\(\int \frac {\sqrt {d+e x}}{x (a+b x+c x^2)} \, dx\) [63]

Optimal result

Integrand size = 25, antiderivative size = 275 \[ \int \frac {\sqrt {d+e x}}{x \left (a+b x+c x^2\right )} \, dx=-\frac {2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a}+\frac {\sqrt {2} \sqrt {c} \left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \sqrt {c} \left (b d-\sqrt {b^2-4 a c} d-2 a e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

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

Mathematica [A] (verified)

Time = 1.55 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d+e x}}{x \left (a+b x+c x^2\right )} \, dx=-\frac {\frac {\sqrt {2} \sqrt {c} \left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (-b d+\sqrt {b^2-4 a c} d+2 a e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}+2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1199, 2009}

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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1199

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e   Subs 
t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* 
d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], 
 x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer 
Q[n] && FractionQ[m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(2 e^{2} \left (-\frac {\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a \,e^{2}}+\frac {4 c \left (\frac {\left (-2 a \,e^{2}+b d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a \,e^{2}-b d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}}\right )\)	\(291\)
default	\(2 e^{2} \left (-\frac {\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a \,e^{2}}+\frac {4 c \left (\frac {\left (-2 a \,e^{2}+b d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (2 a \,e^{2}-b d e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a \,e^{2}}\right )\)	\(291\)
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, d}{2}+e \left (a e -\frac {b d}{2}\right )\right ) \sqrt {2}\, c \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\left (\sqrt {2}\, c \left (\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, d}{2}+e \left (a e -\frac {b d}{2}\right )\right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, a}\)	\(352\)

Input:

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

Output:

2*e^2*(-d^(1/2)/a/e^2*arctanh((e*x+d)^(1/2)/d^(1/2))+4/a/e^2*c*(1/8*(-2*a* 
e^2+b*d*e-(-e^2*(4*a*c-b^2))^(1/2)*d)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b 
*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2) 
/((b*e-2*c*d+(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))-1/8*(2*a*e^2-b*d*e-(-e^2* 
(4*a*c-b^2))^(1/2)*d)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2* 
(4*a*c-b^2))^(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+ 
(-e^2*(4*a*c-b^2))^(1/2))*c)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1217 vs. \(2 (225) = 450\).

Time = 0.32 (sec) , antiderivative size = 2443, normalized size of antiderivative = 8.88 \[ \int \frac {\sqrt {d+e x}}{x \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt( 
(b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) 
*log(sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e + (a^2*b^3 - 4*a^3*b 
*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt(-(a*b* 
e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2* 
e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d - a*c*e)*sqrt(e 
*x + d)) - sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)* 
sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^ 
3*c))*log(-sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e + (a^2*b^3 - 4 
*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))*sqrt( 
-(a*b*e - (b^2 - 2*a*c)*d + (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e 
+ a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d - a*c*e)* 
sqrt(e*x + d)) + sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a 
^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 
- 4*a^3*c))*log(sqrt(2)*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e - (a^2*b^ 
3 - 4*a^3*b*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))* 
sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 - 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b 
*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(a^2*b^2 - 4*a^3*c)) - 4*(b*c*d - a* 
c*e)*sqrt(e*x + d)) - sqrt(2)*a*sqrt(-(a*b*e - (b^2 - 2*a*c)*d - (a^2*b^2 
- 4*a^3*c)*sqrt((b^2*d^2 - 2*a*b*d*e + a^2*e^2)/(a^4*b^2 - 4*a^5*c)))/(...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 719 vs. \(2 (225) = 450\).

Time = 0.29 (sec) , antiderivative size = 719, normalized size of antiderivative = 2.61 \[ \int \frac {\sqrt {d+e x}}{x \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

2*d*arctan(sqrt(e*x + d)/sqrt(-d))/(a*sqrt(-d)) - 1/4*(sqrt(-4*c^2*d + 2*( 
b*c - sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*a^2*d*e^2 - 2*(sqrt(b^2 - 4*a* 
c)*a*c*d^2 - sqrt(b^2 - 4*a*c)*a*b*d*e + sqrt(b^2 - 4*a*c)*a^2*e^2)*sqrt(- 
4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(a)*abs(e) - (2*a^2*b*c*d^2* 
e + 2*a^3*b*e^3 - (a^2*b^2 + 4*a^3*c)*d*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt 
(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*a*c*d - a*b 
*e + sqrt(-4*(a*c*d^2 - a*b*d*e + a^2*e^2)*a*c + (2*a*c*d - a*b*e)^2))/(a* 
c)))/((sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^ 
2 - 4*a*c)*a^3*e^2)*abs(a)*abs(c)*abs(e)) + 1/4*(sqrt(-4*c^2*d + 2*(b*c + 
sqrt(b^2 - 4*a*c)*c)*e)*(b^2 - 4*a*c)*a^2*d*e^2 + 2*(sqrt(b^2 - 4*a*c)*a*c 
*d^2 - sqrt(b^2 - 4*a*c)*a*b*d*e + sqrt(b^2 - 4*a*c)*a^2*e^2)*sqrt(-4*c^2* 
d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*abs(a)*abs(e) - (2*a^2*b*c*d^2*e + 2* 
a^3*b*e^3 - (a^2*b^2 + 4*a^3*c)*d*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 
 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*a*c*d - a*b*e - s 
qrt(-4*(a*c*d^2 - a*b*d*e + a^2*e^2)*a*c + (2*a*c*d - a*b*e)^2))/(a*c)))/( 
(sqrt(b^2 - 4*a*c)*a^2*c*d^2 - sqrt(b^2 - 4*a*c)*a^2*b*d*e + sqrt(b^2 - 4* 
a*c)*a^3*e^2)*abs(a)*abs(c)*abs(e))
 

Mupad [B] (verification not implemented)

Time = 15.36 (sec) , antiderivative size = 10894, normalized size of antiderivative = 39.61 \[ \int \frac {\sqrt {d+e x}}{x \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((b^4*d + 8*a^2*c^2*d - a*b^3*e + a*e*(-(4*a*c - b^2)^3)^(1/2) - 
b*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d + 4*a^2*b*c*e)/(2*(a^2*b^4 + 16 
*a^4*c^2 - 8*a^3*b^2*c)))^(1/2)*((((b^4*d + 8*a^2*c^2*d - a*b^3*e + a*e*(- 
(4*a*c - b^2)^3)^(1/2) - b*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d + 4*a^ 
2*b*c*e)/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)))^(1/2)*((d + e*x)^(1/2)* 
((b^4*d + 8*a^2*c^2*d - a*b^3*e + a*e*(-(4*a*c - b^2)^3)^(1/2) - b*d*(-(4* 
a*c - b^2)^3)^(1/2) - 6*a*b^2*c*d + 4*a^2*b*c*e)/(2*(a^2*b^4 + 16*a^4*c^2 
- 8*a^3*b^2*c)))^(1/2)*(512*a^5*c^4*e^10 + 32*a^3*b^4*c^2*e^10 - 256*a^4*b 
^2*c^3*e^10 + 768*a^4*c^5*d^2*e^8 + 64*a^2*b^4*c^3*d^2*e^8 - 448*a^3*b^2*c 
^4*d^2*e^8 - 896*a^4*b*c^4*d*e^9 - 64*a^2*b^5*c^2*d*e^9 + 480*a^3*b^3*c^3* 
d*e^9) - 384*a^4*c^4*d*e^10 - 384*a^3*c^5*d^3*e^8 + 96*a^2*b^2*c^4*d^3*e^8 
 - 96*a^2*b^3*c^3*d^2*e^9 + 384*a^3*b*c^4*d^2*e^9 + 96*a^3*b^2*c^3*d*e^10) 
 - (d + e*x)^(1/2)*(128*a^3*b*c^3*e^11 + 192*a^3*c^4*d*e^10 - 32*a^2*b^3*c 
^2*e^11 + 576*a^2*c^5*d^3*e^8 + 64*b^4*c^3*d^3*e^8 - 64*b^5*c^2*d^2*e^9 + 
64*a*b^4*c^2*d*e^10 - 384*a*b^2*c^4*d^3*e^8 + 384*a*b^3*c^3*d^2*e^9 - 576* 
a^2*b*c^4*d^2*e^9 - 288*a^2*b^2*c^3*d*e^10))*((b^4*d + 8*a^2*c^2*d - a*b^3 
*e + a*e*(-(4*a*c - b^2)^3)^(1/2) - b*d*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^2 
*c*d + 4*a^2*b*c*e)/(2*(a^2*b^4 + 16*a^4*c^2 - 8*a^3*b^2*c)))^(1/2) + 96*a 
*c^5*d^4*e^8 + 96*a^2*c^4*d^2*e^10 - 32*b^2*c^4*d^4*e^8 + 32*b^4*c^2*d^2*e 
^10 + 64*a*b*c^4*d^3*e^9 - 32*a*b^3*c^2*d*e^11 + 160*a^2*b*c^3*d*e^11 -...
 

Reduce [F]

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

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

Output:

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