c*d^3*f^5*g^3*n+2*A*b^4*d^4*f^6*g^2*n-6*B*ln(b*x+a)*x*a^3*b *d^4*f^2*g^6*n^2+18*B*ln(b*x+a)*x*a^2*b^2*d^4*f^3*g^5*n^2-18*B*ln(b*x+a...
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx=\text {Timed out} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^4,x, algorithm="fricas" )
Output:
Timed out
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx=\text {Timed out} \] Input:
integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(g*x+f)**4,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 852 vs. \(2 (271) = 542\).
Time = 0.08 (sec) , antiderivative size = 852, normalized size of antiderivative = 3.01 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx =\text {Too large to display} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^4,x, algorithm="maxima" )
Output:
1/6*(2*b^3*log(b*x + a)/(b^3*f^3*g - 3*a*b^2*f^2*g^2 + 3*a^2*b*f*g^3 - a^3 *g^4) - 2*d^3*log(d*x + c)/(d^3*f^3*g - 3*c*d^2*f^2*g^2 + 3*c^2*d*f*g^3 - c^3*g^4) + 2*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2 - 3*(b^3*c^2*d - a^2*b*d^3)*f* g + (b^3*c^3 - a^3*d^3)*g^2)*log(g*x + f)/(b^3*d^3*f^6 + a^3*c^3*g^6 - 3*( b^3*c*d^2 + a*b^2*d^3)*f^5*g + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f ^4*g^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^3*g^3 + 3*( a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^2*g^4 - 3*(a^2*b*c^3 + a^3*c^2*d) *f*g^5) - (5*(b^2*c*d - a*b*d^2)*f^2 - 3*(b^2*c^2 - a^2*d^2)*f*g + (a*b*c^ 2 - a^2*c*d)*g^2 + 2*(2*(b^2*c*d - a*b*d^2)*f*g - (b^2*c^2 - a^2*d^2)*g^2) *x)/(b^2*d^2*f^6 + a^2*c^2*f^2*g^4 - 2*(b^2*c*d + a*b*d^2)*f^5*g + (b^2*c^ 2 + 4*a*b*c*d + a^2*d^2)*f^4*g^2 - 2*(a*b*c^2 + a^2*c*d)*f^3*g^3 + (b^2*d^ 2*f^4*g^2 + a^2*c^2*g^6 - 2*(b^2*c*d + a*b*d^2)*f^3*g^3 + (b^2*c^2 + 4*a*b *c*d + a^2*d^2)*f^2*g^4 - 2*(a*b*c^2 + a^2*c*d)*f*g^5)*x^2 + 2*(b^2*d^2*f^ 5*g + a^2*c^2*f*g^5 - 2*(b^2*c*d + a*b*d^2)*f^4*g^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^3*g^3 - 2*(a*b*c^2 + a^2*c*d)*f^2*g^4)*x))*B*n - 1/3*B*log(e *(b*x/(d*x + c) + a/(d*x + c))^n)/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f ^3*g) - 1/3*A/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g)
Leaf count of result is larger than twice the leaf count of optimal. 9692 vs. \(2 (271) = 542\).
Time = 0.69 (sec) , antiderivative size = 9692, normalized size of antiderivative = 34.25 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^4,x, algorithm="giac")
Output:
1/6*(2*(3*B*b^4*c^2*d^2*f^2*n - 6*B*a*b^3*c*d^3*f^2*n + 3*B*a^2*b^2*d^4*f^ 2*n - 3*B*b^4*c^3*d*f*g*n + 3*B*a*b^3*c^2*d^2*f*g*n + 3*B*a^2*b^2*c*d^3*f* g*n - 3*B*a^3*b*d^4*f*g*n + B*b^4*c^4*g^2*n - B*a*b^3*c^3*d*g^2*n - B*a^3* b*c*d^3*g^2*n + B*a^4*d^4*g^2*n)*log(-b*f + (b*x + a)*d*f/(d*x + c) + a*g - (b*x + a)*c*g/(d*x + c))/(b^3*d^3*f^6 - 3*b^3*c*d^2*f^5*g - 3*a*b^2*d^3* f^5*g + 3*b^3*c^2*d*f^4*g^2 + 9*a*b^2*c*d^2*f^4*g^2 + 3*a^2*b*d^3*f^4*g^2 - b^3*c^3*f^3*g^3 - 9*a*b^2*c^2*d*f^3*g^3 - 9*a^2*b*c*d^2*f^3*g^3 - a^3*d^ 3*f^3*g^3 + 3*a*b^2*c^3*f^2*g^4 + 9*a^2*b*c^2*d*f^2*g^4 + 3*a^3*c*d^2*f^2* g^4 - 3*a^2*b*c^3*f*g^5 - 3*a^3*c^2*d*f*g^5 + a^3*c^3*g^6) + 2*(3*B*b^4*c^ 2*d^2*f^2*n - 6*B*a*b^3*c*d^3*f^2*n - 6*(b*x + a)*B*b^3*c^2*d^3*f^2*n/(d*x + c) + 3*B*a^2*b^2*d^4*f^2*n + 12*(b*x + a)*B*a*b^2*c*d^4*f^2*n/(d*x + c) + 3*(b*x + a)^2*B*b^2*c^2*d^4*f^2*n/(d*x + c)^2 - 6*(b*x + a)*B*a^2*b*d^5 *f^2*n/(d*x + c) - 6*(b*x + a)^2*B*a*b*c*d^5*f^2*n/(d*x + c)^2 + 3*(b*x + a)^2*B*a^2*d^6*f^2*n/(d*x + c)^2 - 3*B*b^4*c^3*d*f*g*n + 3*B*a*b^3*c^2*d^2 *f*g*n + 9*(b*x + a)*B*b^3*c^3*d^2*f*g*n/(d*x + c) + 3*B*a^2*b^2*c*d^3*f*g *n - 15*(b*x + a)*B*a*b^2*c^2*d^3*f*g*n/(d*x + c) - 6*(b*x + a)^2*B*b^2*c^ 3*d^3*f*g*n/(d*x + c)^2 - 3*B*a^3*b*d^4*f*g*n + 3*(b*x + a)*B*a^2*b*c*d^4* f*g*n/(d*x + c) + 12*(b*x + a)^2*B*a*b*c^2*d^4*f*g*n/(d*x + c)^2 + 3*(b*x + a)*B*a^3*d^5*f*g*n/(d*x + c) - 6*(b*x + a)^2*B*a^2*c*d^5*f*g*n/(d*x + c) ^2 + B*b^4*c^4*g^2*n - B*a*b^3*c^3*d*g^2*n - 3*(b*x + a)*B*b^3*c^4*d*g^...
Time = 30.66 (sec) , antiderivative size = 1182, normalized size of antiderivative = 4.18 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx =\text {Too large to display} \] Input:
int((A + B*log(e*((a + b*x)/(c + d*x))^n))/(f + g*x)^4,x)
Output:
(B*d^3*n*log(c + d*x))/(3*c^3*g^4 - 3*d^3*f^3*g + 9*c*d^2*f^2*g^2 - 9*c^2* d*f*g^3) - (log(f + g*x)*(g^2*(B*a^3*d^3*n - B*b^3*c^3*n) - g*(3*B*a^2*b*d ^3*f*n - 3*B*b^3*c^2*d*f*n) + 3*B*a*b^2*d^3*f^2*n - 3*B*b^3*c*d^2*f^2*n))/ (3*a^3*c^3*g^6 + 3*b^3*d^3*f^6 - 3*a^3*d^3*f^3*g^3 - 3*b^3*c^3*f^3*g^3 - 9 *a^2*b*c^3*f*g^5 - 9*a*b^2*d^3*f^5*g - 9*a^3*c^2*d*f*g^5 - 9*b^3*c*d^2*f^5 *g + 9*a*b^2*c^3*f^2*g^4 + 9*a^2*b*d^3*f^4*g^2 + 9*a^3*c*d^2*f^2*g^4 + 9*b ^3*c^2*d*f^4*g^2 + 27*a*b^2*c*d^2*f^4*g^2 - 27*a*b^2*c^2*d*f^3*g^3 - 27*a^ 2*b*c*d^2*f^3*g^3 + 27*a^2*b*c^2*d*f^2*g^4) - (B*log(e*((a + b*x)/(c + d*x ))^n))/(3*g*(f^3 + g^3*x^3 + 3*f^2*g*x + 3*f*g^2*x^2)) - (B*b^3*n*log(a + b*x))/(3*a^3*g^4 - 3*b^3*f^3*g + 9*a*b^2*f^2*g^2 - 9*a^2*b*f*g^3) - ((2*A* a^2*c^2*g^4 + 2*A*b^2*d^2*f^4 + 2*A*a^2*d^2*f^2*g^2 + 2*A*b^2*c^2*f^2*g^2 + 3*B*a^2*d^2*f^2*g^2*n - 3*B*b^2*c^2*f^2*g^2*n - 4*A*a*b*c^2*f*g^3 - 4*A* a*b*d^2*f^3*g - 4*A*a^2*c*d*f*g^3 - 4*A*b^2*c*d*f^3*g + 8*A*a*b*c*d*f^2*g^ 2 + B*a*b*c^2*f*g^3*n - 5*B*a*b*d^2*f^3*g*n - B*a^2*c*d*f*g^3*n + 5*B*b^2* c*d*f^3*g*n)/(2*(a^2*c^2*g^4 + b^2*d^2*f^4 + a^2*d^2*f^2*g^2 + b^2*c^2*f^2 *g^2 - 2*a*b*c^2*f*g^3 - 2*a*b*d^2*f^3*g - 2*a^2*c*d*f*g^3 - 2*b^2*c*d*f^3 *g + 4*a*b*c*d*f^2*g^2)) + (x*(B*a*b*c^2*g^4*n - B*a^2*c*d*g^4*n + 5*B*a^2 *d^2*f*g^3*n - 5*B*b^2*c^2*f*g^3*n - 9*B*a*b*d^2*f^2*g^2*n + 9*B*b^2*c*d*f ^2*g^2*n))/(2*(a^2*c^2*g^4 + b^2*d^2*f^4 + a^2*d^2*f^2*g^2 + b^2*c^2*f^2*g ^2 - 2*a*b*c^2*f*g^3 - 2*a*b*d^2*f^3*g - 2*a^2*c*d*f*g^3 - 2*b^2*c*d*f^...
Time = 0.21 (sec) , antiderivative size = 6610, normalized size of antiderivative = 23.36 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(f+g x)^4} \, dx =\text {Too large to display} \] Input:
int((A+B*log(e*((b*x+a)/(d*x+c))^n))/(g*x+f)^4,x)
Output:
( - 6*log(a + b*x)*a**3*b*c**3*f**3*g**6*n - 18*log(a + b*x)*a**3*b*c**3*f **2*g**7*n*x - 18*log(a + b*x)*a**3*b*c**3*f*g**8*n*x**2 - 6*log(a + b*x)* a**3*b*c**3*g**9*n*x**3 + 18*log(a + b*x)*a**3*b*c**2*d*f**4*g**5*n + 54*l og(a + b*x)*a**3*b*c**2*d*f**3*g**6*n*x + 54*log(a + b*x)*a**3*b*c**2*d*f* *2*g**7*n*x**2 + 18*log(a + b*x)*a**3*b*c**2*d*f*g**8*n*x**3 - 18*log(a + b*x)*a**3*b*c*d**2*f**5*g**4*n - 54*log(a + b*x)*a**3*b*c*d**2*f**4*g**5*n *x - 54*log(a + b*x)*a**3*b*c*d**2*f**3*g**6*n*x**2 - 18*log(a + b*x)*a**3 *b*c*d**2*f**2*g**7*n*x**3 + 6*log(a + b*x)*a**3*b*d**3*f**6*g**3*n + 18*l og(a + b*x)*a**3*b*d**3*f**5*g**4*n*x + 18*log(a + b*x)*a**3*b*d**3*f**4*g **5*n*x**2 + 6*log(a + b*x)*a**3*b*d**3*f**3*g**6*n*x**3 + 18*log(a + b*x) *a**2*b**2*c**3*f**4*g**5*n + 54*log(a + b*x)*a**2*b**2*c**3*f**3*g**6*n*x + 54*log(a + b*x)*a**2*b**2*c**3*f**2*g**7*n*x**2 + 18*log(a + b*x)*a**2* b**2*c**3*f*g**8*n*x**3 - 54*log(a + b*x)*a**2*b**2*c**2*d*f**5*g**4*n - 1 62*log(a + b*x)*a**2*b**2*c**2*d*f**4*g**5*n*x - 162*log(a + b*x)*a**2*b** 2*c**2*d*f**3*g**6*n*x**2 - 54*log(a + b*x)*a**2*b**2*c**2*d*f**2*g**7*n*x **3 + 54*log(a + b*x)*a**2*b**2*c*d**2*f**6*g**3*n + 162*log(a + b*x)*a**2 *b**2*c*d**2*f**5*g**4*n*x + 162*log(a + b*x)*a**2*b**2*c*d**2*f**4*g**5*n *x**2 + 54*log(a + b*x)*a**2*b**2*c*d**2*f**3*g**6*n*x**3 - 18*log(a + b*x )*a**2*b**2*d**3*f**7*g**2*n - 54*log(a + b*x)*a**2*b**2*d**3*f**6*g**3*n* x - 54*log(a + b*x)*a**2*b**2*d**3*f**5*g**4*n*x**2 - 18*log(a + b*x)*a...