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

Optimal. Leaf size=291 \[ -\frac{c \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x^2}}{a x} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.670755, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {1295, 264, 1692, 377, 205} \[ -\frac{c \left (\frac{b d-2 a e}{\sqrt{b^2-4 a c}}+d\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-e \left (b-\sqrt{b^2-4 a c}\right )}}-\frac{c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}{\sqrt{\sqrt{b^2-4 a c}+b} \sqrt{d+e x^2}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}-\frac{\sqrt{d+e x^2}}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1295

Int[(((f_.)*(x_))^(m_)*((d_.) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Dist[d/
a, Int[(f*x)^m*(d + e*x^2)^(q - 1), x], x] - Dist[1/(a*f^2), Int[((f*x)^(m + 2)*(d + e*x^2)^(q - 1)*Simp[b*d -
a*e + c*d*x^2, x])/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] &&  !In
tegerQ[q] && GtQ[q, 0] && LtQ[m, 0]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 1692

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandInteg
rand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x^2] && NeQ[b
^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x^2}}{x^2 \left (a+b x^2+c x^4\right )} \, dx &=-\frac{\int \frac{b d-a e+c d x^2}{\sqrt{d+e x^2} \left (a+b x^2+c x^4\right )} \, dx}{a}+\frac{d \int \frac{1}{x^2 \sqrt{d+e x^2}} \, dx}{a}\\ &=-\frac{\sqrt{d+e x^2}}{a x}-\frac{\int \left (\frac{c d+\frac{c (b d-2 a e)}{\sqrt{b^2-4 a c}}}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}}+\frac{c d-\frac{c (b d-2 a e)}{\sqrt{b^2-4 a c}}}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}}\right ) \, dx}{a}\\ &=-\frac{\sqrt{d+e x^2}}{a x}-\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\left (b+\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{a}-\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \int \frac{1}{\left (b-\sqrt{b^2-4 a c}+2 c x^2\right ) \sqrt{d+e x^2}} \, dx}{a}\\ &=-\frac{\sqrt{d+e x^2}}{a x}-\frac{\left (c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b+\sqrt{b^2-4 a c}-\left (-2 c d+\left (b+\sqrt{b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{a}-\frac{\left (c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b-\sqrt{b^2-4 a c}-\left (-2 c d+\left (b-\sqrt{b^2-4 a c}\right ) e\right ) x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )}{a}\\ &=-\frac{\sqrt{d+e x^2}}{a x}-\frac{c \left (d+\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e} x}{\sqrt{b-\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{2 c d-\left (b-\sqrt{b^2-4 a c}\right ) e}}-\frac{c \left (d-\frac{b d-2 a e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} x}{\sqrt{b+\sqrt{b^2-4 a c}} \sqrt{d+e x^2}}\right )}{a \sqrt{b+\sqrt{b^2-4 a c}} \sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}}\\ \end{align*}

Mathematica [B]  time = 6.33564, size = 4644, normalized size = 15.96 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[d + e*x^2]/(a*x)) - (-(b*d*(Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) - Sqr
t[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqr
t[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(
b/c) - Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c
]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (Sq
rt[b^2 - 4*a*c]*d*(Log[Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c
)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c
)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2
- 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sq
rt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (a*e*(Log[Sqrt[-(b/
c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt
[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d
+ ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[
b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/S
qrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (b*d*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2
]) + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*
e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/
c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[
-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 -
4*a*c]/c]/Sqrt[2])) - (Sqrt[b^2 - 4*a*c]*d*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((
-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((
-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[
2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 -
4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]))
- (a*e*(Log[-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]
- Log[2*d + Sqrt[2]*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]
*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) - Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/
c]*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sq
rt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (b*d*(Log[Sqrt[-(b/c) + Sqrt[b^2 -
4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) + Sqrt[
b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sq
rt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/
c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-
(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (Sqrt[b^2 - 4*a*c]*d*(Log[Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]
+ x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x
+ 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*
e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[
-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 -
4*a*c]/c]/Sqrt[2])) + (a*e*(Log[Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 -
4*a*c]/c)*e)/2] - Log[2*d - Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 -
4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(Sqrt[2]*c*Sqrt[-(b/c) + Sq
rt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]
)*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] - Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (b*d*(Log[-(Sqr
t[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[
2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/
Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c
) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*
c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) + (Sqrt[b^2 - 4*a*c]*d*(Log[-(Sqrt[-(b/c) + Sqrt[
b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) +
Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c
) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))/(2*Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4
*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] +
Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])) - (a*e*(Log[-(Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + x]/Sq
rt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2] - Log[2*d + Sqrt[2]*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*e*x + 2*Sq
rt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]*Sqrt[d + e*x^2]]/Sqrt[d + ((-(b/c) + Sqrt[b^2 - 4*a*c]/c)*e)/2]))
/(Sqrt[2]*c*Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]*(-(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2]) + Sqrt[-(b/c) +
Sqrt[b^2 - 4*a*c]/c]/Sqrt[2])*(Sqrt[-(b/c) - Sqrt[b^2 - 4*a*c]/c]/Sqrt[2] + Sqrt[-(b/c) + Sqrt[b^2 - 4*a*c]/c]
/Sqrt[2])))/a

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Maple [C]  time = 0.026, size = 272, normalized size = 0.9 \begin{align*}{\frac{1}{a}\sqrt{e}\ln \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) }+{\frac{1}{2\,a}\sqrt{e}\sum _{{\it \_R}={\it RootOf} \left ( c{{\it \_Z}}^{4}+ \left ( 4\,be-4\,cd \right ){{\it \_Z}}^{3}+ \left ( 16\,a{e}^{2}-8\,deb+6\,c{d}^{2} \right ){{\it \_Z}}^{2}+ \left ( 4\,b{d}^{2}e-4\,c{d}^{3} \right ){\it \_Z}+c{d}^{4} \right ) }{\frac{{{\it \_R}}^{2}cd+2\, \left ( -2\,a{e}^{2}+2\,deb-c{d}^{2} \right ){\it \_R}+c{d}^{3}}{{{\it \_R}}^{3}c+3\,{{\it \_R}}^{2}be-3\,{{\it \_R}}^{2}cd+8\,{\it \_R}\,a{e}^{2}-4\,{\it \_R}\,bde+3\,{\it \_R}\,c{d}^{2}+b{d}^{2}e-c{d}^{3}}\ln \left ( \left ( \sqrt{e{x}^{2}+d}-\sqrt{e}x \right ) ^{2}-{\it \_R} \right ) }}-{\frac{1}{adx} \left ( e{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+{\frac{ex}{ad}\sqrt{e{x}^{2}+d}}+{\frac{1}{a}\sqrt{e}\ln \left ( \sqrt{e}x+\sqrt{e{x}^{2}+d} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*e^(1/2)*ln((e*x^2+d)^(1/2)-e^(1/2)*x)+1/2/a*e^(1/2)*sum((_R^2*c*d+2*(-2*a*e^2+2*b*d*e-c*d^2)*_R+c*d^3)/(_R
^3*c+3*_R^2*b*e-3*_R^2*c*d+8*_R*a*e^2-4*_R*b*d*e+3*_R*c*d^2+b*d^2*e-c*d^3)*ln(((e*x^2+d)^(1/2)-e^(1/2)*x)^2-_R
),_R=RootOf(c*_Z^4+(4*b*e-4*c*d)*_Z^3+(16*a*e^2-8*b*d*e+6*c*d^2)*_Z^2+(4*b*d^2*e-4*c*d^3)*_Z+c*d^4))-1/a/d/x*(
e*x^2+d)^(3/2)+1/a*e/d*x*(e*x^2+d)^(1/2)+1/a*e^(1/2)*ln(e^(1/2)*x+(e*x^2+d)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x^{2} + d}}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 19.6206, size = 4811, normalized size = 16.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(1/2)*a*x*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2*e^2 + (
b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a^
2*b*c*d*e + (a^3*b^2*c - 4*a^4*c^2)*d*x^2*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2
*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - 2*(a*b^2*c - a^2*c^2)*d^2 + (4*a^2*b*c*e^2 + (b^3*c - a*b*c^2)*d^2 - (5*a*b^
2*c - 4*a^2*c^2)*d*e)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((a^4*b^3 - 4*a^5*b*c)*x*sqrt((a^2*b^2*e^2 + (b^4 - 2*
a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - ((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)*d
- (a^2*b^3 - 4*a^3*b*c)*e)*x)*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b
^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))
)/x^2) - sqrt(1/2)*a*x*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2*e^2
+ (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2
*a^2*b*c*d*e + (a^3*b^2*c - 4*a^4*c^2)*d*x^2*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 -
a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - 2*(a*b^2*c - a^2*c^2)*d^2 + (4*a^2*b*c*e^2 + (b^3*c - a*b*c^2)*d^2 - (5*a
*b^2*c - 4*a^2*c^2)*d*e)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((a^4*b^3 - 4*a^5*b*c)*x*sqrt((a^2*b^2*e^2 + (b^4 -
2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - ((a*b^4 - 5*a^2*b^2*c + 4*a^3*c^2)
*d - (a^2*b^3 - 4*a^3*b*c)*e)*x)*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e + (a^3*b^2 - 4*a^4*c)*sqrt((a^
2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*
c)))/x^2) - sqrt(1/2)*a*x*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e - (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2*e
^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log
((2*a^2*b*c*d*e - (a^3*b^2*c - 4*a^4*c^2)*d*x^2*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3
- a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - 2*(a*b^2*c - a^2*c^2)*d^2 + (4*a^2*b*c*e^2 + (b^3*c - a*b*c^2)*d^2 - (
5*a*b^2*c - 4*a^2*c^2)*d*e)*x^2 + 2*sqrt(1/2)*sqrt(e*x^2 + d)*((a^4*b^3 - 4*a^5*b*c)*x*sqrt((a^2*b^2*e^2 + (b^
4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) + ((a*b^4 - 5*a^2*b^2*c + 4*a^3*c
^2)*d - (a^2*b^3 - 4*a^3*b*c)*e)*x)*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e - (a^3*b^2 - 4*a^4*c)*sqrt(
(a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a
^4*c)))/x^2) + sqrt(1/2)*a*x*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e - (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^
2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*
log((2*a^2*b*c*d*e - (a^3*b^2*c - 4*a^4*c^2)*d*x^2*sqrt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*
b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) - 2*(a*b^2*c - a^2*c^2)*d^2 + (4*a^2*b*c*e^2 + (b^3*c - a*b*c^2)*d^2
- (5*a*b^2*c - 4*a^2*c^2)*d*e)*x^2 - 2*sqrt(1/2)*sqrt(e*x^2 + d)*((a^4*b^3 - 4*a^5*b*c)*x*sqrt((a^2*b^2*e^2 +
(b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)) + ((a*b^4 - 5*a^2*b^2*c + 4*a^
3*c^2)*d - (a^2*b^3 - 4*a^3*b*c)*e)*x)*sqrt(-((b^3 - 3*a*b*c)*d - (a*b^2 - 2*a^2*c)*e - (a^3*b^2 - 4*a^4*c)*sq
rt((a^2*b^2*e^2 + (b^4 - 2*a*b^2*c + a^2*c^2)*d^2 - 2*(a*b^3 - a^2*b*c)*d*e)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 -
4*a^4*c)))/x^2) + 4*sqrt(e*x^2 + d))/(a*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x^{2}}}{x^{2} \left (a + b x^{2} + c x^{4}\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError