\(\int \frac {(a d e+(c d^2+a e^2) x+c d e x^2)^{3/2}}{x^2 (d+e x)^3} \, dx\) [44]

Optimal result

Integrand size = 40, antiderivative size = 176 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\frac {3 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d+e x}-\frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{d x (d+e x)^2}-\frac {3 \sqrt {a} \sqrt {e} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {a} \sqrt {e} (d+e x)}\right )}{d^{5/2}} \] Output:

3*(c-a*e^2/d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)-(a*d*e+(a* 
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/x/(e*x+d)^2-3*a^(1/2)*e^(1/2)*(-a*e^2+c*d^ 
2)*arctanh(d^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/a^(1/2)/e^(1/2) 
/(e*x+d))/d^(5/2)
 

Mathematica [A] (verified)

Time = 10.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\frac {-\sqrt {d} (a e+c d x) \left (-2 c d^2 x+a e (d+3 e x)\right )+3 \sqrt {a} \sqrt {e} \left (-c d^2+a e^2\right ) x \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{d^{5/2} x \sqrt {(a e+c d x) (d+e x)}} \] Input:

Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^2*(d + e*x)^3), 
x]
 

Output:

(-(Sqrt[d]*(a*e + c*d*x)*(-2*c*d^2*x + a*e*(d + 3*e*x))) + 3*Sqrt[a]*Sqrt[ 
e]*(-(c*d^2) + a*e^2)*x*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*ArcTanh[(Sqrt[d]*S 
qrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(d^(5/2)*x*Sqrt[(a*e + 
 c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.69 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1214, 25, 27, 1228, 1154, 219}

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.

\(\displaystyle \int \frac {\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx\)

\(\Big \downarrow \) 1214

\(\displaystyle \frac {2 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d+e x}-\frac {\int -\frac {a e^5 \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{d^2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^4}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {a e^5 \left (a d e+\left (2 c d^2-a e^2\right ) x\right )}{d^2 x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{e^4}+\frac {2 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d+e x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a e \int \frac {a d e+\left (2 c d^2-a e^2\right ) x}{x^2 \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{d^2}+\frac {2 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d+e x}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {a e \left (\frac {3}{2} \left (c d^2-a e^2\right ) \int \frac {1}{x \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}\right )}{d^2}+\frac {2 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d+e x}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {a e \left (-3 \left (c d^2-a e^2\right ) \int \frac {1}{4 a d e-\frac {\left (2 a d e+\left (c d^2+a e^2\right ) x\right )^2}{c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}d\frac {2 a d e+\left (c d^2+a e^2\right ) x}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}\right )}{d^2}+\frac {2 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d+e x}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {a e \left (-\frac {3 \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt {a} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt {a} \sqrt {d} \sqrt {e}}-\frac {\sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{x}\right )}{d^2}+\frac {2 \left (c-\frac {a e^2}{d^2}\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{d+e x}\)

Input:

Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^2*(d + e*x)^3),x]
 

Output:

(2*(c - (a*e^2)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(d + e*x 
) + (a*e*(-(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/x) - (3*(c*d^2 - a 
*e^2)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqr 
t[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*Sqrt[a]*Sqrt[d]*Sqrt[e])))/ 
d^2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1214

Int[(x_)^(n_.)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^ 
2)^(p_), x_Symbol] :> Simp[-2*(-d)^n*e^(2*m - n + 3)*(Sqrt[a + b*x + c*x^2] 
/((-2*c*d + b*e)^(m + 2)*(d + e*x))), x] - Simp[e^(2*m + 2)   Int[ExpandToS 
um[(((-d)^n*(-2*c*d + b*e)^(-m - 1))/(e^n*x^n) - ((-c)*d + b*e + c*e*x)^(-m 
 - 1))/(d + e*x), x]/(Sqrt[a + b*x + c*x^2]/x^n), x], x] /; FreeQ[{a, b, c, 
 d, e}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && ILtQ[n, 0] && 
EqQ[m + p, -3/2]
 

rule 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2005\) vs. \(2(158)=316\).

Time = 3.87 (sec) , antiderivative size = 2006, normalized size of antiderivative = 11.40

method	result	size
default	\(\text {Expression too large to display}\)	\(2006\)

Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/x^2/(e*x+d)^3,x,method=_RETURN 
VERBOSE)
 

Output:

1/d^3*(-1/a/d/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(5/2)+3/2*(a*e^2+c*d^2 
)/a/d/e*(1/3*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)+1/2*(a*e^2+c*d^2)*(1/ 
4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/c/d/e+1/ 
8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1/2*a*e^2+1/2*c*d^2+c*d*x*e)/( 
d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/(d*e*c)^(1/2))+a*d*e 
*((a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)+1/2*(a*e^2+c*d^2)*ln((1/2*a*e^2+ 
1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/ 
(d*e*c)^(1/2)-a*d*e/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1 
/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2))/x)))+4*c/a*(1/8*(2*c*d*e*x+a* 
e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(3/2)/c/d/e+3/16*(4*a*c*d^2*e 
^2-(a*e^2+c*d^2)^2)/d/e/c*(1/4*(2*c*d*e*x+a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2 
)*x+c*d*x^2*e)^(1/2)/c/d/e+1/8*(4*a*c*d^2*e^2-(a*e^2+c*d^2)^2)/d/e/c*ln((1 
/2*a*e^2+1/2*c*d^2+c*d*x*e)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e 
)^(1/2))/(d*e*c)^(1/2))))+1/e/d^2*(-2/(a*e^2-c*d^2)/(x+d/e)^3*(d*e*c*(x+d/ 
e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)+4*d*e*c/(a*e^2-c*d^2)*(2/(a*e^2-c*d^2)/( 
x+d/e)^2*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(5/2)-6*d*e*c/(a*e^2-c*d^ 
2)*(1/3*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)+1/2*(a*e^2-c*d^2)*(1 
/4*(2*d*e*c*(x+d/e)+a*e^2-c*d^2)/d/e/c*(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d 
/e))^(1/2)-1/8*(a*e^2-c*d^2)^2/d/e/c*ln((1/2*a*e^2-1/2*c*d^2+d*e*c*(x+d/e) 
)/(d*e*c)^(1/2)+(d*e*c*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(d*e*c)^...
 

Fricas [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.53 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\left [-\frac {3 \, {\left ({\left (c d^{2} e - a e^{3}\right )} x^{2} + {\left (c d^{3} - a d e^{2}\right )} x\right )} \sqrt {\frac {a e}{d}} \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d^{2} e + {\left (c d^{3} + a d e^{2}\right )} x\right )} \sqrt {\frac {a e}{d}} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (a d e - {\left (2 \, c d^{2} - 3 \, a e^{2}\right )} x\right )}}{4 \, {\left (d^{2} e x^{2} + d^{3} x\right )}}, \frac {3 \, {\left ({\left (c d^{2} e - a e^{3}\right )} x^{2} + {\left (c d^{3} - a d e^{2}\right )} x\right )} \sqrt {-\frac {a e}{d}} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-\frac {a e}{d}}}{2 \, {\left (a c d e^{2} x^{2} + a^{2} d e^{2} + {\left (a c d^{2} e + a^{2} e^{3}\right )} x\right )}}\right ) - 2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (a d e - {\left (2 \, c d^{2} - 3 \, a e^{2}\right )} x\right )}}{2 \, {\left (d^{2} e x^{2} + d^{3} x\right )}}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^2/(e*x+d)^3,x, algorit 
hm="fricas")
 

Output:

[-1/4*(3*((c*d^2*e - a*e^3)*x^2 + (c*d^3 - a*d*e^2)*x)*sqrt(a*e/d)*log((8* 
a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 4*sqrt(c*d*e*x^2 + 
 a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) 
+ 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + 
a*e^2)*x)*(a*d*e - (2*c*d^2 - 3*a*e^2)*x))/(d^2*e*x^2 + d^3*x), 1/2*(3*((c 
*d^2*e - a*e^3)*x^2 + (c*d^3 - a*d*e^2)*x)*sqrt(-a*e/d)*arctan(1/2*sqrt(c* 
d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a 
*e/d)/(a*c*d*e^2*x^2 + a^2*d*e^2 + (a*c*d^2*e + a^2*e^3)*x)) - 2*sqrt(c*d* 
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(a*d*e - (2*c*d^2 - 3*a*e^2)*x))/(d^2*e 
*x^2 + d^3*x)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\text {Timed out} \] Input:

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**2/(e*x+d)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\int { \frac {{\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac {3}{2}}}{{\left (e x + d\right )}^{3} x^{2}} \,d x } \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^2/(e*x+d)^3,x, algorit 
hm="maxima")
 

Output:

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)^3*x^2), 
 x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^2/(e*x+d)^3,x, algorit 
hm="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,[0,0,1]%%%},[6,0]%%%}+%%%{%%{[%%%{-2,[0,1,0]%%%},0]: 
[1,0,%%%{
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx=\int \frac {{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^{3/2}}{x^2\,{\left (d+e\,x\right )}^3} \,d x \] Input:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^2*(d + e*x)^3),x)
 

Output:

int((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(3/2)/(x^2*(d + e*x)^3), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.49 (sec) , antiderivative size = 1374, normalized size of antiderivative = 7.81 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^2 (d+e x)^3} \, dx =\text {Too large to display} \] Input:

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^2/(e*x+d)^3,x)
 

Output:

( - 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a**2*d**2*e**4 - 18*sqrt(d + e*x)*sq 
rt(a*e + c*d*x)*a**2*d*e**5*x - 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d**4 
*e**2 + 6*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d**3*e**3*x + 4*sqrt(d + e*x 
)*sqrt(a*e + c*d*x)*c**2*d**5*e*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)* 
sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d 
)*sqrt(c)*sqrt(d + e*x))*a**2*d*e**5*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqr 
t(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + s 
qrt(d)*sqrt(c)*sqrt(d + e*x))*a**2*e**6*x**2 + 6*sqrt(e)*sqrt(d)*sqrt(a)*l 
og(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d** 
2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c*d**3*e**3*x + 6*sqrt(e)*sqrt(d)*sq 
rt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e + a*e**2 
+ c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a*c*d**2*e**4*x**2 + 3*sqrt(e)* 
sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a)*d*e 
 + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*c**2*d**5*e*x + 3*sqr 
t(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) - sqrt(2*sqrt(c)*sqrt(a 
)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*c**2*d**4*e**2*x 
**2 - 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt(2*sqr 
t(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x))*a**2* 
d*e**5*x - 9*sqrt(e)*sqrt(d)*sqrt(a)*log(sqrt(e)*sqrt(a*e + c*d*x) + sqrt( 
2*sqrt(c)*sqrt(a)*d*e + a*e**2 + c*d**2) + sqrt(d)*sqrt(c)*sqrt(d + e*x...