\(\int \frac {\sqrt {a d e+(c d^2+a e^2) x+c d e x^2}}{x (d+e x)} \, dx\) [15]

Optimal result

Integrand size = 40, antiderivative size = 137 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x (d+e x)} \, dx=\frac {2 \sqrt {c} \sqrt {d} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d} (d+e x)}{\sqrt {e} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {e}}-\frac {2 \sqrt {a} \sqrt {e} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e} (d+e x)}{\sqrt {d} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{\sqrt {d}} \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(476\) vs. \(2(137)=274\).

Time = 2.24 (sec) , antiderivative size = 476, normalized size of antiderivative = 3.47 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x (d+e x)} \, dx=-\frac {2 \sqrt {a e+c d x} \sqrt {d+e x} \left (-\left (\left (-\sqrt {c} d+\sqrt {c d^2-a e^2}\right ) \sqrt {-2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2-a e^2}} \arctan \left (\frac {\sqrt {-2 c d^2+a e^2-2 \sqrt {c} d \sqrt {c d^2-a e^2}} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {c} \sqrt {d} \sqrt {e} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )\right )+\left (\sqrt {c} d+\sqrt {c d^2-a e^2}\right ) \sqrt {-2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2-a e^2}} \arctan \left (\frac {\sqrt {-2 c d^2+a e^2+2 \sqrt {c} d \sqrt {c d^2-a e^2}} \sqrt {a e+c d x}}{\sqrt {a} \sqrt {c} \sqrt {d} \sqrt {e} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )+2 \sqrt {a} \sqrt {c} d e \text {arctanh}\left (\frac {\sqrt {e} \sqrt {a e+c d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {d-\frac {a e^2}{c d}}-\sqrt {d+e x}\right )}\right )\right )}{\sqrt {a} \sqrt {d} e^{3/2} \sqrt {(a e+c d x) (d+e x)}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {1215, 1268, 140, 27, 66, 104, 221}

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.

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

\(\Big \downarrow \) 1215

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

\(\Big \downarrow \) 1268

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

\(\Big \downarrow \) 140

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*((2*Sqrt[c]*Sqrt[d]*ArcTanh[(Sqrt[e]*Sqrt 
[a*e + c*d*x])/(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/Sqrt[e] - (2*Sqrt[a]*Sqrt 
[e]*ArcTanh[(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])/(Sqrt[d]*Sqrt[a*e + c*d*x])])/ 
Sqrt[d]))/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 140

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[b*d^(m + n)*f^p   Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] 
, x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x 
)*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, 
b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 
0] || SumSimplerQ[m, -1] ||  !(GtQ[n, 0] || SumSimplerQ[n, -1]))
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

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 1268

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(109)=218\).

Time = 2.34 (sec) , antiderivative size = 308, normalized size of antiderivative = 2.25

method	result	size
default	\(\frac {\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) \ln \left (\frac {\frac {1}{2} a \,e^{2}+\frac {1}{2} c \,d^{2}+c d x e}{\sqrt {d e c}}+\sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}\right )}{2 \sqrt {d e c}}-\frac {a d e \ln \left (\frac {2 a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +2 \sqrt {a d e}\, \sqrt {a d e +\left (a \,e^{2}+c \,d^{2}\right ) x +c d \,x^{2} e}}{x}\right )}{\sqrt {a d e}}}{d}-\frac {\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (a \,e^{2}-c \,d^{2}\right ) \ln \left (\frac {\frac {a \,e^{2}}{2}-\frac {c \,d^{2}}{2}+d e c \left (x +\frac {d}{e}\right )}{\sqrt {d e c}}+\sqrt {d e c \left (x +\frac {d}{e}\right )^{2}+\left (a \,e^{2}-c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}\right )}{2 \sqrt {d e c}}}{d}\)	\(308\)

Input:

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

Output:

1/d*((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))-1/d*((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 = 0.24 (sec) , antiderivative size = 947, normalized size of antiderivative = 6.91 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x (d+e x)} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*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*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e 
^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) + 1/2*sqrt(a*e/d)*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^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) 
 + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2), -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*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) + 1/2*sqrt(a*e/d)*l 
og((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^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a 
*e/d) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2), sqrt(-a*e/d)*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 
*e/d)/(a*c*d*e^2*x^2 + a^2*d*e^2 + (a*c*d^2*e + a^2*e^3)*x)) + 1/2*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*(2*c*d* 
e^2*x + c*d^2*e + a*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt( 
c*d/e) + 8*(c^2*d^3*e + a*c*d*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*x^2 + a*c*d^2*e + (c^2*d^3 + a*c*d*e^2)*x)) + sqrt(-a*e/d)*arc 
tan(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*e/d)/(a*c*d*e^2*x^2 + a^2*d*e^2 + (a*c*d^2*e + a^2*e^3)...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m operator + Error: 
Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x (d+e x)} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \left (\sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}-\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) e +\sqrt {a}\, \mathrm {log}\left (\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {2 \sqrt {c}\, \sqrt {a}\, d e +a \,e^{2}+c \,d^{2}}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\right ) e -\sqrt {a}\, \mathrm {log}\left (2 \sqrt {e}\, \sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}\, \sqrt {c d x +a e}+2 \sqrt {c}\, \sqrt {a}\, d e +2 c d e x \right ) e +2 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {e}\, \sqrt {c d x +a e}+\sqrt {d}\, \sqrt {c}\, \sqrt {e x +d}}{\sqrt {a \,e^{2}-c \,d^{2}}}\right ) d \right )}{d e} \] Input:

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

Output:

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