\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{7/2}}{(d+e x)^{11}} \, dx\) [246]

Optimal result

Integrand size = 37, antiderivative size = 171 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{13 \left (c d^2-a e^2\right ) (d+e x)^{11}}+\frac {8 c d \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{143 \left (c d^2-a e^2\right )^2 (d+e x)^{10}}+\frac {16 c^2 d^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{9/2}}{1287 \left (c d^2-a e^2\right )^3 (d+e x)^9} \] Output:

2/13*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)/(e*x+d)^11+8/1 
43*c*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)^2/(e*x+d)^10 
+16/1287*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(9/2)/(-a*e^2+c*d^2)^3/ 
(e*x+d)^9
 

Mathematica [A] (verified)

Time = 1.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.61 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {2 (a e+c d x)^4 \sqrt {(a e+c d x) (d+e x)} \left (99 a^2 e^4-18 a c d e^2 (13 d+2 e x)+c^2 d^2 \left (143 d^2+52 d e x+8 e^2 x^2\right )\right )}{1287 \left (c d^2-a e^2\right )^3 (d+e x)^7} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^11,x]
 

Output:

(2*(a*e + c*d*x)^4*Sqrt[(a*e + c*d*x)*(d + e*x)]*(99*a^2*e^4 - 18*a*c*d*e^ 
2*(13*d + 2*e*x) + c^2*d^2*(143*d^2 + 52*d*e*x + 8*e^2*x^2)))/(1287*(c*d^2 
 - a*e^2)^3*(d + e*x)^7)
 

Rubi [A] (verified)

Time = 0.53 (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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {4 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}{(d+e x)^{10}}dx}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{13 (d+e x)^{11} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {4 c d \left (\frac {2 c d \int \frac {\left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^{7/2}}{(d+e x)^9}dx}{11 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{11 (d+e x)^{10} \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{13 (d+e x)^{11} \left (c d^2-a e^2\right )}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{13 (d+e x)^{11} \left (c d^2-a e^2\right )}+\frac {4 c d \left (\frac {4 c d \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{99 (d+e x)^9 \left (c d^2-a e^2\right )^2}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{9/2}}{11 (d+e x)^{10} \left (c d^2-a e^2\right )}\right )}{13 \left (c d^2-a e^2\right )}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(7/2)/(d + e*x)^11,x]
 

Output:

(2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(9/2))/(13*(c*d^2 - a*e^2)*(d + 
 e*x)^11) + (4*c*d*((2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(9/2))/(11* 
(c*d^2 - a*e^2)*(d + e*x)^10) + (4*c*d*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e* 
x^2)^(9/2))/(99*(c*d^2 - a*e^2)^2*(d + e*x)^9)))/(13*(c*d^2 - a*e^2))
 

Defintions of rubi rules used

rule 1123

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]
 

rule 1129

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]
 

Maple [A] (verified)

Time = 13.00 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.85

method	result	size
gosper	\(-\frac {2 \left (c d x +a e \right ) \left (8 x^{2} c^{2} d^{2} e^{2}-36 x a c d \,e^{3}+52 x \,c^{2} d^{3} e +99 a^{2} e^{4}-234 a c \,d^{2} e^{2}+143 c^{2} d^{4}\right ) \left (c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e \right )^{\frac {7}{2}}}{1287 \left (e x +d \right )^{10} \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right )}\)	\(146\)
orering	\(-\frac {2 \left (8 x^{2} c^{2} d^{2} e^{2}-36 x a c d \,e^{3}+52 x \,c^{2} d^{3} e +99 a^{2} e^{4}-234 a c \,d^{2} e^{2}+143 c^{2} d^{4}\right ) \left (c d x +a e \right ) {\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {7}{2}}}{1287 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{10}}\)	\(147\)
default	\(\frac {-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{13 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{11}}-\frac {4 d e c \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 \left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{10}}+\frac {4 d e c \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 \left (a \,e^{2}-c \,d^{2}\right )^{2} \left (x +\frac {d}{e}\right )^{9}}\right )}{13 \left (a \,e^{2}-c \,d^{2}\right )}}{e^{11}}\)	\(212\)
trager	\(-\frac {2 \left (8 c^{6} d^{6} e^{2} x^{6}-4 a \,c^{5} d^{5} e^{3} x^{5}+52 c^{6} d^{7} e \,x^{5}+3 a^{2} c^{4} d^{4} e^{4} x^{4}-26 a \,c^{5} d^{6} e^{2} x^{4}+143 c^{6} d^{8} x^{4}+212 a^{3} c^{3} d^{3} e^{5} x^{3}-624 a^{2} c^{4} d^{5} e^{3} x^{3}+572 a \,c^{5} d^{7} e \,x^{3}+458 a^{4} c^{2} d^{2} e^{6} x^{2}-1196 a^{3} c^{3} d^{4} e^{4} x^{2}+858 a^{2} c^{4} d^{6} e^{2} x^{2}+360 a^{5} c d \,e^{7} x -884 a^{4} c^{2} d^{3} e^{5} x +572 a^{3} c^{3} d^{5} e^{3} x +99 a^{6} e^{8}-234 a^{5} c \,d^{2} e^{6}+143 a^{4} c^{2} d^{4} e^{4}\right ) \sqrt {c d \,x^{2} e +a \,e^{2} x +c \,d^{2} x +a d e}}{1287 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} a \,c^{2}-d^{6} c^{3}\right ) \left (e x +d \right )^{7}}\)	\(336\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(7/2)/(e*x+d)^11,x,method=_RETURNVER 
BOSE)
 

Output:

-2/1287*(c*d*x+a*e)*(8*c^2*d^2*e^2*x^2-36*a*c*d*e^3*x+52*c^2*d^3*e*x+99*a^ 
2*e^4-234*a*c*d^2*e^2+143*c^2*d^4)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(7/2) 
/(e*x+d)^10/(a^3*e^6-3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2-c^3*d^6)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 670 vs. \(2 (159) = 318\).

Time = 49.49 (sec) , antiderivative size = 670, normalized size of antiderivative = 3.92 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\frac {2 \, {\left (8 \, c^{6} d^{6} e^{2} x^{6} + 143 \, a^{4} c^{2} d^{4} e^{4} - 234 \, a^{5} c d^{2} e^{6} + 99 \, a^{6} e^{8} + 4 \, {\left (13 \, c^{6} d^{7} e - a c^{5} d^{5} e^{3}\right )} x^{5} + {\left (143 \, c^{6} d^{8} - 26 \, a c^{5} d^{6} e^{2} + 3 \, a^{2} c^{4} d^{4} e^{4}\right )} x^{4} + 4 \, {\left (143 \, a c^{5} d^{7} e - 156 \, a^{2} c^{4} d^{5} e^{3} + 53 \, a^{3} c^{3} d^{3} e^{5}\right )} x^{3} + 2 \, {\left (429 \, a^{2} c^{4} d^{6} e^{2} - 598 \, a^{3} c^{3} d^{4} e^{4} + 229 \, a^{4} c^{2} d^{2} e^{6}\right )} x^{2} + 4 \, {\left (143 \, a^{3} c^{3} d^{5} e^{3} - 221 \, a^{4} c^{2} d^{3} e^{5} + 90 \, a^{5} c d e^{7}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{1287 \, {\left (c^{3} d^{13} - 3 \, a c^{2} d^{11} e^{2} + 3 \, a^{2} c d^{9} e^{4} - a^{3} d^{7} e^{6} + {\left (c^{3} d^{6} e^{7} - 3 \, a c^{2} d^{4} e^{9} + 3 \, a^{2} c d^{2} e^{11} - a^{3} e^{13}\right )} x^{7} + 7 \, {\left (c^{3} d^{7} e^{6} - 3 \, a c^{2} d^{5} e^{8} + 3 \, a^{2} c d^{3} e^{10} - a^{3} d e^{12}\right )} x^{6} + 21 \, {\left (c^{3} d^{8} e^{5} - 3 \, a c^{2} d^{6} e^{7} + 3 \, a^{2} c d^{4} e^{9} - a^{3} d^{2} e^{11}\right )} x^{5} + 35 \, {\left (c^{3} d^{9} e^{4} - 3 \, a c^{2} d^{7} e^{6} + 3 \, a^{2} c d^{5} e^{8} - a^{3} d^{3} e^{10}\right )} x^{4} + 35 \, {\left (c^{3} d^{10} e^{3} - 3 \, a c^{2} d^{8} e^{5} + 3 \, a^{2} c d^{6} e^{7} - a^{3} d^{4} e^{9}\right )} x^{3} + 21 \, {\left (c^{3} d^{11} e^{2} - 3 \, a c^{2} d^{9} e^{4} + 3 \, a^{2} c d^{7} e^{6} - a^{3} d^{5} e^{8}\right )} x^{2} + 7 \, {\left (c^{3} d^{12} e - 3 \, a c^{2} d^{10} e^{3} + 3 \, a^{2} c d^{8} e^{5} - a^{3} d^{6} e^{7}\right )} x\right )}} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^11,x, algorithm= 
"fricas")
 

Output:

2/1287*(8*c^6*d^6*e^2*x^6 + 143*a^4*c^2*d^4*e^4 - 234*a^5*c*d^2*e^6 + 99*a 
^6*e^8 + 4*(13*c^6*d^7*e - a*c^5*d^5*e^3)*x^5 + (143*c^6*d^8 - 26*a*c^5*d^ 
6*e^2 + 3*a^2*c^4*d^4*e^4)*x^4 + 4*(143*a*c^5*d^7*e - 156*a^2*c^4*d^5*e^3 
+ 53*a^3*c^3*d^3*e^5)*x^3 + 2*(429*a^2*c^4*d^6*e^2 - 598*a^3*c^3*d^4*e^4 + 
 229*a^4*c^2*d^2*e^6)*x^2 + 4*(143*a^3*c^3*d^5*e^3 - 221*a^4*c^2*d^3*e^5 + 
 90*a^5*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/(c^3*d^13 
- 3*a*c^2*d^11*e^2 + 3*a^2*c*d^9*e^4 - a^3*d^7*e^6 + (c^3*d^6*e^7 - 3*a*c^ 
2*d^4*e^9 + 3*a^2*c*d^2*e^11 - a^3*e^13)*x^7 + 7*(c^3*d^7*e^6 - 3*a*c^2*d^ 
5*e^8 + 3*a^2*c*d^3*e^10 - a^3*d*e^12)*x^6 + 21*(c^3*d^8*e^5 - 3*a*c^2*d^6 
*e^7 + 3*a^2*c*d^4*e^9 - a^3*d^2*e^11)*x^5 + 35*(c^3*d^9*e^4 - 3*a*c^2*d^7 
*e^6 + 3*a^2*c*d^5*e^8 - a^3*d^3*e^10)*x^4 + 35*(c^3*d^10*e^3 - 3*a*c^2*d^ 
8*e^5 + 3*a^2*c*d^6*e^7 - a^3*d^4*e^9)*x^3 + 21*(c^3*d^11*e^2 - 3*a*c^2*d^ 
9*e^4 + 3*a^2*c*d^7*e^6 - a^3*d^5*e^8)*x^2 + 7*(c^3*d^12*e - 3*a*c^2*d^10* 
e^3 + 3*a^2*c*d^8*e^5 - a^3*d^6*e^7)*x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(7/2)/(e*x+d)**11,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^11,x, algorithm= 
"maxima")
 

Output:

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
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^11,x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{%%%{1,[0,0,7]%%%},[14]%%%}+%%%{%%{[%%%{-14,[0,1,6]%%%},0]: 
[1,0,%%%{
 

Mupad [B] (verification not implemented)

Time = 14.71 (sec) , antiderivative size = 7337, normalized size of antiderivative = 42.91 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx=\text {Too large to display} \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(7/2)/(d + e*x)^11,x)
 

Output:

(((d*((8*c^6*d^7)/(143*e^2*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^2*e)) - (4*c 
^5*d^5*(23*a*e^2 - 19*c*d^2))/(143*e^2*(a*e^2 - c*d^2)^2*(5*a*e^3 - 5*c*d^ 
2*e))))/e + (4*c^4*d^4*(127*a^2*e^4 - 314*c^2*d^4 + 229*a*c*d^2*e^2))/(300 
3*e^3*(a*e^2 - c*d^2)^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 + (((d*((16*c^7*d^8)/(1287*e^2*(a*e^2 - c 
*d^2)^3*(3*a*e^3 - 3*c*d^2*e)) - (8*c^6*d^6*(35*a*e^2 - 31*c*d^2))/(1287*e 
^2*(a*e^2 - c*d^2)^3*(3*a*e^3 - 3*c*d^2*e))))/e - (8*c^5*d^5*(641*a^2*e^4 
+ 872*c^2*d^4 - 1527*a*c*d^2*e^2))/(9009*e^3*(a*e^2 - c*d^2)^3*(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 - 
(((26*c^5*d^6 - 70*a*c^4*d^4*e^2)/(143*e^3*(a*e^2 - c*d^2)*(5*a*e^3 - 5*c* 
d^2*e)) + (4*c^5*d^6)/(13*e^3*(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 - (((6788*c^6*d^7 - 
 7628*a*c^5*d^5*e^2)/(15015*e^3*(a*e^2 - c*d^2)^2*(3*a*e^3 - 3*c*d^2*e)) + 
 (8*c^6*d^7)/(143*e^3*(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 - (((d*((d*((d*((128*c^10 
*d^11)/(135135*e*(a*e^2 - c*d^2)^7) - (128*c^9*d^9*(31*a*e^2 - 27*c*d^2))/ 
(135135*e*(a*e^2 - c*d^2)^7)))/e + (128*c^8*d^8*(409*a^2*e^4 + 322*c^2*d^4 
 - 725*a*c*d^2*e^2))/(135135*e^2*(a*e^2 - c*d^2)^7)))/e - (128*c^7*d^7*(30 
11*a^3*e^6 - 2282*c^3*d^6 + 7490*a*c^2*d^4*e^2 - 8215*a^2*c*d^2*e^4))/(135 
135*e^3*(a*e^2 - c*d^2)^7)))/e + (128*a*c^6*d^6*(2632*a^3*e^6 - 2282*c^...
 

Reduce [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 1125, normalized size of antiderivative = 6.58 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{7/2}}{(d+e x)^{11}} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(7/2)/(e*x+d)^11,x)
 

Output:

(2*( - 99*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**6*e**13 + 234*sqrt(d + e*x)*s 
qrt(a*e + c*d*x)*a**5*c*d**2*e**11 - 360*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a 
**5*c*d*e**12*x - 143*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**4*e**9 
+ 884*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**3*e**10*x - 458*sqrt(d 
+ e*x)*sqrt(a*e + c*d*x)*a**4*c**2*d**2*e**11*x**2 - 572*sqrt(d + e*x)*sqr 
t(a*e + c*d*x)*a**3*c**3*d**5*e**8*x + 1196*sqrt(d + e*x)*sqrt(a*e + c*d*x 
)*a**3*c**3*d**4*e**9*x**2 - 212*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**3*c**3 
*d**3*e**10*x**3 - 858*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**6*e**7 
*x**2 + 624*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**5*e**8*x**3 - 3*s 
qrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*c**4*d**4*e**9*x**4 - 572*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*a*c**5*d**7*e**6*x**3 + 26*sqrt(d + e*x)*sqrt(a*e + c* 
d*x)*a*c**5*d**6*e**7*x**4 + 4*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c**5*d**5 
*e**8*x**5 - 143*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**8*e**5*x**4 - 52* 
sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**6*d**7*e**6*x**5 - 8*sqrt(d + e*x)*sqrt 
(a*e + c*d*x)*c**6*d**6*e**7*x**6 + 8*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**13 + 
 56*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**12*e*x + 168*sqrt(e)*sqrt(d)*sqrt(c)*c 
**6*d**11*e**2*x**2 + 280*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**10*e**3*x**3 + 2 
80*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**9*e**4*x**4 + 168*sqrt(e)*sqrt(d)*sqrt( 
c)*c**6*d**8*e**5*x**5 + 56*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**7*e**6*x**6 + 
8*sqrt(e)*sqrt(d)*sqrt(c)*c**6*d**6*e**7*x**7))/(1287*e**5*(a**3*d**7*e...