\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x^2 (d+e x)} \, dx\) [16]

Optimal result

Integrand size = 40, antiderivative size = 118 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=-\frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{d x}-\frac {\left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {a} d^{3/2} \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 10.15 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\frac {\sqrt {(a e+c d x) (d+e x)} \left (-\frac {\sqrt {d}}{x}+\frac {\left (-c d^2+a e^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {e} \sqrt {d+e x}}\right )}{\sqrt {a} \sqrt {e} \sqrt {a e+c d x} \sqrt {d+e x}}\right )}{d^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1215, 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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-(Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(d*x)) - ((c*d^2 - a*e^2)*Ar 
cTanh[(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]*d^(3/2)*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 1215

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

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. \(707\) vs. \(2(102)=204\).

Time = 2.47 (sec) , antiderivative size = 708, normalized size of antiderivative = 6.00

method	result	size
default	\(\frac {-\frac {{\left (a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e \right )}^{\frac {3}{2}}}{a d e x}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \left (\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 \sqrt {d e c}}-\frac {a d 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}}\right )}{2 a d e}+\frac {2 c \left (\frac {\left (2 c d x e +a \,e^{2}+c \,d^{2}\right ) \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{4 c d e}+\frac {\left (4 a c \,d^{2} e^{2}-\left (a \,e^{2}+c \,d^{2}\right )^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{8 d e c \sqrt {d e c}}\right )}{a}}{d}+\frac {e \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 )}{d^{2}}-\frac {e \left (\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 \sqrt {d e c}}-\frac {a d 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}}\right )}{d^{2}}\)	\(708\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.01 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\left [-\frac {4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e + {\left (c d^{2} - a e^{2}\right )} \sqrt {a d e} x \log \left (\frac {8 \, a^{2} d^{2} e^{2} + {\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {a d e} + 8 \, {\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right )}{4 \, a d^{2} e x}, -\frac {2 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} a d e - {\left (c d^{2} - a e^{2}\right )} \sqrt {-a d e} x \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, a d e + {\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt {-a d e}}{2 \, {\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} + {\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right )}{2 \, a d^{2} e x}\right ] \] Input:

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^2/(e*x+d),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*d^2*e*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)*sq 
rt(-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*d^2*e*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 220 vs. \(2 (102) = 204\).

Time = 0.15 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\frac {{\left (c d^{2} - a e^{2}\right )} \arctan \left (-\frac {\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}}{\sqrt {-a d e}}\right )}{\sqrt {-a d e} d} - \frac {{\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} c d^{2} + {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} a e^{2} + 2 \, \sqrt {c d e} a d e}{{\left (a d e - {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )}^{2}\right )} d} \] Input:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 372, normalized size of antiderivative = 3.15 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^2 (d+e x)} \, dx=\frac {-2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a d e -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{2} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c \,d^{2} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) a \,e^{2} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) c \,d^{2} x +\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) a \,e^{2} x -\sqrt {e}\, \sqrt {d}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) c \,d^{2} x}{2 a \,d^{2} e x} \] Input:

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