\(\int \frac {x (d+e x)}{\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}} \, dx\) [74]

Optimal result

Integrand size = 36, antiderivative size = 155 \[ \int \frac {x (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\left (c d^2-3 a e^2+2 c d e x\right ) \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2 e}-\frac {\left (c d^2-a e^2\right ) \left (c d^2+3 a e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c^{5/2} d^{5/2} e^{3/2}} \] Output:

1/4*(2*c*d*e*x-3*a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/c^2/ 
d^2/e-1/4*(-a*e^2+c*d^2)*(3*a*e^2+c*d^2)*arctanh(c^(1/2)*d^(1/2)*(e*x+d)/e 
^(1/2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/c^(5/2)/d^(5/2)/e^(3/2)
 

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.31 \[ \int \frac {x (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {\sqrt {c} \sqrt {d} \sqrt {e} (d+e x) \left (-3 a^2 e^3+a c d e (d-e x)+c^2 d^2 x (d+2 e x)\right )+2 \left (c^2 d^4+2 a c d^2 e^2-3 a^2 e^4\right ) \sqrt {a e+c d x} \sqrt {d+e x} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )}{4 c^{5/2} d^{5/2} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1225, 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.

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

\(\Big \downarrow \) 1225

\(\displaystyle \frac {\left (-3 a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e}-\frac {\left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right ) \int \frac {1}{\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}dx}{8 c^2 d^2 e}\)

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (-3 a e^2+c d^2+2 c d e x\right ) \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2 e}-\frac {\left (c d^2-a e^2\right ) \left (3 a e^2+c d^2\right ) \text {arctanh}\left (\frac {a e^2+c d^2+2 c d e x}{2 \sqrt {c} \sqrt {d} \sqrt {e} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} e^{3/2}}\)

Input:

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

Output:

((c*d^2 - 3*a*e^2 + 2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2] 
)/(4*c^2*d^2*e) - ((c*d^2 - a*e^2)*(c*d^2 + 3*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])])/(8*c^(5/2)*d^(5/2)*e^(3/2))
 

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 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 1225

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(373\) vs. \(2(135)=270\).

Time = 2.22 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.41

method	result	size
default	\(d \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )+e \left (\frac {x \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{2 d e c}-\frac {3 \left (a \,e^{2}+c \,d^{2}\right ) \left (\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{d e c}-\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 d e c \sqrt {d e c}}\right )}{4 d e c}-\frac {a \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 c \sqrt {d e c}}\right )\)	\(374\)

Input:

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

Output:

d*(1/d/e/c*(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)/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*(1/2*x/d/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*x^ 
2*e)^(1/2)-3/4*(a*e^2+c*d^2)/d/e/c*(1/d/e/c*(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)/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))-1/2*a/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))
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.68 \[ \int \frac {x (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\left [-\frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \sqrt {c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \, \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {c d e} + 8 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, {\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e - 3 \, a c d e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{16 \, c^{3} d^{3} e^{2}}, \frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \sqrt {-c d e} \arctan \left (\frac {\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} {\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt {-c d e}}{2 \, {\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} + {\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \, {\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e - 3 \, a c d e^{3}\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}}{8 \, c^{3} d^{3} e^{2}}\right ] \] Input:

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

Output:

[-1/16*((c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)*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*(2*c^2*d^2*e^2*x + c^2*d^3*e - 3*a*c*d*e^3)*sqrt(c*d*e*x^ 
2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^2), 1/8*((c^2*d^4 + 2*a*c*d^2*e 
^2 - 3*a^2*e^4)*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*(2*c^2*d^2*e^2*x + c^2*d^3*e - 3 
*a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(c^3*d^3*e^2)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (151) = 302\).

Time = 1.31 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.87 \[ \int \frac {x (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\begin {cases} \left (- \frac {a e}{2 c} - \frac {\left (d - \frac {\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}}{2 c d}\right ) \left (a e^{2} + c d^{2}\right )}{2 c d e}\right ) \left (\begin {cases} \frac {\log {\left (a e^{2} + c d^{2} + 2 c d e x + 2 \sqrt {c d e} \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \right )}}{\sqrt {c d e}} & \text {for}\: a d e - \frac {\left (a e^{2} + c d^{2}\right )^{2}}{4 c d e} \neq 0 \\\frac {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right ) \log {\left (x - \frac {- a e^{2} - c d^{2}}{2 c d e} \right )}}{\sqrt {c d e \left (x - \frac {- a e^{2} - c d^{2}}{2 c d e}\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \left (\frac {x}{2 c d} + \frac {d - \frac {\frac {3 a e^{2}}{2} + \frac {3 c d^{2}}{2}}{2 c d}}{c d e}\right ) \sqrt {a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} & \text {for}\: c d e \neq 0 \\\frac {2 \left (- \frac {a c d^{4} e \sqrt {a d e + x \left (a e^{2} + c d^{2}\right )}}{a e^{2} + c d^{2}} + \frac {e \left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {5}{2}}}{5 \left (a e^{2} + c d^{2}\right )} + \frac {\left (a d e + x \left (a e^{2} + c d^{2}\right )\right )^{\frac {3}{2}} \left (- a d e^{2} + c d^{3}\right )}{3 \left (a e^{2} + c d^{2}\right )}\right )}{\left (a e^{2} + c d^{2}\right )^{2}} & \text {for}\: a e^{2} + c d^{2} \neq 0 \\\frac {\frac {d x^{2}}{2} + \frac {e x^{3}}{3}}{\sqrt {a d e}} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise(((-a*e/(2*c) - (d - (3*a*e**2/2 + 3*c*d**2/2)/(2*c*d))*(a*e**2 + 
 c*d**2)/(2*c*d*e))*Piecewise((log(a*e**2 + c*d**2 + 2*c*d*e*x + 2*sqrt(c* 
d*e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)))/sqrt(c*d*e), Ne(a*d*e 
 - (a*e**2 + c*d**2)**2/(4*c*d*e), 0)), ((x - (-a*e**2 - c*d**2)/(2*c*d*e) 
)*log(x - (-a*e**2 - c*d**2)/(2*c*d*e))/sqrt(c*d*e*(x - (-a*e**2 - c*d**2) 
/(2*c*d*e))**2), True)) + (x/(2*c*d) + (d - (3*a*e**2/2 + 3*c*d**2/2)/(2*c 
*d))/(c*d*e))*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2)), Ne(c*d*e, 0) 
), (2*(-a*c*d**4*e*sqrt(a*d*e + x*(a*e**2 + c*d**2))/(a*e**2 + c*d**2) + e 
*(a*d*e + x*(a*e**2 + c*d**2))**(5/2)/(5*(a*e**2 + c*d**2)) + (a*d*e + x*( 
a*e**2 + c*d**2))**(3/2)*(-a*d*e**2 + c*d**3)/(3*(a*e**2 + c*d**2)))/(a*e* 
*2 + c*d**2)**2, Ne(a*e**2 + c*d**2, 0)), ((d*x**2/2 + e*x**3/3)/sqrt(a*d* 
e), True))
 

Maxima [F(-2)]

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

integrate(x*(e*x+d)/(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(a*e^2-c*d^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.06 \[ \int \frac {x (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {1}{4} \, \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e} {\left (\frac {2 \, x}{c d} + \frac {c d^{2} - 3 \, a e^{2}}{c^{2} d^{2} e}\right )} + \frac {{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \log \left ({\left | -c d^{2} - a e^{2} - 2 \, \sqrt {c d e} {\left (\sqrt {c d e} x - \sqrt {c d e x^{2} + c d^{2} x + a e^{2} x + a d e}\right )} \right |}\right )}{8 \, \sqrt {c d e} c^{2} d^{2} e} \] Input:

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

Output:

1/4*sqrt(c*d*e*x^2 + c*d^2*x + a*e^2*x + a*d*e)*(2*x/(c*d) + (c*d^2 - 3*a* 
e^2)/(c^2*d^2*e)) + 1/8*(c^2*d^4 + 2*a*c*d^2*e^2 - 3*a^2*e^4)*log(abs(-c*d 
^2 - a*e^2 - 2*sqrt(c*d*e)*(sqrt(c*d*e)*x - sqrt(c*d*e*x^2 + c*d^2*x + a*e 
^2*x + a*d*e))))/(sqrt(c*d*e)*c^2*d^2*e)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.63 \[ \int \frac {x (d+e x)}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {-3 \sqrt {e x +d}\, \sqrt {c d x +a e}\, a c d \,e^{3}+\sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{3} e +2 \sqrt {e x +d}\, \sqrt {c d x +a e}\, c^{2} d^{2} e^{2} x +3 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a^{2} e^{4}-2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) a c \,d^{2} e^{2}-\sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) c^{2} d^{4}}{4 c^{3} d^{3} e^{2}} \] Input:

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

Output:

( - 3*sqrt(d + e*x)*sqrt(a*e + c*d*x)*a*c*d*e**3 + sqrt(d + e*x)*sqrt(a*e 
+ c*d*x)*c**2*d**3*e + 2*sqrt(d + e*x)*sqrt(a*e + c*d*x)*c**2*d**2*e**2*x 
+ 3*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**2*e**4 - 2*sqrt(e)*sqrt(d)*sqr 
t(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*c*d**2*e**2 - sqrt(e)*sqrt(d)*sqrt(c)*log((sqrt(e)*sqr 
t(a*e + c*d*x) + sqrt(d)*sqrt(c)*sqrt(d + e*x))/sqrt(a*e**2 - c*d**2))*c** 
2*d**4)/(4*c**3*d**3*e**2)