3.4.58 \(\int \frac {\sqrt {d+e x^2}}{x (a+b x^2+c x^4)} \, dx\) [358]

3.4.58.1 Optimal result

Integrand size = 29, antiderivative size = 281 \[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{a}+\frac {\sqrt {c} \left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {c} \left (b d-\sqrt {b^2-4 a c} d-2 a e\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2} a \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]

output

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

3.4.58.2 Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=-\frac {\frac {\sqrt {2} \sqrt {c} \left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {\sqrt {2} \sqrt {c} \left (-b d+\sqrt {b^2-4 a c} d+2 a e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x^2}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}+2 \sqrt {d} \text {arctanh}\left (\frac {\sqrt {d+e x^2}}{\sqrt {d}}\right )}{2 a} \]

input

Integrate[Sqrt[d + e*x^2]/(x*(a + b*x^2 + c*x^4)),x]
 

output

-1/2*((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^2])/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^2])/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^2]/S 
qrt[d]])/a
 

3.4.58.3 Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1578, 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.

\(\displaystyle \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx\)

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

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

input

Int[Sqrt[d + e*x^2]/(x*(a + b*x^2 + c*x^4)),x]
 

output

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

3.4.58.3.1 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 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 2009

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

3.4.58.4 Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.27

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \left (a \,e^{2}-\frac {b d e}{2}-\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, d}{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\sqrt {2}\, c \left (\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, d}{2}+e \left (a e -\frac {b d}{2}\right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}}{\sqrt {d}}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a}\)	\(358\)
default	\(\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {e \,x^{2}+d}}{x}\right )}{a}-\frac {\sqrt {2}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, c \left (a \,e^{2}-\frac {b d e}{2}-\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, d}{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\sqrt {2}\, c \left (\frac {\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, d}{2}+e \left (a e -\frac {b d}{2}\right )\right ) \arctan \left (\frac {c \sqrt {e \,x^{2}+d}\, \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {e \,x^{2}+d}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\right )}{a \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}\)	\(393\)

input

int((e*x^2+d)^(1/2)/x/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)
 

output

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

3.4.58.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1557 vs. \(2 (232) = 464\).

Time = 49.74 (sec) , antiderivative size = 3126, normalized size of antiderivative = 11.12 \[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]

input

integrate((e*x^2+d)^(1/2)/x/(c*x^4+b*x^2+a),x, algorithm="fricas")
 

output

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

3.4.58.6 Sympy [F]

\[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=\int \frac {\sqrt {d + e x^{2}}}{x \left (a + b x^{2} + c x^{4}\right )}\, dx \]

input

integrate((e*x**2+d)**(1/2)/x/(c*x**4+b*x**2+a),x)
 

output

Integral(sqrt(d + e*x**2)/(x*(a + b*x**2 + c*x**4)), x)
 

3.4.58.7 Maxima [F]

\[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {\sqrt {e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x} \,d x } \]

input

integrate((e*x^2+d)^(1/2)/x/(c*x^4+b*x^2+a),x, algorithm="maxima")
 

output

integrate(sqrt(e*x^2 + d)/((c*x^4 + b*x^2 + a)*x), x)
 

3.4.58.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 724 vs. \(2 (232) = 464\).

Time = 0.31 (sec) , antiderivative size = 724, normalized size of antiderivative = 2.58 \[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=\frac {d \arctan \left (\frac {\sqrt {e x^{2} + d}}{\sqrt {-d}}\right )}{a \sqrt {-d}} - \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} a^{2} d e^{2} - 2 \, {\left (\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}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} {\left | e \right |} - {\left (2 \, a^{2} b c d^{2} e + 2 \, a^{3} b e^{3} - {\left (a^{2} b^{2} + 4 \, a^{3} c\right )} d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c - \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, a c d - a b e + \sqrt {-4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} a c + {\left (2 \, a c d - a b e\right )}^{2}}}{a c}}}\right )}{8 \, {\left (\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}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}} + \frac {{\left (\sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left (b^{2} - 4 \, a c\right )} a^{2} d e^{2} + 2 \, {\left (\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}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e} {\left | a \right |} {\left | e \right |} - {\left (2 \, a^{2} b c d^{2} e + 2 \, a^{3} b e^{3} - {\left (a^{2} b^{2} + 4 \, a^{3} c\right )} d e^{2}\right )} \sqrt {-4 \, c^{2} d + 2 \, {\left (b c + \sqrt {b^{2} - 4 \, a c} c\right )} e}\right )} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \sqrt {e x^{2} + d}}{\sqrt {-\frac {2 \, a c d - a b e - \sqrt {-4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} a c + {\left (2 \, a c d - a b e\right )}^{2}}}{a c}}}\right )}{8 \, {\left (\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}\right )} {\left | a \right |} {\left | c \right |} {\left | e \right |}} \]

input

integrate((e*x^2+d)^(1/2)/x/(c*x^4+b*x^2+a),x, algorithm="giac")
 

output

d*arctan(sqrt(e*x^2 + d)/sqrt(-d))/(a*sqrt(-d)) - 1/8*(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^2 + 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)) + 1/8*(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^2 + 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))
 

3.4.58.9 Mupad [B] (verification not implemented)

Time = 11.49 (sec) , antiderivative size = 10964, normalized size of antiderivative = 39.02 \[ \int \frac {\sqrt {d+e x^2}}{x \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]

input

int((d + e*x^2)^(1/2)/(x*(a + b*x^2 + c*x^4)),x)
 

output

atan((((d + e*x^2)^(1/2)*(2*a^2*c^3*e^12 + 6*c^5*d^4*e^8 - 8*b*c^4*d^3*e^9 
 + 4*b^2*c^3*d^2*e^10 - 4*a*b*c^3*d*e^11) + ((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)/(8*(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)/(8*(a^2*b^4 + 16*a^4*c^2 - 8*a 
^3*b^2*c)))^(1/2)*((d + e*x^2)^(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)/(8*(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 + 6 
4*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) - 192*a^4*c^4*d*e^10 - 192*a^3 
*c^5*d^3*e^8 + 48*a^2*b^2*c^4*d^3*e^8 - 48*a^2*b^3*c^3*d^2*e^9 + 192*a^3*b 
*c^4*d^2*e^9 + 48*a^3*b^2*c^3*d*e^10) - (d + e*x^2)^(1/2)*(32*a^3*b*c^3*e^ 
11 + 48*a^3*c^4*d*e^10 - 8*a^2*b^3*c^2*e^11 + 144*a^2*c^5*d^3*e^8 + 16*b^4 
*c^3*d^3*e^8 - 16*b^5*c^2*d^2*e^9 + 16*a*b^4*c^2*d*e^10 - 96*a*b^2*c^4*d^3 
*e^8 + 96*a*b^3*c^3*d^2*e^9 - 144*a^2*b*c^4*d^2*e^9 - 72*a^2*b^2*c^3*d*e^1 
0))*((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)/(8*(a^2*b^4 + 16*a^4* 
c^2 - 8*a^3*b^2*c)))^(1/2) + 12*a*c^5*d^4*e^8 + 12*a^2*c^4*d^2*e^10 - 4...