\(\int \frac {d+e x}{x^2 \sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [77]

Optimal result

Integrand size = 38, antiderivative size = 120 \[ \int \frac {d+e x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{a e x}+\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{a^{3/2} \sqrt {d} e^{3/2}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(728\) vs. \(2(120)=240\).

Time = 14.14 (sec) , antiderivative size = 728, normalized size of antiderivative = 6.07 \[ \int \frac {d+e x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {a e+c d x} \sqrt {d+e x} \left (\frac {\sqrt {a} \sqrt {e} \sqrt {a e+c d x} \left (a e^2 \left (-3 d-3 e x+\sqrt {d-\frac {a e^2}{c d}} \sqrt {d+e x}\right )+c \left (4 d^3+d^2 \left (5 e x-4 \sqrt {d-\frac {a e^2}{c d}} \sqrt {d+e x}\right )+d \left (e^2 x^2-3 e \sqrt {d-\frac {a e^2}{c d}} x \sqrt {d+e x}\right )\right )\right )}{x \left (c d^2 \left (4 \sqrt {d-\frac {a e^2}{c d}}-4 \sqrt {d+e x}\right )+c d e x \left (3 \sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )+a e^2 \left (-\sqrt {d-\frac {a e^2}{c d}}+3 \sqrt {d+e x}\right )\right )}-\frac {\sqrt {c d^2-a e^2} \left (c d^2-a e^2+\sqrt {c} d \sqrt {c d^2-a e^2}\right ) \arctan \left (\frac {\sqrt {-2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2-a e^2}} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {c} \sqrt {d} \sqrt {e} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )}{\sqrt {d} \sqrt {-2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2-a e^2}}}-\frac {\sqrt {c d^2-a e^2} \left (-c d^2+a e^2+\sqrt {c} d \sqrt {c d^2-a e^2}\right ) \arctan \left (\frac {\sqrt {-2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2-a e^2}} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {c} \sqrt {d} \sqrt {e} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )}{\sqrt {d} \sqrt {-2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2-a e^2}}}\right )}{a^{3/2} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*((Sqrt[a]*Sqrt[e]*Sqrt[a*e + c*d*x]*(a*e^ 
2*(-3*d - 3*e*x + Sqrt[d - (a*e^2)/(c*d)]*Sqrt[d + e*x]) + c*(4*d^3 + d^2* 
(5*e*x - 4*Sqrt[d - (a*e^2)/(c*d)]*Sqrt[d + e*x]) + d*(e^2*x^2 - 3*e*Sqrt[ 
d - (a*e^2)/(c*d)]*x*Sqrt[d + e*x]))))/(x*(c*d^2*(4*Sqrt[d - (a*e^2)/(c*d) 
] - 4*Sqrt[d + e*x]) + c*d*e*x*(3*Sqrt[d - (a*e^2)/(c*d)] - Sqrt[d + e*x]) 
 + a*e^2*(-Sqrt[d - (a*e^2)/(c*d)] + 3*Sqrt[d + e*x]))) - (Sqrt[c*d^2 - a* 
e^2]*(c*d^2 - a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 - a*e^2])*ArcTan[(Sqrt[-2*c*d^2 
 + a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 - a*e^2]]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sq 
rt[c]*Sqrt[d]*Sqrt[e]*(Sqrt[d - (a*e^2)/(c*d)] - Sqrt[d + e*x]))])/(Sqrt[d 
]*Sqrt[-2*c*d^2 + a*e^2 - 2*Sqrt[c]*d*Sqrt[c*d^2 - a*e^2]]) - (Sqrt[c*d^2 
- a*e^2]*(-(c*d^2) + a*e^2 + Sqrt[c]*d*Sqrt[c*d^2 - a*e^2])*ArcTan[(Sqrt[- 
2*c*d^2 + a*e^2 + 2*Sqrt[c]*d*Sqrt[c*d^2 - a*e^2]]*Sqrt[a*e + c*d*x])/(Sqr 
t[a]*Sqrt[c]*Sqrt[d]*Sqrt[e]*(Sqrt[d - (a*e^2)/(c*d)] - Sqrt[d + e*x]))])/ 
(Sqrt[d]*Sqrt[-2*c*d^2 + a*e^2 + 2*Sqrt[c]*d*Sqrt[c*d^2 - a*e^2]])))/(a^(3 
/2)*e^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {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.

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 [A] (verified)

Time = 2.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.68

method	result	size
default	\(d \left (-\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{2 a d e \sqrt {a d e}}\right )-\frac {e \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{\sqrt {a d e}}\)	\(201\)

Input:

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

Output:

d*(-1/a/d/e/x*(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)/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))-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)
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 354, normalized size of antiderivative = 2.95 \[ \int \frac {d+e x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e + {\left (c d^{2} - a e^{2}\right )} \sqrt {a d e} x \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} - 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right )}{4 \, a^{2} d e^{2} x}, -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e + {\left (c d^{2} - a e^{2}\right )} \sqrt {-a d e} x \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right )}{2 \, a^{2} d e^{2} x}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((e*x+d)/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),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>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 227 vs. \(2 (104) = 208\).

Time = 0.14 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.89 \[ \int \frac {d+e x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=-\frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{\sqrt {-a d e} a e} - \frac {{\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} c d^{2} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a e^{2} + 2 \, \sqrt {c d e} a d e}{{\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )} a e} \] Input:

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

Output:

-(c*d^2 - a*e^2)*arctan(-(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2 
*x + a*d*e))/sqrt(-a*d*e))/(sqrt(-a*d*e)*a*e) - ((sqrt(c*d*e)*x - sqrt(c*d 
*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))*c*d^2 + (sqrt(c*d*e)*x - sqrt(c*d*e*x 
^2 + c*d^2*x + a*e^2*x + a*d*e))*a*e^2 + 2*sqrt(c*d*e)*a*d*e)/((a*d*e - (s 
qrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e))^2)*a*e)
 

Mupad [B] (verification not implemented)

Time = 6.13 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.50 \[ \int \frac {d+e x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {d\,\mathrm {atanh}\left (\frac {\frac {x\,\left (c\,d^2+a\,e^2\right )}{2}+a\,d\,e}{\sqrt {a\,d\,e}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}\right )\,\left (c\,d^2+a\,e^2\right )}{2\,{\left (a\,d\,e\right )}^{3/2}}-\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{a\,e\,x}-\frac {e\,\ln \left (\frac {a\,e^2}{2}+\frac {c\,d^2}{2}+\frac {\sqrt {a\,d\,e}\,\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}}{x}+\frac {a\,d\,e}{x}\right )}{\sqrt {a\,d\,e}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.10 \[ \int \frac {d+e x}{x^2 \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a d e +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{2} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c \,d^{2} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{2} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c \,d^{2} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) a \,e^{2} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) c \,d^{2} x}{2 a^{2} d \,e^{2} x} \] Input:

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

Output:

( - 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*d*e + 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*e**2*x - sqrt(e)*sqrt(d)*sqrt(a)*log(sq 
rt(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*d**2*x + 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) + sq 
rt(d)*sqrt(c)*sqrt(d + e*x))*a*e**2*x - 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*d**2*x - sqrt(e)*sqrt(d)*sqrt(a)*log(2*sqrt(e 
)*sqrt(d)*sqrt(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e 
+ 2*c*d*e*x)*a*e**2*x + sqrt(e)*sqrt(d)*sqrt(a)*log(2*sqrt(e)*sqrt(d)*sqrt 
(c)*sqrt(d + e*x)*sqrt(a*e + c*d*x) + 2*sqrt(c)*sqrt(a)*d*e + 2*c*d*e*x)*c 
*d**2*x)/(2*a**2*d*e**2*x)