Integrand size = 37, antiderivative size = 171 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{7 \left (c d^2-a e^2\right ) (d+e x)^5}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{35 \left (c d^2-a e^2\right )^2 (d+e x)^4}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 \left (c d^2-a e^2\right )^3 (d+e x)^3} \]
2/7*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)/(e*x+d)^5+8/35* c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^2/(e*x+d)^4+16/ 105*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/(-a*e^2+c*d^2)^3/(e*x+ d)^3
Time = 1.38 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \left (15 a^3 e^5+3 a^2 c d e^3 (-14 d+e x)+a c^2 d^2 e \left (35 d^2-14 d e x-4 e^2 x^2\right )+c^3 d^3 x \left (35 d^2+28 d e x+8 e^2 x^2\right )\right )}{105 \left (c d^2-a e^2\right )^3 (d+e x)^4} \]
(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(15*a^3*e^5 + 3*a^2*c*d*e^3*(-14*d + e*x) + a*c^2*d^2*e*(35*d^2 - 14*d*e*x - 4*e^2*x^2) + c^3*d^3*x*(35*d^2 + 28*d* e*x + 8*e^2*x^2)))/(105*(c*d^2 - a*e^2)^3*(d + e*x)^4)
Time = 0.33 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {1129, 1129, 1123}
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.
| \(\displaystyle \int \frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^5} \, dx\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {4 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{(d+e x)^4}dx}{7 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {4 c d \left (\frac {2 c d \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{(d+e x)^3}dx}{5 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )}\right )}{7 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 (d+e x)^5 \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 (d+e x)^3 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 (d+e x)^4 \left (c d^2-a e^2\right )}\right )}{7 \left (c d^2-a e^2\right )}\) |
(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(7*(c*d^2 - a*e^2)*(d + e*x)^5) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*(c* d^2 - a*e^2)*(d + e*x)^4) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2) ^(3/2))/(15*(c*d^2 - a*e^2)^2*(d + e*x)^3)))/(7*(c*d^2 - a*e^2))
3.20.16.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Time = 3.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.85
| method | result | size |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (8 x^{2} c^{2} d^{2} e^{2}-12 x a c d \,e^{3}+28 x \,c^{2} d^{3} e +15 a^{2} e^{4}-42 a c \,d^{2} e^{2}+35 c^{2} d^{4}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (e x +d \right )^{4} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}\) | \(146\) |
| trager | \(-\frac {2 \left (8 e^{2} c^{3} d^{3} x^{3}-4 e^{3} a \,c^{2} d^{2} x^{2}+28 d^{4} e \,c^{3} x^{2}+3 d \,e^{4} a^{2} c x -14 d^{3} e^{2} c^{2} a x +35 d^{5} c^{3} x +15 a^{3} e^{5}-42 d^{2} e^{3} a^{2} c +35 d^{4} a \,c^{2} e \right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{105 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right ) \left (e x +d \right )^{4}}\) | \(183\) |
| default | \(\frac {-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{7 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{5}}-\frac {4 c d e \left (-\frac {2 \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{5 \left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{4}}+\frac {4 c d e \left (c d e \left (x +\frac {d}{e}\right )^{2}+\left (e^{2} a -c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{15 \left (e^{2} a -c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{3}}\right )}{7 \left (e^{2} a -c \,d^{2}\right )}}{e^{5}}\) | \(212\) |
-2/105*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-12*a*c*d*e^3*x+28*c^2*d^3*e*x+15*a^2 *e^4-42*a*c*d^2*e^2+35*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)/(e *x+d)^4/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (159) = 318\).
Time = 2.51 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.18 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {2 \, {\left (8 \, c^{3} d^{3} e^{2} x^{3} + 35 \, a c^{2} d^{4} e - 42 \, a^{2} c d^{2} e^{3} + 15 \, a^{3} e^{5} + 4 \, {\left (7 \, c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x^{2} + {\left (35 \, c^{3} d^{5} - 14 \, a c^{2} d^{3} e^{2} + 3 \, a^{2} c d e^{4}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{105 \, {\left (c^{3} d^{10} - 3 \, a c^{2} d^{8} e^{2} + 3 \, a^{2} c d^{6} e^{4} - a^{3} d^{4} e^{6} + {\left (c^{3} d^{6} e^{4} - 3 \, a c^{2} d^{4} e^{6} + 3 \, a^{2} c d^{2} e^{8} - a^{3} e^{10}\right )} x^{4} + 4 \, {\left (c^{3} d^{7} e^{3} - 3 \, a c^{2} d^{5} e^{5} + 3 \, a^{2} c d^{3} e^{7} - a^{3} d e^{9}\right )} x^{3} + 6 \, {\left (c^{3} d^{8} e^{2} - 3 \, a c^{2} d^{6} e^{4} + 3 \, a^{2} c d^{4} e^{6} - a^{3} d^{2} e^{8}\right )} x^{2} + 4 \, {\left (c^{3} d^{9} e - 3 \, a c^{2} d^{7} e^{3} + 3 \, a^{2} c d^{5} e^{5} - a^{3} d^{3} e^{7}\right )} x\right )}} \]
2/105*(8*c^3*d^3*e^2*x^3 + 35*a*c^2*d^4*e - 42*a^2*c*d^2*e^3 + 15*a^3*e^5 + 4*(7*c^3*d^4*e - a*c^2*d^2*e^3)*x^2 + (35*c^3*d^5 - 14*a*c^2*d^3*e^2 + 3 *a^2*c*d*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^10 - 3 *a*c^2*d^8*e^2 + 3*a^2*c*d^6*e^4 - a^3*d^4*e^6 + (c^3*d^6*e^4 - 3*a*c^2*d^ 4*e^6 + 3*a^2*c*d^2*e^8 - a^3*e^10)*x^4 + 4*(c^3*d^7*e^3 - 3*a*c^2*d^5*e^5 + 3*a^2*c*d^3*e^7 - a^3*d*e^9)*x^3 + 6*(c^3*d^8*e^2 - 3*a*c^2*d^6*e^4 + 3 *a^2*c*d^4*e^6 - a^3*d^2*e^8)*x^2 + 4*(c^3*d^9*e - 3*a*c^2*d^7*e^3 + 3*a^2 *c*d^5*e^5 - a^3*d^3*e^7)*x)
Timed out. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*(a*e^2-c*d^2)>0)', see `assume ?` for mor
Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (159) = 318\).
Time = 0.33 (sec) , antiderivative size = 711, normalized size of antiderivative = 4.16 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=-\frac {2}{105} \, {\left (\frac {8 \, \sqrt {c d e} c^{3} d^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{3} d^{6} e^{2} {\left | e \right |} - 3 \, a c^{2} d^{4} e^{4} {\left | e \right |} + 3 \, a^{2} c d^{2} e^{6} {\left | e \right |} - a^{3} e^{8} {\left | e \right |}} + \frac {\frac {3 \, {\left (35 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{3} d^{3} e^{3} - 35 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} + 21 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}} c d e - 5 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {7}{2}}\right )} c d^{2} e \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}} - \frac {3 \, {\left (35 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{3} d^{3} e^{3} - 35 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c^{2} d^{2} e^{2} + 21 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}} c d e - 5 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {7}{2}}\right )} a e^{3} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{3} d^{6} e^{6} - 3 \, a c^{2} d^{4} e^{8} + 3 \, a^{2} c d^{2} e^{10} - a^{3} e^{12}} - \frac {7 \, {\left (15 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c^{2} d^{2} e^{2} - 10 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {3}{2}} c d e + 3 \, {\left (c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}\right )}^{\frac {5}{2}}\right )} c d \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{c^{2} d^{4} e^{4} - 2 \, a c d^{2} e^{6} + a^{2} e^{8}}}{c d^{2} {\left | e \right |} - a e^{2} {\left | e \right |}}\right )} {\left | e \right |} \]
-2/105*(8*sqrt(c*d*e)*c^3*d^3*sgn(1/(e*x + d))*sgn(e)/(c^3*d^6*e^2*abs(e) - 3*a*c^2*d^4*e^4*abs(e) + 3*a^2*c*d^2*e^6*abs(e) - a^3*e^8*abs(e)) + (3*( 35*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^3*d^3*e^3 - 35*(c*d *e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c^2*d^2*e^2 + 21*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*c*d*e - 5*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(7/2))*c*d^2*e*sgn(1/(e*x + d))*sgn(e)/(c^3*d^6*e ^6 - 3*a*c^2*d^4*e^8 + 3*a^2*c*d^2*e^10 - a^3*e^12) - 3*(35*sqrt(c*d*e - c *d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^3*d^3*e^3 - 35*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c^2*d^2*e^2 + 21*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)*c*d*e - 5*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e* x + d))^(7/2))*a*e^3*sgn(1/(e*x + d))*sgn(e)/(c^3*d^6*e^6 - 3*a*c^2*d^4*e^ 8 + 3*a^2*c*d^2*e^10 - a^3*e^12) - 7*(15*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*c^2*d^2*e^2 - 10*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(3/2)*c*d*e + 3*(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))^(5/2)) *c*d*sgn(1/(e*x + d))*sgn(e)/(c^2*d^4*e^4 - 2*a*c*d^2*e^6 + a^2*e^8))/(c*d ^2*abs(e) - a*e^2*abs(e)))*abs(e)
Time = 11.31 (sec) , antiderivative size = 877, normalized size of antiderivative = 5.13 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^5} \, dx=\frac {\left (\frac {4\,c^2\,d^3}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {4\,a\,c\,d\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}-\frac {\left (\frac {2\,a\,e^2}{7\,a\,e^3-7\,c\,d^2\,e}-\frac {2\,c\,d^2}{7\,a\,e^3-7\,c\,d^2\,e}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^4}+\frac {\left (\frac {4\,c^3\,d^4+4\,a\,c^2\,d^2\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {8\,c^3\,d^4}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {8\,c^4\,d^5+8\,a\,c^3\,d^3\,e^2}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,c^4\,d^5}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {\left (\frac {2\,c^2\,d^3+2\,a\,c\,d\,e^2}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}-\frac {4\,c^2\,d^3}{7\,\left (a\,e^2-c\,d^2\right )\,\left (5\,a\,e^3-5\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^3}+\frac {\left (\frac {8\,c^3\,d^4}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}-\frac {8\,a\,c^2\,d^2\,e^2}{35\,{\left (a\,e^2-c\,d^2\right )}^2\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{{\left (d+e\,x\right )}^2}+\frac {\left (\frac {16\,c^4\,d^5}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^4}-\frac {16\,a\,c^3\,d^3\,e}{105\,{\left (a\,e^2-c\,d^2\right )}^4}\right )\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{d+e\,x}+\frac {12\,c^2\,d^2\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{35\,\left (a\,e^2-c\,d^2\right )\,\left (3\,a\,e^3-3\,c\,d^2\,e\right )\,{\left (d+e\,x\right )}^2}-\frac {8\,c^3\,d^3\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{105\,e\,{\left (a\,e^2-c\,d^2\right )}^3\,\left (d+e\,x\right )} \]
(((4*c^2*d^3)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*a*c*d*e^2)/(7 *(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d* e*x^2)^(1/2))/(d + e*x)^3 - (((2*a*e^2)/(7*a*e^3 - 7*c*d^2*e) - (2*c*d^2)/ (7*a*e^3 - 7*c*d^2*e))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^4 + (((4*c^3*d^4 + 4*a*c^2*d^2*e^2)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) - (8*c^3*d^4)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)))*( x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((8*c^4*d^5 + 8*a*c^3*d^3*e^2)/(105*e*(a*e^2 - c*d^2)^4) - (16*c^4*d^5)/(105*e*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + ((( 2*c^2*d^3 + 2*a*c*d*e^2)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)) - (4*c^ 2*d^3)/(7*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d *e + c*d*e*x^2)^(1/2))/(d + e*x)^3 + (((8*c^3*d^4)/(35*(a*e^2 - c*d^2)^2*( 3*a*e^3 - 3*c*d^2*e)) - (8*a*c^2*d^2*e^2)/(35*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2 + (((16*c^4*d^5)/(105*e*(a*e^2 - c*d^2)^4) - (16*a*c^3*d^3*e)/(105*(a*e^2 - c*d^2)^4))*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x) + (12 *c^2*d^2*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(35*(a*e^2 - c*d^2 )*(3*a*e^3 - 3*c*d^2*e)*(d + e*x)^2) - (8*c^3*d^3*(x*(a*e^2 + c*d^2) + a*d *e + c*d*e*x^2)^(1/2))/(105*e*(a*e^2 - c*d^2)^3*(d + e*x))