Integrand size = 27, antiderivative size = 275 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=-\frac {B (b c-a d)}{6 (b f-a g) (d f-c g) (f+g x)^2}-\frac {B (b c-a d) (2 b d f-b c g-a d g)}{3 (b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {b^3 B \log (a+b x)}{3 g (b f-a g)^3}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{3 g (f+g x)^3}-\frac {B d^3 \log (c+d x)}{3 g (d f-c g)^3}+\frac {B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{3 (b f-a g)^3 (d f-c g)^3} \]
-1/6*B*(-a*d+b*c)/(-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)/(-a*g+b*f)^2/(-c*g+d*f)^2/(g*x+f)+1/3*b^3*B*ln(b*x+a)/g/(- a*g+b*f)^3+1/3*(-A-B*ln(e*(b*x+a)/(d*x+c)))/g/(g*x+f)^3-1/3*B*d^3*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))*ln(g*x+f)/(-a*g+b*f)^3/(-c*g+d*f)^3
Time = 0.39 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^3}+B (b c-a d) \left (-\frac {g}{2 (b f-a g) (d f-c g) (f+g x)^2}+\frac {g (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac {b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}+\frac {d^3 \log (c+d x)}{(b c-a d) (-d f+c g)^3}+\frac {g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \log (f+g x)}{(b f-a g)^3 (d f-c g)^3}\right )}{3 g} \]
(-((A + B*Log[(e*(a + b*x))/(c + d*x)])/(f + g*x)^3) + B*(b*c - a*d)*(-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 = 0.55 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2948, 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.
| \(\displaystyle \int \frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{(f+g x)^4} \, dx\) |
| \(\Big \downarrow \) 2948 |
| \(\displaystyle \frac {B (b c-a d) \int \frac {1}{(a+b x) (c+d x) (f+g x)^3}dx}{3 g}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 g (f+g x)^3}\) |
| \(\Big \downarrow \) 93 |
| \(\displaystyle \frac {B (b c-a d) \int \left (\frac {b^4}{(b c-a d) (b f-a g)^3 (a+b x)}+\frac {d^4}{(b c-a d) (c g-d f)^3 (c+d x)}+\frac {g^2 \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}-\frac {g^2 (-2 b d f+b c g+a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)^2}+\frac {g^2}{(b f-a g) (d f-c g) (f+g x)^3}\right )dx}{3 g}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 g (f+g x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B (b c-a d) \left (\frac {g \log (f+g x) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right )}{(b f-a g)^3 (d f-c g)^3}+\frac {b^3 \log (a+b x)}{(b c-a d) (b f-a g)^3}-\frac {d^3 \log (c+d x)}{(b c-a d) (d f-c g)^3}-\frac {g (-a d g-b c g+2 b d f)}{(f+g x) (b f-a g)^2 (d f-c g)^2}-\frac {g}{2 (f+g x)^2 (b f-a g) (d f-c g)}\right )}{3 g}-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{3 g (f+g x)^3}\) |
-1/3*(A + B*Log[(e*(a + b*x))/(c + d*x)])/(g*(f + g*x)^3) + (B*(b*c - a*d) *(-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)
3.3.38.3.1 Defintions of rubi rules used
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_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*( (A + B*Log[e*((a + b*x)^n/(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] / ; FreeQ[{a, b, c, d, e, f, g, A, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && !(EqQ[m, -2] && IntegerQ[n])
Leaf count of result is larger than twice the leaf count of optimal. \(1503\) vs. \(2(266)=532\).
Time = 3.05 (sec) , antiderivative size = 1504, normalized size of antiderivative = 5.47
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1504\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1821\) |
| default | \(\text {Expression too large to display}\) | \(1821\) |
| risch | \(\text {Expression too large to display}\) | \(2293\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2896\) |
-1/3*A/(g*x+f)^3/g-B/d^2*(a*d-b*c)*e*(2*d^3*e*g*(a*d-b*c)/(c*g-d*f)^2*(-1/ 2/(a*g-b*f)^2/e^2*(1/(c*g-d*f)*ln(c*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-d*f*(b *e/d+(a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)+e*(a*g-b*f)/(c*g-d*f)/(c*g*(b*e/d +(a*d-b*c)*e/d/(d*x+c))-d*f*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f))+1/ 2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(c*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-d*f*( b*e/d+(a*d-b*c)*e/d/(d*x+c))-2*a*e*g+2*b*e*f)*(b*e/d+(a*d-b*c)*e/d/(d*x+c) )/(c*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-d*f*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-a*e *g+b*e*f)^2/(a*g-b*f)^2/e^2)+d^2*e^2*(a^2*d^2-2*a*b*c*d+b^2*c^2)*g^2/(c*g- d*f)^2*(-1/3/(a*g-b*f)/(a^2*g^2-2*a*b*f*g+b^2*f^2)/e^3*(1/2*e^2*(a^2*g^2-2 *a*b*f*g+b^2*f^2)/(c*g-d*f)/(c*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-d*f*(b*e/d+ (a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)^2-1/(c*g-d*f)*ln(c*g*(b*e/d+(a*d-b*c)* e/d/(d*x+c))-d*f*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)-e*(a*g-b*f)/(c *g-d*f)/(c*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-d*f*(b*e/d+(a*d-b*c)*e/d/(d*x+c ))-a*e*g+b*e*f))-1/3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*(3*a^2*e^2*g^2-6*a*b* e^2*f*g-3*a*c*e*g^2*(b*e/d+(a*d-b*c)*e/d/(d*x+c))+3*a*d*e*f*g*(b*e/d+(a*d- b*c)*e/d/(d*x+c))+3*b^2*e^2*f^2+3*b*c*e*f*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))- 3*b*d*e*f^2*(b*e/d+(a*d-b*c)*e/d/(d*x+c))+c^2*g^2*(b*e/d+(a*d-b*c)*e/d/(d* x+c))^2-2*c*d*f*g*(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2+d^2*f^2*(b*e/d+(a*d-b*c) *e/d/(d*x+c))^2)*(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(c*g*(b*e/d+(a*d-b*c)*e/d/( d*x+c))-d*f*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-a*e*g+b*e*f)^3/(a*g-b*f)/(a^2...
Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (263) = 526\).
Time = 0.26 (sec) , antiderivative size = 848, normalized size of antiderivative = 3.08 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=\frac {1}{6} \, {\left (\frac {2 \, b^{3} \log \left (b x + a\right )}{b^{3} f^{3} g - 3 \, a b^{2} f^{2} g^{2} + 3 \, a^{2} b f g^{3} - a^{3} g^{4}} - \frac {2 \, d^{3} \log \left (d x + c\right )}{d^{3} f^{3} g - 3 \, c d^{2} f^{2} g^{2} + 3 \, c^{2} d f g^{3} - c^{3} g^{4}} + \frac {2 \, {\left (3 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f^{2} - 3 \, {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} f g + {\left (b^{3} c^{3} - a^{3} d^{3}\right )} g^{2}\right )} \log \left (g x + f\right )}{b^{3} d^{3} f^{6} + a^{3} c^{3} g^{6} - 3 \, {\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} f^{5} g + 3 \, {\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} f^{4} g^{2} - {\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} f^{3} g^{3} + 3 \, {\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} f^{2} g^{4} - 3 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} f g^{5}} - \frac {5 \, {\left (b^{2} c d - a b d^{2}\right )} f^{2} - 3 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} f g + {\left (a b c^{2} - a^{2} c d\right )} g^{2} + 2 \, {\left (2 \, {\left (b^{2} c d - a b d^{2}\right )} f g - {\left (b^{2} c^{2} - a^{2} d^{2}\right )} g^{2}\right )} x}{b^{2} d^{2} f^{6} + a^{2} c^{2} f^{2} g^{4} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{5} g + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{4} g^{2} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{3} g^{3} + {\left (b^{2} d^{2} f^{4} g^{2} + a^{2} c^{2} g^{6} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{3} g^{3} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{2} g^{4} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f g^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{2} f^{5} g + a^{2} c^{2} f g^{5} - 2 \, {\left (b^{2} c d + a b d^{2}\right )} f^{4} g^{2} + {\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} f^{3} g^{3} - 2 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} g^{4}\right )} x} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g}\right )} B - \frac {A}{3 \, {\left (g^{4} x^{3} + 3 \, f g^{3} x^{2} + 3 \, f^{2} g^{2} x + f^{3} g\right )}} \]
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) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g))*B - 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. 9339 vs. \(2 (263) = 526\).
Time = 0.78 (sec) , antiderivative size = 9339, normalized size of antiderivative = 33.96 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=\text {Too large to display} \]
1/6*(2*(3*B*b^4*c^2*d^2*e*f^2 - 6*B*a*b^3*c*d^3*e*f^2 + 3*B*a^2*b^2*d^4*e* f^2 - 3*B*b^4*c^3*d*e*f*g + 3*B*a*b^3*c^2*d^2*e*f*g + 3*B*a^2*b^2*c*d^3*e* f*g - 3*B*a^3*b*d^4*e*f*g + B*b^4*c^4*e*g^2 - B*a*b^3*c^3*d*e*g^2 - B*a^3* b*c*d^3*e*g^2 + B*a^4*d^4*e*g^2)*log(-b*e*f + a*e*g + (b*e*x + a*e)*d*f/(d *x + c) - (b*e*x + a*e)*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*e^4*f^2 - 6*B*a*b^3*c*d^3*e^4*f^2 + 3*B*a^2*b^2*d^4*e^4* f^2 - 3*B*b^4*c^3*d*e^4*f*g + 3*B*a*b^3*c^2*d^2*e^4*f*g + 3*B*a^2*b^2*c*d^ 3*e^4*f*g - 3*B*a^3*b*d^4*e^4*f*g + B*b^4*c^4*e^4*g^2 - B*a*b^3*c^3*d*e^4* g^2 - B*a^3*b*c*d^3*e^4*g^2 + B*a^4*d^4*e^4*g^2 - 6*(b*e*x + a*e)*B*b^3*c^ 2*d^3*e^3*f^2/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^2*c*d^4*e^3*f^2/(d*x + c) - 6*(b*e*x + a*e)*B*a^2*b*d^5*e^3*f^2/(d*x + c) + 9*(b*e*x + a*e)*B*b^3*c ^3*d^2*e^3*f*g/(d*x + c) - 15*(b*e*x + a*e)*B*a*b^2*c^2*d^3*e^3*f*g/(d*x + c) + 3*(b*e*x + a*e)*B*a^2*b*c*d^4*e^3*f*g/(d*x + c) + 3*(b*e*x + a*e)*B* a^3*d^5*e^3*f*g/(d*x + c) - 3*(b*e*x + a*e)*B*b^3*c^4*d*e^3*g^2/(d*x + c) + 3*(b*e*x + a*e)*B*a*b^2*c^3*d^2*e^3*g^2/(d*x + c) + 3*(b*e*x + a*e)*B*a^ 2*b*c^2*d^3*e^3*g^2/(d*x + c) - 3*(b*e*x + a*e)*B*a^3*c*d^4*e^3*g^2/(d*...
Time = 7.32 (sec) , antiderivative size = 1154, normalized size of antiderivative = 4.20 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(f+g x)^4} \, dx=\frac {\ln \left (f+g\,x\right )\,\left (g\,\left (3\,B\,a^2\,b\,d^3\,f-3\,B\,b^3\,c^2\,d\,f\right )-g^2\,\left (B\,a^3\,d^3-B\,b^3\,c^3\right )-3\,B\,a\,b^2\,d^3\,f^2+3\,B\,b^3\,c\,d^2\,f^2\right )}{3\,a^3\,c^3\,g^6-9\,a^3\,c^2\,d\,f\,g^5+9\,a^3\,c\,d^2\,f^2\,g^4-3\,a^3\,d^3\,f^3\,g^3-9\,a^2\,b\,c^3\,f\,g^5+27\,a^2\,b\,c^2\,d\,f^2\,g^4-27\,a^2\,b\,c\,d^2\,f^3\,g^3+9\,a^2\,b\,d^3\,f^4\,g^2+9\,a\,b^2\,c^3\,f^2\,g^4-27\,a\,b^2\,c^2\,d\,f^3\,g^3+27\,a\,b^2\,c\,d^2\,f^4\,g^2-9\,a\,b^2\,d^3\,f^5\,g-3\,b^3\,c^3\,f^3\,g^3+9\,b^3\,c^2\,d\,f^4\,g^2-9\,b^3\,c\,d^2\,f^5\,g+3\,b^3\,d^3\,f^6}-\frac {\frac {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^2-3\,B\,b^2\,c^2\,f^2\,g^2-4\,A\,a\,b\,c^2\,f\,g^3-4\,A\,a\,b\,d^2\,f^3\,g+B\,a\,b\,c^2\,f\,g^3-4\,A\,a^2\,c\,d\,f\,g^3-5\,B\,a\,b\,d^2\,f^3\,g-4\,A\,b^2\,c\,d\,f^3\,g-B\,a^2\,c\,d\,f\,g^3+5\,B\,b^2\,c\,d\,f^3\,g+8\,A\,a\,b\,c\,d\,f^2\,g^2}{2\,\left (a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4\right )}+\frac {x^2\,\left (B\,a^2\,d^2\,g^4-2\,B\,f\,a\,b\,d^2\,g^3-B\,b^2\,c^2\,g^4+2\,B\,f\,b^2\,c\,d\,g^3\right )}{a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4}+\frac {x\,\left (-B\,a^2\,c\,d\,g^4+5\,B\,a^2\,d^2\,f\,g^3+B\,a\,b\,c^2\,g^4-9\,B\,a\,b\,d^2\,f^2\,g^2-5\,B\,b^2\,c^2\,f\,g^3+9\,B\,b^2\,c\,d\,f^2\,g^2\right )}{2\,\left (a^2\,c^2\,g^4-2\,a^2\,c\,d\,f\,g^3+a^2\,d^2\,f^2\,g^2-2\,a\,b\,c^2\,f\,g^3+4\,a\,b\,c\,d\,f^2\,g^2-2\,a\,b\,d^2\,f^3\,g+b^2\,c^2\,f^2\,g^2-2\,b^2\,c\,d\,f^3\,g+b^2\,d^2\,f^4\right )}}{3\,f^3\,g+9\,f^2\,g^2\,x+9\,f\,g^3\,x^2+3\,g^4\,x^3}-\frac {B\,b^3\,\ln \left (a+b\,x\right )}{3\,a^3\,g^4-9\,a^2\,b\,f\,g^3+9\,a\,b^2\,f^2\,g^2-3\,b^3\,f^3\,g}+\frac {B\,d^3\,\ln \left (c+d\,x\right )}{3\,c^3\,g^4-9\,c^2\,d\,f\,g^3+9\,c\,d^2\,f^2\,g^2-3\,d^3\,f^3\,g}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{3\,g\,\left (f^3+3\,f^2\,g\,x+3\,f\,g^2\,x^2+g^3\,x^3\right )} \]
(log(f + g*x)*(g*(3*B*a^2*b*d^3*f - 3*B*b^3*c^2*d*f) - g^2*(B*a^3*d^3 - B* b^3*c^3) - 3*B*a*b^2*d^3*f^2 + 3*B*b^3*c*d^2*f^2))/(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) - ((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 - 3*B*b^2*c^2*f^2*g^2 - 4*A*a*b*c^2*f*g^3 - 4*A*a*b*d^2*f^3*g + B*a*b*c^2*f*g^3 - 4*A*a^2*c*d*f*g ^3 - 5*B*a*b*d^2*f^3*g - 4*A*b^2*c*d*f^3*g - B*a^2*c*d*f*g^3 + 5*B*b^2*c*d *f^3*g + 8*A*a*b*c*d*f^2*g^2)/(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^2*(B*a^2*d^2*g^4 - B*b^2*c^ 2*g^4 - 2*B*a*b*d^2*f*g^3 + 2*B*b^2*c*d*f*g^3))/(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*(5*B*a^2*d^2* f*g^3 - 5*B*b^2*c^2*f*g^3 + B*a*b*c^2*g^4 - B*a^2*c*d*g^4 - 9*B*a*b*d^2*f^ 2*g^2 + 9*B*b^2*c*d*f^2*g^2))/(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)))/(3*f^3*g + 3*g^4*x^3 + 9*f^2...