\(\int \frac {x \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{(d+e x)^2} \, dx\) [22]

Optimal result

Integrand size = 38, antiderivative size = 159 \[ \int \frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2}+\frac {2 d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{e^2 (d+e x)}-\frac {\left (3 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {c} \sqrt {d} (d+e x)}\right )}{\sqrt {c} \sqrt {d} e^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.81 \[ \int \frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (\frac {\sqrt {e} (3 d+e x)}{d+e x}-\frac {\left (3 c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \sqrt {d+e x}}\right )}{\sqrt {c} \sqrt {d} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{e^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {1213, 27, 1160, 1092, 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.

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

\(\Big \downarrow \) 1213

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

rule 1213

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 - n + 2)   Int[Expan 
dToSum[((-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], x] /; FreeQ[{a, b, c, d, e} 
, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[m, 0] && IGtQ[n, 0] && EqQ[m 
+ p, -3/2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(141)=282\).

Time = 2.42 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.17

method	result	size
default	\(\frac {\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}}{e^{2}}-\frac {d \left (-\frac {2 \left (d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {2 d e c \left (\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}\right )}{a \,e^{2}-c \,d^{2}}\right )}{e^{3}}\)	\(345\)

Input:

int(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^2*e)^(1/2)/(e*x+d)^2,x,method=_RETURNVE 
RBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.60 \[ \int \frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\left [-\frac {{\left (3 \, c d^{3} - a d e^{2} + {\left (3 \, c d^{2} e - a e^{3}\right )} x\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (c d e^{2} x + 3 \, c d^{2} e\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{4 \, {\left (c d e^{4} x + c d^{2} e^{3}\right )}}, \frac {{\left (3 \, c d^{3} - a d e^{2} + {\left (3 \, c d^{2} e - a e^{3}\right )} x\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (c d e^{2} x + 3 \, c d^{2} e\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{2 \, {\left (c d e^{4} x + c d^{2} e^{3}\right )}}\right ] \] Input:

integrate(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x, algorithm 
="fricas")
 

Output:

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

Sympy [F]

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

integrate(x*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/(e*x+d)**2,x)
 

Output:

Integral(x*sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**2, x)
 

Maxima [F(-2)]

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

integrate(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (141) = 282\).

Time = 0.20 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.79 \[ \int \frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {{\left (\frac {2 \, \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} d \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{e^{2} {\left | e \right |}} + \frac {{\left (3 \, c d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - a e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )\right )} \arctan \left (\frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}}}{\sqrt {-c d e}}\right )}{\sqrt {-c d e} e {\left | e \right |}} + \frac {\sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} c d^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right ) - \sqrt {c d e - \frac {c d^{2} e}{e x + d} + \frac {a e^{3}}{e x + d}} a e^{2} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{{\left (\frac {c d^{2} e}{e x + d} - \frac {a e^{3}}{e x + d}\right )} e {\left | e \right |}}\right )} {\left | e \right |}}{e} \] Input:

integrate(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x, algorithm 
="giac")
 

Output:

(2*sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))*d*sgn(1/(e*x + d))*sg 
n(e)/(e^2*abs(e)) + (3*c*d^2*sgn(1/(e*x + d))*sgn(e) - a*e^2*sgn(1/(e*x + 
d))*sgn(e))*arctan(sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3/(e*x + d))/sqrt( 
-c*d*e))/(sqrt(-c*d*e)*e*abs(e)) + (sqrt(c*d*e - c*d^2*e/(e*x + d) + a*e^3 
/(e*x + d))*c*d^2*sgn(1/(e*x + d))*sgn(e) - sqrt(c*d*e - c*d^2*e/(e*x + d) 
 + a*e^3/(e*x + d))*a*e^2*sgn(1/(e*x + d))*sgn(e))/((c*d^2*e/(e*x + d) - a 
*e^3/(e*x + d))*e*abs(e)))*abs(e)/e
 

Mupad [F(-1)]

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

int((x*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2,x)
 

Output:

int((x*(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2))/(d + e*x)^2, x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.09 \[ \int \frac {x \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx=\frac {12 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c \,d^{2} e +4 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c d \,e^{2} x +4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a d \,e^{2}+4 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a \,e^{3} x -12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{3}-12 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c \,d^{2} e x -\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a d \,e^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, a \,e^{3} x +9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c \,d^{3}+9 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, c \,d^{2} e x}{4 c d \,e^{3} \left (e x +d \right )} \] Input:

int(x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/(e*x+d)^2,x)
 

Output:

(12*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c*d**2*e + 4*sqrt(d + e*x)*sqrt(a*e + 
c*d*x)*c*d*e**2*x + 4*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d* 
x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*a*d*e**2 + 4*sq 
rt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqr 
t(d + e*x))/sqrt(a*e**2 - c*d**2))*a*e**3*x - 12*sqrt(e)*sqrt(d)*sqrt(c)*l 
og((sqrt(e)*sqrt(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 
 - c*d**2))*c*d**3 - 12*sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqrt(a*e + c* 
d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c*d**2*e*x - 
sqrt(e)*sqrt(d)*sqrt(c)*a*d*e**2 - sqrt(e)*sqrt(d)*sqrt(c)*a*e**3*x + 9*sq 
rt(e)*sqrt(d)*sqrt(c)*c*d**3 + 9*sqrt(e)*sqrt(d)*sqrt(c)*c*d**2*e*x)/(4*c* 
d*e**3*(d + e*x))