Integrand size = 26, antiderivative size = 28 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=\frac {(a+b x)^7}{7 (b d-a e) (d+e x)^7} \]
Leaf count is larger than twice the leaf count of optimal. \(271\) vs. \(2(28)=56\).
Time = 0.06 (sec) , antiderivative size = 271, normalized size of antiderivative = 9.68 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=-\frac {a^6 e^6+a^5 b e^5 (d+7 e x)+a^4 b^2 e^4 \left (d^2+7 d e x+21 e^2 x^2\right )+a^3 b^3 e^3 \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+a^2 b^4 e^2 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )+a b^5 e \left (d^5+7 d^4 e x+21 d^3 e^2 x^2+35 d^2 e^3 x^3+35 d e^4 x^4+21 e^5 x^5\right )+b^6 \left (d^6+7 d^5 e x+21 d^4 e^2 x^2+35 d^3 e^3 x^3+35 d^2 e^4 x^4+21 d e^5 x^5+7 e^6 x^6\right )}{7 e^7 (d+e x)^7} \]
-1/7*(a^6*e^6 + a^5*b*e^5*(d + 7*e*x) + a^4*b^2*e^4*(d^2 + 7*d*e*x + 21*e^ 2*x^2) + a^3*b^3*e^3*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + a^2*b ^4*e^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + a* b^5*e*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4 + 21*e^5*x^5) + b^6*(d^6 + 7*d^5*e*x + 21*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 35* d^2*e^4*x^4 + 21*d*e^5*x^5 + 7*e^6*x^6))/(e^7*(d + e*x)^7)
Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1098, 27, 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.
| \(\displaystyle \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx\) |
| \(\Big \downarrow \) 1098 |
| \(\displaystyle \frac {\int \frac {b^6 (a+b x)^6}{(d+e x)^8}dx}{b^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^6}{(d+e x)^8}dx\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {(a+b x)^7}{7 (d+e x)^7 (b d-a e)}\) |
3.15.97.3.1 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] :> 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[((d_.) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[ {a, b, c, d, e, m}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(26)=52\).
Time = 2.62 (sec) , antiderivative size = 314, normalized size of antiderivative = 11.21
| method | result | size |
| risch | \(\frac {-\frac {b^{6} x^{6}}{e}-\frac {3 b^{5} \left (a e +b d \right ) x^{5}}{e^{2}}-\frac {5 b^{4} \left (a^{2} e^{2}+a b d e +b^{2} d^{2}\right ) x^{4}}{e^{3}}-\frac {5 b^{3} \left (a^{3} e^{3}+a^{2} b d \,e^{2}+a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{e^{4}}-\frac {3 b^{2} \left (e^{4} a^{4}+b \,e^{3} d \,a^{3}+b^{2} e^{2} d^{2} a^{2}+a \,b^{3} d^{3} e +b^{4} d^{4}\right ) x^{2}}{e^{5}}-\frac {b \left (a^{5} e^{5}+a^{4} b d \,e^{4}+a^{3} b^{2} d^{2} e^{3}+a^{2} b^{3} d^{3} e^{2}+a \,b^{4} d^{4} e +b^{5} d^{5}\right ) x}{e^{6}}-\frac {a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}}{7 e^{7}}}{\left (e x +d \right )^{7}}\) | \(314\) |
| norman | \(\frac {-\frac {b^{6} x^{6}}{e}-\frac {3 \left (e a \,b^{5}+d \,b^{6}\right ) x^{5}}{e^{2}}-\frac {5 \left (e^{2} a^{2} b^{4}+d e a \,b^{5}+b^{6} d^{2}\right ) x^{4}}{e^{3}}-\frac {5 \left (e^{3} a^{3} b^{3}+d \,e^{2} a^{2} b^{4}+d^{2} e a \,b^{5}+d^{3} b^{6}\right ) x^{3}}{e^{4}}-\frac {3 \left (e^{4} a^{4} b^{2}+d \,e^{3} a^{3} b^{3}+d^{2} e^{2} a^{2} b^{4}+d^{3} e a \,b^{5}+d^{4} b^{6}\right ) x^{2}}{e^{5}}-\frac {\left (a^{5} b \,e^{5}+d \,e^{4} a^{4} b^{2}+d^{2} e^{3} a^{3} b^{3}+d^{3} e^{2} a^{2} b^{4}+d^{4} e a \,b^{5}+d^{5} b^{6}\right ) x}{e^{6}}-\frac {a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}}{7 e^{7}}}{\left (e x +d \right )^{7}}\) | \(324\) |
| default | \(-\frac {3 b^{2} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right )}{e^{7} \left (e x +d \right )^{5}}-\frac {b^{6}}{e^{7} \left (e x +d \right )}-\frac {5 b^{4} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{e^{7} \left (e x +d \right )^{3}}-\frac {a^{6} e^{6}-6 a^{5} b d \,e^{5}+15 a^{4} b^{2} d^{2} e^{4}-20 a^{3} b^{3} d^{3} e^{3}+15 a^{2} b^{4} d^{4} e^{2}-6 a \,b^{5} d^{5} e +b^{6} d^{6}}{7 e^{7} \left (e x +d \right )^{7}}-\frac {5 b^{3} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{e^{7} \left (e x +d \right )^{4}}-\frac {3 b^{5} \left (a e -b d \right )}{e^{7} \left (e x +d \right )^{2}}-\frac {b \left (a^{5} e^{5}-5 a^{4} b d \,e^{4}+10 a^{3} b^{2} d^{2} e^{3}-10 a^{2} b^{3} d^{3} e^{2}+5 a \,b^{4} d^{4} e -b^{5} d^{5}\right )}{e^{7} \left (e x +d \right )^{6}}\) | \(357\) |
| gosper | \(-\frac {7 x^{6} b^{6} e^{6}+21 x^{5} a \,b^{5} e^{6}+21 x^{5} b^{6} d \,e^{5}+35 x^{4} a^{2} b^{4} e^{6}+35 x^{4} a \,b^{5} d \,e^{5}+35 x^{4} b^{6} d^{2} e^{4}+35 x^{3} a^{3} b^{3} e^{6}+35 x^{3} a^{2} b^{4} d \,e^{5}+35 x^{3} a \,b^{5} d^{2} e^{4}+35 x^{3} b^{6} d^{3} e^{3}+21 x^{2} a^{4} b^{2} e^{6}+21 x^{2} a^{3} b^{3} d \,e^{5}+21 x^{2} a^{2} b^{4} d^{2} e^{4}+21 x^{2} a \,b^{5} d^{3} e^{3}+21 x^{2} b^{6} d^{4} e^{2}+7 x \,a^{5} b \,e^{6}+7 x \,a^{4} b^{2} d \,e^{5}+7 x \,a^{3} b^{3} d^{2} e^{4}+7 x \,a^{2} b^{4} d^{3} e^{3}+7 x a \,b^{5} d^{4} e^{2}+7 x \,b^{6} d^{5} e +a^{6} e^{6}+a^{5} b d \,e^{5}+a^{4} b^{2} d^{2} e^{4}+a^{3} b^{3} d^{3} e^{3}+a^{2} b^{4} d^{4} e^{2}+a \,b^{5} d^{5} e +b^{6} d^{6}}{7 e^{7} \left (e x +d \right )^{7}}\) | \(370\) |
| parallelrisch | \(\frac {-7 x^{6} b^{6} e^{6}-21 x^{5} a \,b^{5} e^{6}-21 x^{5} b^{6} d \,e^{5}-35 x^{4} a^{2} b^{4} e^{6}-35 x^{4} a \,b^{5} d \,e^{5}-35 x^{4} b^{6} d^{2} e^{4}-35 x^{3} a^{3} b^{3} e^{6}-35 x^{3} a^{2} b^{4} d \,e^{5}-35 x^{3} a \,b^{5} d^{2} e^{4}-35 x^{3} b^{6} d^{3} e^{3}-21 x^{2} a^{4} b^{2} e^{6}-21 x^{2} a^{3} b^{3} d \,e^{5}-21 x^{2} a^{2} b^{4} d^{2} e^{4}-21 x^{2} a \,b^{5} d^{3} e^{3}-21 x^{2} b^{6} d^{4} e^{2}-7 x \,a^{5} b \,e^{6}-7 x \,a^{4} b^{2} d \,e^{5}-7 x \,a^{3} b^{3} d^{2} e^{4}-7 x \,a^{2} b^{4} d^{3} e^{3}-7 x a \,b^{5} d^{4} e^{2}-7 x \,b^{6} d^{5} e -a^{6} e^{6}-a^{5} b d \,e^{5}-a^{4} b^{2} d^{2} e^{4}-a^{3} b^{3} d^{3} e^{3}-a^{2} b^{4} d^{4} e^{2}-a \,b^{5} d^{5} e -b^{6} d^{6}}{7 e^{7} \left (e x +d \right )^{7}}\) | \(377\) |
(-b^6/e*x^6-3*b^5*(a*e+b*d)/e^2*x^5-5*b^4*(a^2*e^2+a*b*d*e+b^2*d^2)/e^3*x^ 4-5*b^3*(a^3*e^3+a^2*b*d*e^2+a*b^2*d^2*e+b^3*d^3)/e^4*x^3-3*b^2*(a^4*e^4+a ^3*b*d*e^3+a^2*b^2*d^2*e^2+a*b^3*d^3*e+b^4*d^4)/e^5*x^2-b*(a^5*e^5+a^4*b*d *e^4+a^3*b^2*d^2*e^3+a^2*b^3*d^3*e^2+a*b^4*d^4*e+b^5*d^5)/e^6*x-1/7*(a^6*e ^6+a^5*b*d*e^5+a^4*b^2*d^2*e^4+a^3*b^3*d^3*e^3+a^2*b^4*d^4*e^2+a*b^5*d^5*e +b^6*d^6)/e^7)/(e*x+d)^7
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (26) = 52\).
Time = 0.32 (sec) , antiderivative size = 398, normalized size of antiderivative = 14.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=-\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^ 3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^ 6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*e^ 3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a *b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6 *d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11* x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)
Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (26) = 52\).
Time = 0.24 (sec) , antiderivative size = 398, normalized size of antiderivative = 14.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=-\frac {7 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6} + 21 \, {\left (b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (b^{6} d^{2} e^{4} + a b^{5} d e^{5} + a^{2} b^{4} e^{6}\right )} x^{4} + 35 \, {\left (b^{6} d^{3} e^{3} + a b^{5} d^{2} e^{4} + a^{2} b^{4} d e^{5} + a^{3} b^{3} e^{6}\right )} x^{3} + 21 \, {\left (b^{6} d^{4} e^{2} + a b^{5} d^{3} e^{3} + a^{2} b^{4} d^{2} e^{4} + a^{3} b^{3} d e^{5} + a^{4} b^{2} e^{6}\right )} x^{2} + 7 \, {\left (b^{6} d^{5} e + a b^{5} d^{4} e^{2} + a^{2} b^{4} d^{3} e^{3} + a^{3} b^{3} d^{2} e^{4} + a^{4} b^{2} d e^{5} + a^{5} b e^{6}\right )} x}{7 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
-1/7*(7*b^6*e^6*x^6 + b^6*d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^ 3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^5 + a^6*e^6 + 21*(b^6*d*e^5 + a*b^5*e^ 6)*x^5 + 35*(b^6*d^2*e^4 + a*b^5*d*e^5 + a^2*b^4*e^6)*x^4 + 35*(b^6*d^3*e^ 3 + a*b^5*d^2*e^4 + a^2*b^4*d*e^5 + a^3*b^3*e^6)*x^3 + 21*(b^6*d^4*e^2 + a *b^5*d^3*e^3 + a^2*b^4*d^2*e^4 + a^3*b^3*d*e^5 + a^4*b^2*e^6)*x^2 + 7*(b^6 *d^5*e + a*b^5*d^4*e^2 + a^2*b^4*d^3*e^3 + a^3*b^3*d^2*e^4 + a^4*b^2*d*e^5 + a^5*b*e^6)*x)/(e^14*x^7 + 7*d*e^13*x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11* x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 369, normalized size of antiderivative = 13.18 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=-\frac {7 \, b^{6} e^{6} x^{6} + 21 \, b^{6} d e^{5} x^{5} + 21 \, a b^{5} e^{6} x^{5} + 35 \, b^{6} d^{2} e^{4} x^{4} + 35 \, a b^{5} d e^{5} x^{4} + 35 \, a^{2} b^{4} e^{6} x^{4} + 35 \, b^{6} d^{3} e^{3} x^{3} + 35 \, a b^{5} d^{2} e^{4} x^{3} + 35 \, a^{2} b^{4} d e^{5} x^{3} + 35 \, a^{3} b^{3} e^{6} x^{3} + 21 \, b^{6} d^{4} e^{2} x^{2} + 21 \, a b^{5} d^{3} e^{3} x^{2} + 21 \, a^{2} b^{4} d^{2} e^{4} x^{2} + 21 \, a^{3} b^{3} d e^{5} x^{2} + 21 \, a^{4} b^{2} e^{6} x^{2} + 7 \, b^{6} d^{5} e x + 7 \, a b^{5} d^{4} e^{2} x + 7 \, a^{2} b^{4} d^{3} e^{3} x + 7 \, a^{3} b^{3} d^{2} e^{4} x + 7 \, a^{4} b^{2} d e^{5} x + 7 \, a^{5} b e^{6} x + b^{6} d^{6} + a b^{5} d^{5} e + a^{2} b^{4} d^{4} e^{2} + a^{3} b^{3} d^{3} e^{3} + a^{4} b^{2} d^{2} e^{4} + a^{5} b d e^{5} + a^{6} e^{6}}{7 \, {\left (e x + d\right )}^{7} e^{7}} \]
-1/7*(7*b^6*e^6*x^6 + 21*b^6*d*e^5*x^5 + 21*a*b^5*e^6*x^5 + 35*b^6*d^2*e^4 *x^4 + 35*a*b^5*d*e^5*x^4 + 35*a^2*b^4*e^6*x^4 + 35*b^6*d^3*e^3*x^3 + 35*a *b^5*d^2*e^4*x^3 + 35*a^2*b^4*d*e^5*x^3 + 35*a^3*b^3*e^6*x^3 + 21*b^6*d^4* e^2*x^2 + 21*a*b^5*d^3*e^3*x^2 + 21*a^2*b^4*d^2*e^4*x^2 + 21*a^3*b^3*d*e^5 *x^2 + 21*a^4*b^2*e^6*x^2 + 7*b^6*d^5*e*x + 7*a*b^5*d^4*e^2*x + 7*a^2*b^4* d^3*e^3*x + 7*a^3*b^3*d^2*e^4*x + 7*a^4*b^2*d*e^5*x + 7*a^5*b*e^6*x + b^6* d^6 + a*b^5*d^5*e + a^2*b^4*d^4*e^2 + a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + a^5*b*d*e^5 + a^6*e^6)/((e*x + d)^7*e^7)
Time = 9.71 (sec) , antiderivative size = 378, normalized size of antiderivative = 13.50 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^8} \, dx=-\frac {\frac {a^6\,e^6+a^5\,b\,d\,e^5+a^4\,b^2\,d^2\,e^4+a^3\,b^3\,d^3\,e^3+a^2\,b^4\,d^4\,e^2+a\,b^5\,d^5\,e+b^6\,d^6}{7\,e^7}+\frac {b^6\,x^6}{e}+\frac {5\,b^3\,x^3\,\left (a^3\,e^3+a^2\,b\,d\,e^2+a\,b^2\,d^2\,e+b^3\,d^3\right )}{e^4}+\frac {b\,x\,\left (a^5\,e^5+a^4\,b\,d\,e^4+a^3\,b^2\,d^2\,e^3+a^2\,b^3\,d^3\,e^2+a\,b^4\,d^4\,e+b^5\,d^5\right )}{e^6}+\frac {3\,b^5\,x^5\,\left (a\,e+b\,d\right )}{e^2}+\frac {3\,b^2\,x^2\,\left (a^4\,e^4+a^3\,b\,d\,e^3+a^2\,b^2\,d^2\,e^2+a\,b^3\,d^3\,e+b^4\,d^4\right )}{e^5}+\frac {5\,b^4\,x^4\,\left (a^2\,e^2+a\,b\,d\,e+b^2\,d^2\right )}{e^3}}{d^7+7\,d^6\,e\,x+21\,d^5\,e^2\,x^2+35\,d^4\,e^3\,x^3+35\,d^3\,e^4\,x^4+21\,d^2\,e^5\,x^5+7\,d\,e^6\,x^6+e^7\,x^7} \]
-((a^6*e^6 + b^6*d^6 + a^2*b^4*d^4*e^2 + a^3*b^3*d^3*e^3 + a^4*b^2*d^2*e^4 + a*b^5*d^5*e + a^5*b*d*e^5)/(7*e^7) + (b^6*x^6)/e + (5*b^3*x^3*(a^3*e^3 + b^3*d^3 + a*b^2*d^2*e + a^2*b*d*e^2))/e^4 + (b*x*(a^5*e^5 + b^5*d^5 + a^ 2*b^3*d^3*e^2 + a^3*b^2*d^2*e^3 + a*b^4*d^4*e + a^4*b*d*e^4))/e^6 + (3*b^5 *x^5*(a*e + b*d))/e^2 + (3*b^2*x^2*(a^4*e^4 + b^4*d^4 + a^2*b^2*d^2*e^2 + a*b^3*d^3*e + a^3*b*d*e^3))/e^5 + (5*b^4*x^4*(a^2*e^2 + b^2*d^2 + a*b*d*e) )/e^3)/(d^7 + e^7*x^7 + 7*d*e^6*x^6 + 21*d^5*e^2*x^2 + 35*d^4*e^3*x^3 + 35 *d^3*e^4*x^4 + 21*d^2*e^5*x^5 + 7*d^6*e*x)