Integrand size = 33, antiderivative size = 183 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=\frac {B n}{9 d g^4 (c+d x)^3}+\frac {b B n}{6 d (b c-a d) g^4 (c+d x)^2}+\frac {b^2 B n}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {b^3 B n \log (a+b x)}{3 d (b c-a d)^3 g^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d g^4 (c+d x)^3}-\frac {b^3 B n \log (c+d x)}{3 d (b c-a d)^3 g^4} \]
1/9*B*n/d/g^4/(d*x+c)^3+1/6*b*B*n/d/(-a*d+b*c)/g^4/(d*x+c)^2+1/3*b^2*B*n/d /(-a*d+b*c)^2/g^4/(d*x+c)+1/3*b^3*B*n*ln(b*x+a)/d/(-a*d+b*c)^3/g^4+1/3*(-A -B*ln(e*((b*x+a)/(d*x+c))^n))/d/g^4/(d*x+c)^3-1/3*b^3*B*n*ln(d*x+c)/d/(-a* d+b*c)^3/g^4
Time = 0.10 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.80 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=\frac {-6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {B n \left ((b c-a d) \left (2 a^2 d^2-a b d (7 c+3 d x)+b^2 \left (11 c^2+15 c d x+6 d^2 x^2\right )\right )+6 b^3 (c+d x)^3 \log (a+b x)-6 b^3 (c+d x)^3 \log (c+d x)\right )}{(b c-a d)^3}}{18 d g^4 (c+d x)^3} \]
(-6*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + (B*n*((b*c - a*d)*(2*a^2*d^2 - a*b*d*(7*c + 3*d*x) + b^2*(11*c^2 + 15*c*d*x + 6*d^2*x^2)) + 6*b^3*(c + d*x)^3*Log[a + b*x] - 6*b^3*(c + d*x)^3*Log[c + d*x]))/(b*c - a*d)^3)/(18* d*g^4*(c + d*x)^3)
Time = 0.35 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2947, 27, 54, 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.
| \(\displaystyle \int \frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{(c g+d g x)^4} \, dx\) |
| \(\Big \downarrow \) 2947 |
| \(\displaystyle \frac {B n (b c-a d) \int \frac {1}{g^3 (a+b x) (c+d x)^4}dx}{3 d g}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d g^4 (c+d x)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B n (b c-a d) \int \frac {1}{(a+b x) (c+d x)^4}dx}{3 d g^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d g^4 (c+d x)^3}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {B n (b c-a d) \int \left (\frac {b^4}{(b c-a d)^4 (a+b x)}-\frac {d b^3}{(b c-a d)^4 (c+d x)}-\frac {d b^2}{(b c-a d)^3 (c+d x)^2}-\frac {d b}{(b c-a d)^2 (c+d x)^3}-\frac {d}{(b c-a d) (c+d x)^4}\right )dx}{3 d g^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d g^4 (c+d x)^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {B n (b c-a d) \left (\frac {b^3 \log (a+b x)}{(b c-a d)^4}-\frac {b^3 \log (c+d x)}{(b c-a d)^4}+\frac {b^2}{(c+d x) (b c-a d)^3}+\frac {b}{2 (c+d x)^2 (b c-a d)^2}+\frac {1}{3 (c+d x)^3 (b c-a d)}\right )}{3 d g^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d g^4 (c+d x)^3}\) |
-1/3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])/(d*g^4*(c + d*x)^3) + (B*(b*c - a*d)*n*(1/(3*(b*c - a*d)*(c + d*x)^3) + b/(2*(b*c - a*d)^2*(c + d*x)^2) + b^2/((b*c - a*d)^3*(c + d*x)) + (b^3*Log[a + b*x])/(b*c - a*d)^4 - (b^3* Log[c + d*x])/(b*c - a*d)^4))/(3*d*g^4)
3.1.36.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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. \(439\) vs. \(2(174)=348\).
Time = 16.01 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.40
| method | result | size |
| parallelrisch | \(-\frac {9 B \,a^{2} b^{2} c \,d^{6} n^{2}-18 B a \,b^{3} c^{2} d^{5} n^{2}-18 A \,a^{2} b^{2} c \,d^{6} n +18 A a \,b^{3} c^{2} d^{5} n +18 B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{6} n +18 B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d^{5} n -18 B x a \,b^{3} c \,d^{6} n^{2}-18 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{2} b^{2} c \,d^{6} n +18 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a \,b^{3} c^{2} d^{5} n -2 B \,a^{3} b \,d^{7} n^{2}+11 B \,b^{4} c^{3} d^{4} n^{2}+6 A \,a^{3} b \,d^{7} n -6 A \,b^{4} c^{3} d^{4} n +6 B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) a^{3} b \,d^{7} n +6 B \,x^{3} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{7} n -6 B \,x^{2} a \,b^{3} d^{7} n^{2}+6 B \,x^{2} b^{4} c \,d^{6} n^{2}+3 B x \,a^{2} b^{2} d^{7} n^{2}+15 B x \,b^{4} c^{2} d^{5} n^{2}}{18 g^{4} \left (d x +c \right )^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) n b \,d^{5}}\) | \(440\) |
-1/18*(9*B*a^2*b^2*c*d^6*n^2-18*B*a*b^3*c^2*d^5*n^2-18*A*a^2*b^2*c*d^6*n+1 8*A*a*b^3*c^2*d^5*n+18*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^6*n+18*B*x* ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d^5*n-18*B*x*a*b^3*c*d^6*n^2-18*B*ln(e*( (b*x+a)/(d*x+c))^n)*a^2*b^2*c*d^6*n+18*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^3*c ^2*d^5*n-2*B*a^3*b*d^7*n^2+11*B*b^4*c^3*d^4*n^2+6*A*a^3*b*d^7*n-6*A*b^4*c^ 3*d^4*n+6*B*ln(e*((b*x+a)/(d*x+c))^n)*a^3*b*d^7*n+6*B*x^3*ln(e*((b*x+a)/(d *x+c))^n)*b^4*d^7*n-6*B*x^2*a*b^3*d^7*n^2+6*B*x^2*b^4*c*d^6*n^2+3*B*x*a^2* b^2*d^7*n^2+15*B*x*b^4*c^2*d^5*n^2)/g^4/(d*x+c)^3/(a^3*d^3-3*a^2*b*c*d^2+3 *a*b^2*c^2*d-b^3*c^3)/n/b/d^5
Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (171) = 342\).
Time = 0.28 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.64 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=-\frac {6 \, A b^{3} c^{3} - 18 \, A a b^{2} c^{2} d + 18 \, A a^{2} b c d^{2} - 6 \, A a^{3} d^{3} - 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (5 \, B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + B a^{2} b d^{3}\right )} n x - {\left (11 \, B b^{3} c^{3} - 18 \, B a b^{2} c^{2} d + 9 \, B a^{2} b c d^{2} - 2 \, B a^{3} d^{3}\right )} n + 6 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} \log \left (e\right ) - 6 \, {\left (B b^{3} d^{3} n x^{3} + 3 \, B b^{3} c d^{2} n x^{2} + 3 \, B b^{3} c^{2} d n x + {\left (3 \, B a b^{2} c^{2} d - 3 \, B a^{2} b c d^{2} + B a^{3} d^{3}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{18 \, {\left ({\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} g^{4} x^{3} + 3 \, {\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} g^{4} x^{2} + 3 \, {\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} g^{4} x + {\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )} g^{4}\right )}} \]
-1/18*(6*A*b^3*c^3 - 18*A*a*b^2*c^2*d + 18*A*a^2*b*c*d^2 - 6*A*a^3*d^3 - 6 *(B*b^3*c*d^2 - B*a*b^2*d^3)*n*x^2 - 3*(5*B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + B*a^2*b*d^3)*n*x - (11*B*b^3*c^3 - 18*B*a*b^2*c^2*d + 9*B*a^2*b*c*d^2 - 2* B*a^3*d^3)*n + 6*(B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2 - B*a^3*d^ 3)*log(e) - 6*(B*b^3*d^3*n*x^3 + 3*B*b^3*c*d^2*n*x^2 + 3*B*b^3*c^2*d*n*x + (3*B*a*b^2*c^2*d - 3*B*a^2*b*c*d^2 + B*a^3*d^3)*n)*log((b*x + a)/(d*x + c )))/((b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*g^4*x^3 + 3 *(b^3*c^4*d^3 - 3*a*b^2*c^3*d^4 + 3*a^2*b*c^2*d^5 - a^3*c*d^6)*g^4*x^2 + 3 *(b^3*c^5*d^2 - 3*a*b^2*c^4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)*g^4*x + ( b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4)*g^4)
Timed out. \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (171) = 342\).
Time = 0.21 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.37 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=\frac {1}{18} \, B n {\left (\frac {6 \, b^{2} d^{2} x^{2} + 11 \, b^{2} c^{2} - 7 \, a b c d + 2 \, a^{2} d^{2} + 3 \, {\left (5 \, b^{2} c d - a b d^{2}\right )} x}{{\left (b^{2} c^{2} d^{4} - 2 \, a b c d^{5} + a^{2} d^{6}\right )} g^{4} x^{3} + 3 \, {\left (b^{2} c^{3} d^{3} - 2 \, a b c^{2} d^{4} + a^{2} c d^{5}\right )} g^{4} x^{2} + 3 \, {\left (b^{2} c^{4} d^{2} - 2 \, a b c^{3} d^{3} + a^{2} c^{2} d^{4}\right )} g^{4} x + {\left (b^{2} c^{5} d - 2 \, a b c^{4} d^{2} + a^{2} c^{3} d^{3}\right )} g^{4}} + \frac {6 \, b^{3} \log \left (b x + a\right )}{{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} g^{4}} - \frac {6 \, b^{3} \log \left (d x + c\right )}{{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} g^{4}}\right )} - \frac {B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{3 \, {\left (d^{4} g^{4} x^{3} + 3 \, c d^{3} g^{4} x^{2} + 3 \, c^{2} d^{2} g^{4} x + c^{3} d g^{4}\right )}} - \frac {A}{3 \, {\left (d^{4} g^{4} x^{3} + 3 \, c d^{3} g^{4} x^{2} + 3 \, c^{2} d^{2} g^{4} x + c^{3} d g^{4}\right )}} \]
1/18*B*n*((6*b^2*d^2*x^2 + 11*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2 + 3*(5*b^2*c *d - a*b*d^2)*x)/((b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*g^4*x^3 + 3*(b^2*c ^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*g^4*x^2 + 3*(b^2*c^4*d^2 - 2*a*b*c^3*d ^3 + a^2*c^2*d^4)*g^4*x + (b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*g^4) + 6*b^3*log(b*x + a)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^ 4)*g^4) - 6*b^3*log(d*x + c)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*g^4)) - 1/3*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^4*g^4* x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4) - 1/3*A/(d^4*g^4*x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4)
Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (171) = 342\).
Time = 0.68 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.21 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=\frac {1}{18} \, {\left (6 \, {\left (\frac {3 \, {\left (b x + a\right )} B b^{2} n}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}} - \frac {3 \, {\left (b x + a\right )}^{2} B b d n}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{2}} + \frac {{\left (b x + a\right )}^{3} B d^{2} n}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{3}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) - \frac {2 \, {\left (B d^{2} n - 3 \, B d^{2} \log \left (e\right ) - 3 \, A d^{2}\right )} {\left (b x + a\right )}^{3}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{3}} + \frac {9 \, {\left (B b d n - 2 \, B b d \log \left (e\right ) - 2 \, A b d\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{2}} - \frac {18 \, {\left (B b^{2} n - B b^{2} \log \left (e\right ) - A b^{2}\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
1/18*(6*(3*(b*x + a)*B*b^2*n/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)* (d*x + c)) - 3*(b*x + a)^2*B*b*d*n/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2 *g^4)*(d*x + c)^2) + (b*x + a)^3*B*d^2*n/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a ^2*d^2*g^4)*(d*x + c)^3))*log((b*x + a)/(d*x + c)) - 2*(B*d^2*n - 3*B*d^2* log(e) - 3*A*d^2)*(b*x + a)^3/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4) *(d*x + c)^3) + 9*(B*b*d*n - 2*B*b*d*log(e) - 2*A*b*d)*(b*x + a)^2/((b^2*c ^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^2) - 18*(B*b^2*n - B*b^2*l og(e) - A*b^2)*(b*x + a)/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 1.62 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.91 \[ \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c g+d g x)^4} \, dx=\frac {B\,a^2\,d\,n}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}-\frac {A\,a^2\,d}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}-\frac {A\,b^2\,c^2}{3\,d\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}-\frac {B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{3\,d\,g^4\,{\left (c+d\,x\right )}^3}+\frac {2\,A\,a\,b\,c}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}+\frac {B\,b^2\,d\,n\,x^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}-\frac {7\,B\,a\,b\,c\,n}{18\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}+\frac {11\,B\,b^2\,c^2\,n}{18\,d\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}+\frac {5\,B\,b^2\,c\,n\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}-\frac {B\,a\,b\,d\,n\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^3}+\frac {B\,b^3\,n\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,d\,g^4\,{\left (a\,d-b\,c\right )}^3} \]
(B*a^2*d*n)/(9*g^4*(a*d - b*c)^2*(c + d*x)^3) - (A*a^2*d)/(3*g^4*(a*d - b* c)^2*(c + d*x)^3) - (A*b^2*c^2)/(3*d*g^4*(a*d - b*c)^2*(c + d*x)^3) - (B*l og(e*((a + b*x)/(c + d*x))^n))/(3*d*g^4*(c + d*x)^3) + (B*b^3*n*atan((a*d* 1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(3*d*g^4*(a*d - b*c)^3) + (2*A*a* b*c)/(3*g^4*(a*d - b*c)^2*(c + d*x)^3) + (B*b^2*d*n*x^2)/(3*g^4*(a*d - b*c )^2*(c + d*x)^3) - (7*B*a*b*c*n)/(18*g^4*(a*d - b*c)^2*(c + d*x)^3) + (11* B*b^2*c^2*n)/(18*d*g^4*(a*d - b*c)^2*(c + d*x)^3) + (5*B*b^2*c*n*x)/(6*g^4 *(a*d - b*c)^2*(c + d*x)^3) - (B*a*b*d*n*x)/(6*g^4*(a*d - b*c)^2*(c + d*x) ^3)