Integrand size = 24, antiderivative size = 154 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {10 x}{99 d^3 \left (d^2-e^2 x^2\right )^{9/2}}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}+\frac {80 x}{693 d^5 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {32 x}{231 d^7 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {128 x}{693 d^9 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {256 x}{693 d^{11} \sqrt {d^2-e^2 x^2}} \] Output:
10/99*x/d^3/(-e^2*x^2+d^2)^(9/2)-1/11/d/e/(e*x+d)/(-e^2*x^2+d^2)^(9/2)+80/ 693*x/d^5/(-e^2*x^2+d^2)^(7/2)+32/231*x/d^7/(-e^2*x^2+d^2)^(5/2)+128/693*x /d^9/(-e^2*x^2+d^2)^(3/2)+256/693*x/d^11/(-e^2*x^2+d^2)^(1/2)
Time = 0.53 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-63 d^{10}+630 d^9 e x+630 d^8 e^2 x^2-1680 d^7 e^3 x^3-1680 d^6 e^4 x^4+2016 d^5 e^5 x^5+2016 d^4 e^6 x^6-1152 d^3 e^7 x^7-1152 d^2 e^8 x^8+256 d e^9 x^9+256 e^{10} x^{10}\right )}{693 d^{11} e (d-e x)^5 (d+e x)^6} \] Input:
Integrate[1/((d + e*x)*(d^2 - e^2*x^2)^(11/2)),x]
Output:
(Sqrt[d^2 - e^2*x^2]*(-63*d^10 + 630*d^9*e*x + 630*d^8*e^2*x^2 - 1680*d^7* e^3*x^3 - 1680*d^6*e^4*x^4 + 2016*d^5*e^5*x^5 + 2016*d^4*e^6*x^6 - 1152*d^ 3*e^7*x^7 - 1152*d^2*e^8*x^8 + 256*d*e^9*x^9 + 256*e^10*x^10))/(693*d^11*e *(d - e*x)^5*(d + e*x)^6)
Time = 0.40 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {470, 209, 209, 209, 209, 208}
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.
| \(\displaystyle \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx\) |
| \(\Big \downarrow \) 470 |
| \(\displaystyle \frac {10 \int \frac {1}{\left (d^2-e^2 x^2\right )^{11/2}}dx}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {10 \left (\frac {8 \int \frac {1}{\left (d^2-e^2 x^2\right )^{9/2}}dx}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {10 \left (\frac {8 \left (\frac {6 \int \frac {1}{\left (d^2-e^2 x^2\right )^{7/2}}dx}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {10 \left (\frac {8 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {10 \left (\frac {8 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}+\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 d^2}+\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}\right )}{7 d^2}+\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}\right )}{9 d^2}+\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {10 \left (\frac {x}{9 d^2 \left (d^2-e^2 x^2\right )^{9/2}}+\frac {8 \left (\frac {x}{7 d^2 \left (d^2-e^2 x^2\right )^{7/2}}+\frac {6 \left (\frac {x}{5 d^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \left (\frac {x}{3 d^2 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 x}{3 d^4 \sqrt {d^2-e^2 x^2}}\right )}{5 d^2}\right )}{7 d^2}\right )}{9 d^2}\right )}{11 d}-\frac {1}{11 d e (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}\) |
Input:
Int[1/((d + e*x)*(d^2 - e^2*x^2)^(11/2)),x]
Output:
-1/11*1/(d*e*(d + e*x)*(d^2 - e^2*x^2)^(9/2)) + (10*(x/(9*d^2*(d^2 - e^2*x ^2)^(9/2)) + (8*(x/(7*d^2*(d^2 - e^2*x^2)^(7/2)) + (6*(x/(5*d^2*(d^2 - e^2 *x^2)^(5/2)) + (4*(x/(3*d^2*(d^2 - e^2*x^2)^(3/2)) + (2*x)/(3*d^4*Sqrt[d^2 - e^2*x^2])))/(5*d^2)))/(7*d^2)))/(9*d^2)))/(11*d)
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[(n + 2*p + 2)/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c^2 + a*d^2, 0] && LtQ[n, 0] && NeQ[n + p + 1, 0] && IntegerQ[2*p]
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88
| method | result | size |
| gosper | \(-\frac {\left (-e x +d \right ) \left (-256 e^{10} x^{10}-256 d \,e^{9} x^{9}+1152 d^{2} e^{8} x^{8}+1152 d^{3} e^{7} x^{7}-2016 d^{4} e^{6} x^{6}-2016 d^{5} e^{5} x^{5}+1680 d^{6} e^{4} x^{4}+1680 d^{7} e^{3} x^{3}-630 d^{8} e^{2} x^{2}-630 d^{9} e x +63 d^{10}\right )}{693 d^{11} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(136\) |
| orering | \(-\frac {\left (-e x +d \right ) \left (-256 e^{10} x^{10}-256 d \,e^{9} x^{9}+1152 d^{2} e^{8} x^{8}+1152 d^{3} e^{7} x^{7}-2016 d^{4} e^{6} x^{6}-2016 d^{5} e^{5} x^{5}+1680 d^{6} e^{4} x^{4}+1680 d^{7} e^{3} x^{3}-630 d^{8} e^{2} x^{2}-630 d^{9} e x +63 d^{10}\right )}{693 d^{11} e \left (-e^{2} x^{2}+d^{2}\right )^{\frac {11}{2}}}\) | \(136\) |
| trager | \(-\frac {\left (-256 e^{10} x^{10}-256 d \,e^{9} x^{9}+1152 d^{2} e^{8} x^{8}+1152 d^{3} e^{7} x^{7}-2016 d^{4} e^{6} x^{6}-2016 d^{5} e^{5} x^{5}+1680 d^{6} e^{4} x^{4}+1680 d^{7} e^{3} x^{3}-630 d^{8} e^{2} x^{2}-630 d^{9} e x +63 d^{10}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{693 d^{11} \left (e x +d \right )^{6} \left (-e x +d \right )^{5} e}\) | \(145\) |
| default | \(\frac {-\frac {1}{11 d e \left (x +\frac {d}{e}\right ) \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {10 e \left (-\frac {-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e}{18 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}+\frac {-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{63 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}+\frac {8 \left (-\frac {3 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{35 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}+\frac {6 \left (-\frac {2 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 d^{2} e^{2} \left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}-\frac {4 \left (-2 e^{2} \left (x +\frac {d}{e}\right )+2 d e \right )}{15 e^{2} d^{4} \sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{7 d^{2}}\right )}{9 d^{2}}}{d^{2}}\right )}{11 d}}{e}\) | \(341\) |
Input:
int(1/(e*x+d)/(-e^2*x^2+d^2)^(11/2),x,method=_RETURNVERBOSE)
Output:
-1/693*(-e*x+d)*(-256*e^10*x^10-256*d*e^9*x^9+1152*d^2*e^8*x^8+1152*d^3*e^ 7*x^7-2016*d^4*e^6*x^6-2016*d^5*e^5*x^5+1680*d^6*e^4*x^4+1680*d^7*e^3*x^3- 630*d^8*e^2*x^2-630*d^9*e*x+63*d^10)/d^11/e/(-e^2*x^2+d^2)^(11/2)
Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (130) = 260\).
Time = 0.74 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.39 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {63 \, e^{11} x^{11} + 63 \, d e^{10} x^{10} - 315 \, d^{2} e^{9} x^{9} - 315 \, d^{3} e^{8} x^{8} + 630 \, d^{4} e^{7} x^{7} + 630 \, d^{5} e^{6} x^{6} - 630 \, d^{6} e^{5} x^{5} - 630 \, d^{7} e^{4} x^{4} + 315 \, d^{8} e^{3} x^{3} + 315 \, d^{9} e^{2} x^{2} - 63 \, d^{10} e x - 63 \, d^{11} + {\left (256 \, e^{10} x^{10} + 256 \, d e^{9} x^{9} - 1152 \, d^{2} e^{8} x^{8} - 1152 \, d^{3} e^{7} x^{7} + 2016 \, d^{4} e^{6} x^{6} + 2016 \, d^{5} e^{5} x^{5} - 1680 \, d^{6} e^{4} x^{4} - 1680 \, d^{7} e^{3} x^{3} + 630 \, d^{8} e^{2} x^{2} + 630 \, d^{9} e x - 63 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{693 \, {\left (d^{11} e^{12} x^{11} + d^{12} e^{11} x^{10} - 5 \, d^{13} e^{10} x^{9} - 5 \, d^{14} e^{9} x^{8} + 10 \, d^{15} e^{8} x^{7} + 10 \, d^{16} e^{7} x^{6} - 10 \, d^{17} e^{6} x^{5} - 10 \, d^{18} e^{5} x^{4} + 5 \, d^{19} e^{4} x^{3} + 5 \, d^{20} e^{3} x^{2} - d^{21} e^{2} x - d^{22} e\right )}} \] Input:
integrate(1/(e*x+d)/(-e^2*x^2+d^2)^(11/2),x, algorithm="fricas")
Output:
-1/693*(63*e^11*x^11 + 63*d*e^10*x^10 - 315*d^2*e^9*x^9 - 315*d^3*e^8*x^8 + 630*d^4*e^7*x^7 + 630*d^5*e^6*x^6 - 630*d^6*e^5*x^5 - 630*d^7*e^4*x^4 + 315*d^8*e^3*x^3 + 315*d^9*e^2*x^2 - 63*d^10*e*x - 63*d^11 + (256*e^10*x^10 + 256*d*e^9*x^9 - 1152*d^2*e^8*x^8 - 1152*d^3*e^7*x^7 + 2016*d^4*e^6*x^6 + 2016*d^5*e^5*x^5 - 1680*d^6*e^4*x^4 - 1680*d^7*e^3*x^3 + 630*d^8*e^2*x^2 + 630*d^9*e*x - 63*d^10)*sqrt(-e^2*x^2 + d^2))/(d^11*e^12*x^11 + d^12*e^1 1*x^10 - 5*d^13*e^10*x^9 - 5*d^14*e^9*x^8 + 10*d^15*e^8*x^7 + 10*d^16*e^7* x^6 - 10*d^17*e^6*x^5 - 10*d^18*e^5*x^4 + 5*d^19*e^4*x^3 + 5*d^20*e^3*x^2 - d^21*e^2*x - d^22*e)
\[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int \frac {1}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {11}{2}} \left (d + e x\right )}\, dx \] Input:
integrate(1/(e*x+d)/(-e**2*x**2+d**2)**(11/2),x)
Output:
Integral(1/((-(-d + e*x)*(d + e*x))**(11/2)*(d + e*x)), x)
Time = 0.04 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=-\frac {1}{11 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d e^{2} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2} e\right )}} + \frac {10 \, x}{99 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{3}} + \frac {80 \, x}{693 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5}} + \frac {32 \, x}{231 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7}} + \frac {128 \, x}{693 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9}} + \frac {256 \, x}{693 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{11}} \] Input:
integrate(1/(e*x+d)/(-e^2*x^2+d^2)^(11/2),x, algorithm="maxima")
Output:
-1/11/((-e^2*x^2 + d^2)^(9/2)*d*e^2*x + (-e^2*x^2 + d^2)^(9/2)*d^2*e) + 10 /99*x/((-e^2*x^2 + d^2)^(9/2)*d^3) + 80/693*x/((-e^2*x^2 + d^2)^(7/2)*d^5) + 32/231*x/((-e^2*x^2 + d^2)^(5/2)*d^7) + 128/693*x/((-e^2*x^2 + d^2)^(3/ 2)*d^9) + 256/693*x/(sqrt(-e^2*x^2 + d^2)*d^11)
\[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\int { \frac {1}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {11}{2}} {\left (e x + d\right )}} \,d x } \] Input:
integrate(1/(e*x+d)/(-e^2*x^2+d^2)^(11/2),x, algorithm="giac")
Output:
integrate(1/((-e^2*x^2 + d^2)^(11/2)*(e*x + d)), x)
Time = 7.01 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.53 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {19\,x}{99\,d^3}-\frac {3}{22\,d^2\,e}\right )}{{\left (d+e\,x\right )}^5\,{\left (d-e\,x\right )}^5}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {32\,x}{231\,d^7}-\frac {1}{352\,d^6\,e}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}+\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1091\,x}{11088\,d^5}+\frac {9}{176\,d^4\,e}\right )}{{\left (d+e\,x\right )}^4\,{\left (d-e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{352\,d^6\,e\,{\left (d+e\,x\right )}^6}+\frac {128\,x\,\sqrt {d^2-e^2\,x^2}}{693\,d^9\,{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}+\frac {256\,x\,\sqrt {d^2-e^2\,x^2}}{693\,d^{11}\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \] Input:
int(1/((d^2 - e^2*x^2)^(11/2)*(d + e*x)),x)
Output:
((d^2 - e^2*x^2)^(1/2)*((19*x)/(99*d^3) - 3/(22*d^2*e)))/((d + e*x)^5*(d - e*x)^5) + ((d^2 - e^2*x^2)^(1/2)*((32*x)/(231*d^7) - 1/(352*d^6*e)))/((d + e*x)^3*(d - e*x)^3) + ((d^2 - e^2*x^2)^(1/2)*((1091*x)/(11088*d^5) + 9/( 176*d^4*e)))/((d + e*x)^4*(d - e*x)^4) - (d^2 - e^2*x^2)^(1/2)/(352*d^6*e* (d + e*x)^6) + (128*x*(d^2 - e^2*x^2)^(1/2))/(693*d^9*(d + e*x)^2*(d - e*x )^2) + (256*x*(d^2 - e^2*x^2)^(1/2))/(693*d^11*(d + e*x)*(d - e*x))
Time = 0.24 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.91 \[ \int \frac {1}{(d+e x) \left (d^2-e^2 x^2\right )^{11/2}} \, dx=\frac {630 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{9}+630 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} e x -2520 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} e^{2} x^{2}-2520 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{6} e^{3} x^{3}+3780 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} e^{4} x^{4}+3780 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e^{5} x^{5}-2520 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{6} x^{6}-2520 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{7} x^{7}+630 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{8} x^{8}+630 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{9} x^{9}-63 d^{10}+630 d^{9} e x +630 d^{8} e^{2} x^{2}-1680 d^{7} e^{3} x^{3}-1680 d^{6} e^{4} x^{4}+2016 d^{5} e^{5} x^{5}+2016 d^{4} e^{6} x^{6}-1152 d^{3} e^{7} x^{7}-1152 d^{2} e^{8} x^{8}+256 d \,e^{9} x^{9}+256 e^{10} x^{10}}{693 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{11} e \left (e^{9} x^{9}+d \,e^{8} x^{8}-4 d^{2} e^{7} x^{7}-4 d^{3} e^{6} x^{6}+6 d^{4} e^{5} x^{5}+6 d^{5} e^{4} x^{4}-4 d^{6} e^{3} x^{3}-4 d^{7} e^{2} x^{2}+d^{8} e x +d^{9}\right )} \] Input:
int(1/(e*x+d)/(-e^2*x^2+d^2)^(11/2),x)
Output:
(630*sqrt(d**2 - e**2*x**2)*d**9 + 630*sqrt(d**2 - e**2*x**2)*d**8*e*x - 2 520*sqrt(d**2 - e**2*x**2)*d**7*e**2*x**2 - 2520*sqrt(d**2 - e**2*x**2)*d* *6*e**3*x**3 + 3780*sqrt(d**2 - e**2*x**2)*d**5*e**4*x**4 + 3780*sqrt(d**2 - e**2*x**2)*d**4*e**5*x**5 - 2520*sqrt(d**2 - e**2*x**2)*d**3*e**6*x**6 - 2520*sqrt(d**2 - e**2*x**2)*d**2*e**7*x**7 + 630*sqrt(d**2 - e**2*x**2)* d*e**8*x**8 + 630*sqrt(d**2 - e**2*x**2)*e**9*x**9 - 63*d**10 + 630*d**9*e *x + 630*d**8*e**2*x**2 - 1680*d**7*e**3*x**3 - 1680*d**6*e**4*x**4 + 2016 *d**5*e**5*x**5 + 2016*d**4*e**6*x**6 - 1152*d**3*e**7*x**7 - 1152*d**2*e* *8*x**8 + 256*d*e**9*x**9 + 256*e**10*x**10)/(693*sqrt(d**2 - e**2*x**2)*d **11*e*(d**9 + d**8*e*x - 4*d**7*e**2*x**2 - 4*d**6*e**3*x**3 + 6*d**5*e** 4*x**4 + 6*d**4*e**5*x**5 - 4*d**3*e**6*x**6 - 4*d**2*e**7*x**7 + d*e**8*x **8 + e**9*x**9))