\(\int \frac {\sqrt {d+e x}}{x^2 (a+b x+c x^2)} \, dx\) [64]

Optimal result

Integrand size = 25, antiderivative size = 330 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {d+e x}}{a x}+\frac {(2 b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{a^2 \sqrt {d}}-\frac {\sqrt {2} \sqrt {c} \left (b^2 d-2 a c d-a b e+\sqrt {b^2-4 a c} (b d-a e)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d-2 a c d-a b e-\sqrt {b^2-4 a c} (b d-a e)\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-(e*x+d)^(1/2)/a/x+(-a*e+2*b*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/a^2/d^(1/2) 
-2^(1/2)*c^(1/2)*(b^2*d-2*a*c*d-a*b*e+(-4*a*c+b^2)^(1/2)*(-a*e+b*d))*arcta 
nh(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 
^2/(-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^2*d-2*a*c*d-a*b*e-(-4*a*c+b^2)^(1/2)*(-a*e+b*d))*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^2/(-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 = 1.89 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {a \sqrt {d+e x}}{x}+\frac {\sqrt {2} \sqrt {c} \left (b^2 d-2 a c d+b \sqrt {b^2-4 a c} d-a b e-a \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (-b^2 d+2 a c d+b \sqrt {b^2-4 a c} d+a b e-a \sqrt {b^2-4 a c} e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {(2 b d-a e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{a^2} \] Input:

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

Output:

(-((a*Sqrt[d + e*x])/x) + (Sqrt[2]*Sqrt[c]*(b^2*d - 2*a*c*d + b*Sqrt[b^2 - 
 4*a*c]*d - a*b*e - a*Sqrt[b^2 - 4*a*c]*e)*ArcTan[(Sqrt[2]*Sqrt[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^2*d) + 2*a*c 
*d + b*Sqrt[b^2 - 4*a*c]*d + a*b*e - a*Sqrt[b^2 - 4*a*c]*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*b*d - a*e)*Ar 
cTanh[Sqrt[d + e*x]/Sqrt[d]])/Sqrt[d])/a^2
 

Rubi [A] (verified)

Time = 2.46 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.12, 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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(2*(-1/2*(e*Sqrt[d + e*x])/(a*x) + (e^2*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(2 
*a*Sqrt[d]) + (e*(b*d - a*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(a^2*Sqrt[d]) 
 - (Sqrt[c]*e*(b^2*d - 2*a*c*d - a*b*e + Sqrt[b^2 - 4*a*c]*(b*d - a*e))*Ar 
cTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c]) 
*e]])/(Sqrt[2]*a^2*Sqrt[b^2 - 4*a*c]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])* 
e]) + (Sqrt[c]*e*(b^2*d - 2*a*c*d - a*b*e - Sqrt[b^2 - 4*a*c]*(b*d - 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^2*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 = 1.66 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.13

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {4 c \left (\frac {\left (a b \,e^{2}+2 a c d e -d e \,b^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-a b \,e^{2}-2 a c d e +d e \,b^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a^{2} e^{3}}+\frac {-\frac {a \sqrt {e x +d}}{2 x}-\frac {\left (a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{a^{2} e^{3}}\right )\)	\(373\)
default	\(2 e^{3} \left (\frac {4 c \left (\frac {\left (a b \,e^{2}+2 a c d e -d e \,b^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}-\frac {\left (-a b \,e^{2}-2 a c d e +d e \,b^{2}-\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a^{2} e^{3}}+\frac {-\frac {a \sqrt {e x +d}}{2 x}-\frac {\left (a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{a^{2} e^{3}}\right )\)	\(373\)
risch	\(-\frac {\sqrt {e x +d}}{a x}-\frac {e \left (-\frac {\left (-a e +2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{a e \sqrt {d}}+\frac {8 c \left (-\frac {\left (a b \,e^{2}+2 a c d e -d e \,b^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (-a b \,e^{2}-2 a c d e +d e \,b^{2}+\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a e -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e a}\right )}{a}\)	\(376\)
pseudoelliptic	\(\frac {2 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, x \sqrt {2}\, \left (\frac {\left (\sqrt {d}\, a e -b \,d^{\frac {3}{2}}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (\left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}+\frac {b \sqrt {d}\, a e}{2}\right )\right ) c \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+2 \left (x \left (\frac {\left (-\sqrt {d}\, a e +b \,d^{\frac {3}{2}}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+e \left (\left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}+\frac {b \sqrt {d}\, a e}{2}\right )\right ) \sqrt {2}\, c \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )-\frac {\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (x \left (a e -2 b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+a \sqrt {e x +d}\, \sqrt {d}\right )}{2}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}{\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, x \,a^{2}}\)	\(433\)

Input:

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

Output:

2*e^3*(4/a^2/e^3*c*(1/8*(a*b*e^2+2*a*c*d*e-d*e*b^2-(-e^2*(4*a*c-b^2))^(1/2 
)*a*e+(-e^2*(4*a*c-b^2))^(1/2)*b*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*(-a*b*e^2-2*a*c*d*e+d*e 
*b^2-(-e^2*(4*a*c-b^2))^(1/2)*a*e+(-e^2*(4*a*c-b^2))^(1/2)*b*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)*arc 
tanh((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/a^2/e^3*(-1/2*a*(e*x+d)^(1/2)/x-1/2*(a*e-2*b*d)/d^(1/2)*arctanh((e* 
x+d)^(1/2)/d^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2424 vs. \(2 (278) = 556\).

Time = 5.45 (sec) , antiderivative size = 4857, normalized size of antiderivative = 14.72 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 776 vs. \(2 (278) = 556\).

Time = 0.33 (sec) , antiderivative size = 776, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 14.85 (sec) , antiderivative size = 19887, normalized size of antiderivative = 60.26 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((((a*e - 2*b*d)*((8*(d + e*x)^(1/2)*(6*a^4*c^5*e^12 + 4*a^2*c^7*d^4* 
e^8 + 6*a^3*c^6*d^2*e^10 + 4*b^4*c^5*d^4*e^8 + 21*a^2*b^2*c^5*d^2*e^10 - 1 
8*a^3*b*c^5*d*e^11 - 8*a*b^2*c^6*d^4*e^8 - 12*a*b^3*c^5*d^3*e^9))/a^4 - (( 
a*e - 2*b*d)*((8*(16*a^5*b*c^4*e^12 + 20*a^5*c^5*d*e^11 + a^3*b^5*c^2*e^12 
 - 8*a^4*b^3*c^3*e^12 + 20*a^4*c^6*d^3*e^9 + 40*a^2*b^3*c^5*d^4*e^8 - 20*a 
^2*b^4*c^4*d^3*e^9 - 27*a^2*b^5*c^3*d^2*e^10 - 20*a^3*b^2*c^5*d^3*e^9 + 84 
*a^3*b^3*c^4*d^2*e^10 - 8*a*b^5*c^4*d^4*e^8 + 6*a*b^6*c^3*d^3*e^9 + 2*a*b^ 
7*c^2*d^2*e^10 - 3*a^2*b^6*c^2*d*e^11 - 32*a^3*b*c^6*d^4*e^8 + 28*a^3*b^4* 
c^3*d*e^11 - 36*a^4*b*c^5*d^2*e^10 - 68*a^4*b^2*c^4*d*e^11))/a^4 - ((a*e - 
 2*b*d)*((8*(d + e*x)^(1/2)*(60*a^6*b*c^4*e^11 + 16*a^6*c^5*d*e^10 + 5*a^4 
*b^5*c^2*e^11 - 35*a^5*b^3*c^3*e^11 + 40*a^5*c^6*d^3*e^8 - 8*a^2*b^6*c^3*d 
^3*e^8 + 8*a^2*b^7*c^2*d^2*e^9 + 56*a^3*b^4*c^4*d^3*e^8 - 52*a^3*b^5*c^3*d 
^2*e^9 - 108*a^4*b^2*c^5*d^3*e^8 + 68*a^4*b^3*c^4*d^2*e^9 - 12*a^3*b^6*c^2 
*d*e^10 + 87*a^4*b^4*c^3*d*e^10 + 56*a^5*b*c^5*d^2*e^9 - 162*a^5*b^2*c^4*d 
*e^10))/a^4 - (((8*(32*a^8*c^4*e^11 + 2*a^6*b^4*c^2*e^11 - 16*a^7*b^2*c^3* 
e^11 + 32*a^7*c^5*d^2*e^9 + 8*a^5*b^3*c^4*d^3*e^8 - 6*a^5*b^4*c^3*d^2*e^9 
+ 16*a^6*b^2*c^4*d^2*e^9 - 64*a^7*b*c^4*d*e^10 - 2*a^5*b^5*c^2*d*e^10 - 32 
*a^6*b*c^5*d^3*e^8 + 24*a^6*b^3*c^3*d*e^10))/a^4 - (4*(a*e - 2*b*d)*(d + e 
*x)^(1/2)*(64*a^9*c^4*e^10 + 4*a^7*b^4*c^2*e^10 - 32*a^8*b^2*c^3*e^10 + 96 
*a^8*c^5*d^2*e^8 + 8*a^6*b^4*c^3*d^2*e^8 - 56*a^7*b^2*c^4*d^2*e^8 - 112...
 

Reduce [F]

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

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

Output:

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