Integrand size = 20, antiderivative size = 133 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {(B d-A e) (a+b x)^4}{6 e (b d-a e) (d+e x)^6}+\frac {(2 b B d+A b e-3 a B e) (a+b x)^4}{15 e (b d-a e)^2 (d+e x)^5}+\frac {b (2 b B d+A b e-3 a B e) (a+b x)^4}{60 e (b d-a e)^3 (d+e x)^4} \]
-1/6*(-A*e+B*d)*(b*x+a)^4/e/(-a*e+b*d)/(e*x+d)^6+1/15*(A*b*e-3*B*a*e+2*B*b *d)*(b*x+a)^4/e/(-a*e+b*d)^2/(e*x+d)^5+1/60*b*(A*b*e-3*B*a*e+2*B*b*d)*(b*x +a)^4/e/(-a*e+b*d)^3/(e*x+d)^4
Time = 0.06 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {2 a^3 e^3 (5 A e+B (d+6 e x))+3 a^2 b e^2 \left (2 A e (d+6 e x)+B \left (d^2+6 d e x+15 e^2 x^2\right )\right )+3 a b^2 e \left (A e \left (d^2+6 d e x+15 e^2 x^2\right )+B \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )+2 B \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )\right )}{60 e^5 (d+e x)^6} \]
-1/60*(2*a^3*e^3*(5*A*e + B*(d + 6*e*x)) + 3*a^2*b*e^2*(2*A*e*(d + 6*e*x) + B*(d^2 + 6*d*e*x + 15*e^2*x^2)) + 3*a*b^2*e*(A*e*(d^2 + 6*d*e*x + 15*e^2 *x^2) + B*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3)) + b^3*(A*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*B*(d^4 + 6*d^3*e*x + 15*d^2*e^ 2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)))/(e^5*(d + e*x)^6)
Time = 0.22 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.98, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {87, 55, 48}
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.
| \(\displaystyle \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \int \frac {(a+b x)^3}{(d+e x)^6}dx}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {(-3 a B e+A b e+2 b B d) \left (\frac {b \int \frac {(a+b x)^3}{(d+e x)^5}dx}{5 (b d-a e)}+\frac {(a+b x)^4}{5 (d+e x)^5 (b d-a e)}\right )}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {\left (\frac {b (a+b x)^4}{20 (d+e x)^4 (b d-a e)^2}+\frac {(a+b x)^4}{5 (d+e x)^5 (b d-a e)}\right ) (-3 a B e+A b e+2 b B d)}{3 e (b d-a e)}-\frac {(a+b x)^4 (B d-A e)}{6 e (d+e x)^6 (b d-a e)}\) |
-1/6*((B*d - A*e)*(a + b*x)^4)/(e*(b*d - a*e)*(d + e*x)^6) + ((2*b*B*d + A *b*e - 3*a*B*e)*((a + b*x)^4/(5*(b*d - a*e)*(d + e*x)^5) + (b*(a + b*x)^4) /(20*(b*d - a*e)^2*(d + e*x)^4)))/(3*e*(b*d - a*e))
3.11.47.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Leaf count of result is larger than twice the leaf count of optimal. \(265\) vs. \(2(127)=254\).
Time = 0.70 (sec) , antiderivative size = 266, normalized size of antiderivative = 2.00
| method | result | size |
| risch | \(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {b^{2} \left (A b e +3 B a e +2 B b d \right ) x^{3}}{3 e^{2}}-\frac {b \left (3 A a b \,e^{2}+A \,b^{2} d e +3 B \,a^{2} e^{2}+3 B a b d e +2 b^{2} B \,d^{2}\right ) x^{2}}{4 e^{3}}-\frac {\left (6 A \,a^{2} b \,e^{3}+3 A a \,b^{2} d \,e^{2}+A \,b^{3} d^{2} e +2 B \,a^{3} e^{3}+3 B \,a^{2} b d \,e^{2}+3 B a \,b^{2} d^{2} e +2 b^{3} B \,d^{3}\right ) x}{10 e^{4}}-\frac {10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5}}}{\left (e x +d \right )^{6}}\) | \(266\) |
| default | \(-\frac {3 A \,a^{2} b \,e^{3}-6 A a \,b^{2} d \,e^{2}+3 A \,b^{3} d^{2} e +B \,a^{3} e^{3}-6 B \,a^{2} b d \,e^{2}+9 B a \,b^{2} d^{2} e -4 b^{3} B \,d^{3}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b^{2} \left (A b e +3 B a e -4 B b d \right )}{3 e^{5} \left (e x +d \right )^{3}}-\frac {b^{3} B}{2 e^{5} \left (e x +d \right )^{2}}-\frac {a^{3} A \,e^{4}-3 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}-A \,b^{3} d^{3} e -B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}-3 B a \,b^{2} d^{3} e +b^{3} B \,d^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {3 b \left (A a b \,e^{2}-A \,b^{2} d e +B \,a^{2} e^{2}-3 B a b d e +2 b^{2} B \,d^{2}\right )}{4 e^{5} \left (e x +d \right )^{4}}\) | \(281\) |
| norman | \(\frac {-\frac {b^{3} B \,x^{4}}{2 e}-\frac {\left (A \,b^{3} e^{2}+3 B a \,b^{2} e^{2}+2 b^{3} B d e \right ) x^{3}}{3 e^{3}}-\frac {\left (3 A a \,b^{2} e^{3}+A \,b^{3} d \,e^{2}+3 B \,a^{2} b \,e^{3}+3 B a \,b^{2} d \,e^{2}+2 b^{3} B \,d^{2} e \right ) x^{2}}{4 e^{4}}-\frac {\left (6 A \,a^{2} b \,e^{4}+3 A a \,b^{2} d \,e^{3}+A \,b^{3} d^{2} e^{2}+2 B \,a^{3} e^{4}+3 B \,a^{2} b d \,e^{3}+3 B a \,b^{2} d^{2} e^{2}+2 b^{3} B \,d^{3} e \right ) x}{10 e^{5}}-\frac {10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6}}}{\left (e x +d \right )^{6}}\) | \(294\) |
| gosper | \(-\frac {30 B \,x^{4} b^{3} e^{4}+20 A \,x^{3} b^{3} e^{4}+60 B \,x^{3} a \,b^{2} e^{4}+40 B \,x^{3} b^{3} d \,e^{3}+45 A \,x^{2} a \,b^{2} e^{4}+15 A \,x^{2} b^{3} d \,e^{3}+45 B \,x^{2} a^{2} b \,e^{4}+45 B \,x^{2} a \,b^{2} d \,e^{3}+30 B \,x^{2} b^{3} d^{2} e^{2}+36 A x \,a^{2} b \,e^{4}+18 A x a \,b^{2} d \,e^{3}+6 A x \,b^{3} d^{2} e^{2}+12 B x \,a^{3} e^{4}+18 B x \,a^{2} b d \,e^{3}+18 B x a \,b^{2} d^{2} e^{2}+12 B x \,b^{3} d^{3} e +10 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+3 A a \,b^{2} d^{2} e^{2}+A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+3 B \,a^{2} b \,d^{2} e^{2}+3 B a \,b^{2} d^{3} e +2 b^{3} B \,d^{4}}{60 e^{5} \left (e x +d \right )^{6}}\) | \(300\) |
| parallelrisch | \(-\frac {30 b^{3} B \,x^{4} e^{5}+20 A \,b^{3} e^{5} x^{3}+60 B a \,b^{2} e^{5} x^{3}+40 B \,b^{3} d \,e^{4} x^{3}+45 A a \,b^{2} e^{5} x^{2}+15 A \,b^{3} d \,e^{4} x^{2}+45 B \,a^{2} b \,e^{5} x^{2}+45 B a \,b^{2} d \,e^{4} x^{2}+30 B \,b^{3} d^{2} e^{3} x^{2}+36 A \,a^{2} b \,e^{5} x +18 A a \,b^{2} d \,e^{4} x +6 A \,b^{3} d^{2} e^{3} x +12 B \,a^{3} e^{5} x +18 B \,a^{2} b d \,e^{4} x +18 B a \,b^{2} d^{2} e^{3} x +12 B \,b^{3} d^{3} e^{2} x +10 a^{3} A \,e^{5}+6 A \,a^{2} b d \,e^{4}+3 A a \,b^{2} d^{2} e^{3}+A \,b^{3} d^{3} e^{2}+2 B \,a^{3} d \,e^{4}+3 B \,a^{2} b \,d^{2} e^{3}+3 B a \,b^{2} d^{3} e^{2}+2 b^{3} B \,d^{4} e}{60 e^{6} \left (e x +d \right )^{6}}\) | \(307\) |
(-1/2*b^3*B/e*x^4-1/3*b^2/e^2*(A*b*e+3*B*a*e+2*B*b*d)*x^3-1/4*b/e^3*(3*A*a *b*e^2+A*b^2*d*e+3*B*a^2*e^2+3*B*a*b*d*e+2*B*b^2*d^2)*x^2-1/10/e^4*(6*A*a^ 2*b*e^3+3*A*a*b^2*d*e^2+A*b^3*d^2*e+2*B*a^3*e^3+3*B*a^2*b*d*e^2+3*B*a*b^2* d^2*e+2*B*b^3*d^3)*x-1/60/e^5*(10*A*a^3*e^4+6*A*a^2*b*d*e^3+3*A*a*b^2*d^2* e^2+A*b^3*d^3*e+2*B*a^3*d*e^3+3*B*a^2*b*d^2*e^2+3*B*a*b^2*d^3*e+2*B*b^3*d^ 4))/(e*x+d)^6
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (127) = 254\).
Time = 0.22 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3) *d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20* (2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3*B *a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3 *A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x ^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
Timed out. \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 317 vs. \(2 (127) = 254\).
Time = 0.24 (sec) , antiderivative size = 317, normalized size of antiderivative = 2.38 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 2 \, B b^{3} d^{4} + 10 \, A a^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 20 \, {\left (2 \, B b^{3} d e^{3} + {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 15 \, {\left (2 \, B b^{3} d^{2} e^{2} + {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 6 \, {\left (2 \, B b^{3} d^{3} e + {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{60 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
-1/60*(30*B*b^3*e^4*x^4 + 2*B*b^3*d^4 + 10*A*a^3*e^4 + (3*B*a*b^2 + A*b^3) *d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 + 2*(B*a^3 + 3*A*a^2*b)*d*e^3 + 20* (2*B*b^3*d*e^3 + (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 15*(2*B*b^3*d^2*e^2 + (3*B *a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 6*(2*B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 + 3*(B*a^2*b + A*a*b^2)*d*e^3 + 2*(B*a^3 + 3 *A*a^2*b)*e^4)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x ^3 + 15*d^4*e^7*x^2 + 6*d^5*e^6*x + d^6*e^5)
Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (127) = 254\).
Time = 0.30 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {30 \, B b^{3} e^{4} x^{4} + 40 \, B b^{3} d e^{3} x^{3} + 60 \, B a b^{2} e^{4} x^{3} + 20 \, A b^{3} e^{4} x^{3} + 30 \, B b^{3} d^{2} e^{2} x^{2} + 45 \, B a b^{2} d e^{3} x^{2} + 15 \, A b^{3} d e^{3} x^{2} + 45 \, B a^{2} b e^{4} x^{2} + 45 \, A a b^{2} e^{4} x^{2} + 12 \, B b^{3} d^{3} e x + 18 \, B a b^{2} d^{2} e^{2} x + 6 \, A b^{3} d^{2} e^{2} x + 18 \, B a^{2} b d e^{3} x + 18 \, A a b^{2} d e^{3} x + 12 \, B a^{3} e^{4} x + 36 \, A a^{2} b e^{4} x + 2 \, B b^{3} d^{4} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 3 \, B a^{2} b d^{2} e^{2} + 3 \, A a b^{2} d^{2} e^{2} + 2 \, B a^{3} d e^{3} + 6 \, A a^{2} b d e^{3} + 10 \, A a^{3} e^{4}}{60 \, {\left (e x + d\right )}^{6} e^{5}} \]
-1/60*(30*B*b^3*e^4*x^4 + 40*B*b^3*d*e^3*x^3 + 60*B*a*b^2*e^4*x^3 + 20*A*b ^3*e^4*x^3 + 30*B*b^3*d^2*e^2*x^2 + 45*B*a*b^2*d*e^3*x^2 + 15*A*b^3*d*e^3* x^2 + 45*B*a^2*b*e^4*x^2 + 45*A*a*b^2*e^4*x^2 + 12*B*b^3*d^3*e*x + 18*B*a* b^2*d^2*e^2*x + 6*A*b^3*d^2*e^2*x + 18*B*a^2*b*d*e^3*x + 18*A*a*b^2*d*e^3* x + 12*B*a^3*e^4*x + 36*A*a^2*b*e^4*x + 2*B*b^3*d^4 + 3*B*a*b^2*d^3*e + A* b^3*d^3*e + 3*B*a^2*b*d^2*e^2 + 3*A*a*b^2*d^2*e^2 + 2*B*a^3*d*e^3 + 6*A*a^ 2*b*d*e^3 + 10*A*a^3*e^4)/((e*x + d)^6*e^5)
Time = 1.44 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.41 \[ \int \frac {(a+b x)^3 (A+B x)}{(d+e x)^7} \, dx=-\frac {\frac {2\,B\,a^3\,d\,e^3+10\,A\,a^3\,e^4+3\,B\,a^2\,b\,d^2\,e^2+6\,A\,a^2\,b\,d\,e^3+3\,B\,a\,b^2\,d^3\,e+3\,A\,a\,b^2\,d^2\,e^2+2\,B\,b^3\,d^4+A\,b^3\,d^3\,e}{60\,e^5}+\frac {x\,\left (2\,B\,a^3\,e^3+3\,B\,a^2\,b\,d\,e^2+6\,A\,a^2\,b\,e^3+3\,B\,a\,b^2\,d^2\,e+3\,A\,a\,b^2\,d\,e^2+2\,B\,b^3\,d^3+A\,b^3\,d^2\,e\right )}{10\,e^4}+\frac {b^2\,x^3\,\left (A\,b\,e+3\,B\,a\,e+2\,B\,b\,d\right )}{3\,e^2}+\frac {b\,x^2\,\left (3\,B\,a^2\,e^2+3\,B\,a\,b\,d\,e+3\,A\,a\,b\,e^2+2\,B\,b^2\,d^2+A\,b^2\,d\,e\right )}{4\,e^3}+\frac {B\,b^3\,x^4}{2\,e}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]
-((10*A*a^3*e^4 + 2*B*b^3*d^4 + A*b^3*d^3*e + 2*B*a^3*d*e^3 + 3*A*a*b^2*d^ 2*e^2 + 3*B*a^2*b*d^2*e^2 + 6*A*a^2*b*d*e^3 + 3*B*a*b^2*d^3*e)/(60*e^5) + (x*(2*B*a^3*e^3 + 2*B*b^3*d^3 + 6*A*a^2*b*e^3 + A*b^3*d^2*e + 3*A*a*b^2*d* e^2 + 3*B*a*b^2*d^2*e + 3*B*a^2*b*d*e^2))/(10*e^4) + (b^2*x^3*(A*b*e + 3*B *a*e + 2*B*b*d))/(3*e^2) + (b*x^2*(3*B*a^2*e^2 + 2*B*b^2*d^2 + 3*A*a*b*e^2 + A*b^2*d*e + 3*B*a*b*d*e))/(4*e^3) + (B*b^3*x^4)/(2*e))/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3 + 15*d^2*e^4*x^4 + 6*d^5*e* x)