\(\int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx\) [134]

Optimal result

Integrand size = 21, antiderivative size = 121 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=-\frac {\sqrt {b x+c x^2}}{e (d+e x)}+\frac {2 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e^2}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{\sqrt {d} e^2 \sqrt {c d-b e}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\frac {\sqrt {x (b+c x)} \left (-\frac {e}{d+e x}+\frac {(2 c d-b e) \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{\sqrt {d} \sqrt {-c d+b e} \sqrt {x} \sqrt {b+c x}}-\frac {2 \sqrt {c} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{e^2} \] Input:

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

Output:

(Sqrt[x*(b + c*x)]*(-(e/(d + e*x)) + ((2*c*d - b*e)*ArcTan[(-(e*Sqrt[x]*Sq 
rt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]*Sqrt[-(c*d) + b*e])])/(Sqrt[d]* 
Sqrt[-(c*d) + b*e]*Sqrt[x]*Sqrt[b + c*x]) - (2*Sqrt[c]*Log[-(Sqrt[c]*Sqrt[ 
x]) + Sqrt[b + c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/e^2
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1161, 1269, 1091, 219, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\frac {4 c \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {(2 c d-b e) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\)

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {4 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{e}-\frac {(2 c d-b e) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{\sqrt {d} e \sqrt {c d-b e}}}{2 e}-\frac {\sqrt {b x+c x^2}}{e (d+e x)}\)

Input:

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

Output:

-(Sqrt[b*x + c*x^2]/(e*(d + e*x))) + ((4*Sqrt[c]*ArcTanh[(Sqrt[c]*x)/Sqrt[ 
b*x + c*x^2]])/e - ((2*c*d - b*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[ 
d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(Sqrt[d]*e*Sqrt[c*d - b*e]))/(2*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 1091

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

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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1269

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {\left (e x +d \right ) \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )-2 \left (\sqrt {c}\, \left (e x +d \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )-\frac {e \sqrt {x \left (c x +b \right )}}{2}\right ) \sqrt {d \left (b e -c d \right )}}{\sqrt {d \left (b e -c d \right )}\, e^{2} \left (e x +d \right )}\)	\(120\)
default	\(\frac {\frac {e^{2} \left (c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}\right )^{\frac {3}{2}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \left (\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{2 e \sqrt {c}}+\frac {d \left (b e -c d \right ) \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}-\frac {2 c \,e^{2} \left (\frac {\left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{4 c}+\frac {\left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}\right )}{8 c^{\frac {3}{2}}}\right )}{d \left (b e -c d \right )}}{e^{2}}\)	\(589\)

Input:

int((c*x^2+b*x)^(1/2)/(e*x+d)^2,x,method=_RETURNVERBOSE)
 

Output:

-((e*x+d)*(b*e-2*c*d)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))-2* 
(c^(1/2)*(e*x+d)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))-1/2*e*(x*(c*x+b))^(1 
/2))*(d*(b*e-c*d))^(1/2))/(d*(b*e-c*d))^(1/2)/e^2/(e*x+d)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 839, normalized size of antiderivative = 6.93 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*(2*(c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(c)*log(2*c*x + b + 
2*sqrt(c*x^2 + b*x)*sqrt(c)) - (2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqr 
t(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c 
*x^2 + b*x))/(e*x + d)) - 2*(c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/(c*d^3* 
e^2 - b*d^2*e^3 + (c*d^2*e^3 - b*d*e^4)*x), ((2*c*d^2 - b*d*e + (2*c*d*e - 
 b*e^2)*x)*sqrt(-c*d^2 + b*d*e)*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b 
*x)/(c*d*x + b*d)) + (c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(c)*log 
(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c)) - (c*d^2*e - b*d*e^2)*sqrt(c*x^2 
 + b*x))/(c*d^3*e^2 - b*d^2*e^3 + (c*d^2*e^3 - b*d*e^4)*x), -1/2*(4*(c*d^3 
 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(-c)*arctan(sqrt(c*x^2 + b*x)*sqrt 
(-c)/(c*x + b)) + (2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sqrt(c*d^2 - b*d 
*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/ 
(e*x + d)) + 2*(c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/(c*d^3*e^2 - b*d^2*e 
^3 + (c*d^2*e^3 - b*d*e^4)*x), ((2*c*d^2 - b*d*e + (2*c*d*e - b*e^2)*x)*sq 
rt(-c*d^2 + b*d*e)*arctan(sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + 
b*d)) - 2*(c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^2)*x)*sqrt(-c)*arctan(sqrt(c 
*x^2 + b*x)*sqrt(-c)/(c*x + b)) - (c*d^2*e - b*d*e^2)*sqrt(c*x^2 + b*x))/( 
c*d^3*e^2 - b*d^2*e^3 + (c*d^2*e^3 - b*d*e^4)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x)^(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(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 603, normalized size of antiderivative = 4.98 \[ \int \frac {\sqrt {b x+c x^2}}{(d+e x)^2} \, dx=\frac {-\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b d e -\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b \,e^{2} x +2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c \,d^{2}+2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c d e x -\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b d e -\sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b \,e^{2} x +2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c \,d^{2}+2 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c d e x -\sqrt {x}\, \sqrt {c x +b}\, b d \,e^{2}+\sqrt {x}\, \sqrt {c x +b}\, c \,d^{2} e +2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b \,d^{2} e +2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b d \,e^{2} x -2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) c \,d^{3}-2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) c \,d^{2} e x}{d \,e^{2} \left (b \,e^{2} x -c d e x +b d e -c \,d^{2}\right )} \] Input:

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

Output:

( - sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) 
- sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*d*e - sqrt(d)*sqrt(b*e - c 
*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c 
))/(sqrt(d)*sqrt(c)))*b*e**2*x + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e 
- c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c) 
))*c*d**2 + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt 
(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c*d*e*x - sqrt(d)* 
sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sq 
rt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*d*e - sqrt(d)*sqrt(b*e - c*d)*atan((sq 
rt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)* 
sqrt(c)))*b*e**2*x + 2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqr 
t(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c*d**2 + 
2*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + 
sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c*d*e*x - sqrt(x)*sqrt(b + c*x 
)*b*d*e**2 + sqrt(x)*sqrt(b + c*x)*c*d**2*e + 2*sqrt(c)*log((sqrt(b + c*x) 
 + sqrt(x)*sqrt(c))/sqrt(b))*b*d**2*e + 2*sqrt(c)*log((sqrt(b + c*x) + sqr 
t(x)*sqrt(c))/sqrt(b))*b*d*e**2*x - 2*sqrt(c)*log((sqrt(b + c*x) + sqrt(x) 
*sqrt(c))/sqrt(b))*c*d**3 - 2*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c) 
)/sqrt(b))*c*d**2*e*x)/(d*e**2*(b*d*e + b*e**2*x - c*d**2 - c*d*e*x))