3.384 \(\int \frac{\sqrt{1-x^2}}{x^2 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=265 \[ -\frac{c \left (\frac{2 a+b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{c \left (1-\frac{2 a+b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{1-x^2}}{a x} \]

[Out]

-(Sqrt[1 - x^2]/(a*x)) - (c*(1 + (2*a + b)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*x)/(Sq
rt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/(a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) -
(c*(1 - (2*a + b)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]
]*Sqrt[1 - x^2])])/(a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.779562, antiderivative size = 265, 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} \[ -\frac{c \left (\frac{2 a+b}{\sqrt{b^2-4 a c}}+1\right ) \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{a \sqrt{b-\sqrt{b^2-4 a c}} \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}-\frac{c \left (1-\frac{2 a+b}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{a \sqrt{\sqrt{b^2-4 a c}+b} \sqrt{\sqrt{b^2-4 a c}+b+2 c}}-\frac{\sqrt{1-x^2}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - x^2]/(x^2*(a + b*x^2 + c*x^4)),x]

[Out]

-(Sqrt[1 - x^2]/(a*x)) - (c*(1 + (2*a + b)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]*x)/(Sq
rt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[1 - x^2])])/(a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c - Sqrt[b^2 - 4*a*c]]) -
(c*(1 - (2*a + b)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]
]*Sqrt[1 - x^2])])/(a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[b + 2*c + Sqrt[b^2 - 4*a*c]])

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

Mathematica [B]  time = 5.47949, size = 2661, normalized size = 10.04 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[1 - x^2]/(x^2*(a + b*x^2 + c*x^4)),x]

[Out]

-(4*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*Sqrt[-((b^2
+ c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*Sqrt[1 - x^2] + Sqrt[2]*(2*a + b + Sqrt[b^2
- 4*a*c])*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[-(Sqrt[(-b + Sqr
t[b^2 - 4*a*c])/c]/Sqrt[2]) + x] - Sqrt[2]*(2*a + b + Sqrt[b^2 - 4*a*c])*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*
a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]/Sqrt[2] + x] - 2*Sqrt[2]*a*Sqr
t[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c]
)/c)]/Sqrt[2]) + x] - Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*
Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[
b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[-(Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2]) + x] + 2*
Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[Sqrt[-((b + Sqrt[b
^2 - 4*a*c])/c)]/Sqrt[2] + x] + Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]
))/c^2]*x*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x] - Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a
+ Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]/Sqrt[2] + x]
- 2*Sqrt[2]*a*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 - Sqrt[2]*
Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*b*
Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 - Sqrt[2]*Sqrt[(-b + Sqr
t[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*Sqrt[b^2 - 4*a*c
]*Sqrt[-((b^2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 - Sqrt[2]*Sqrt[(-b + S
qrt[b^2 - 4*a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + 2*Sqrt[2]*a*Sqrt[-((b^
2 + c*(-2*a + Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*
a*c])/c]*x + Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*b*Sqrt[-((b^2 + c*(-2*a +
Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x + S
qrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[-((b^2 + c*(-2*a
+ Sqrt[b^2 - 4*a*c]) + b*(c + Sqrt[b^2 - 4*a*c]))/c^2)]*x*Log[2 + Sqrt[2]*Sqrt[(-b + Sqrt[b^2 - 4*a*c])/c]*x +
Sqrt[2]*Sqrt[(b + 2*c - Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + 2*Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4
*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqr
t[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] + Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c
+ Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + S
qrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(
-c + Sqrt[b^2 - 4*a*c]))/c^2]*x*Log[2 - Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c +
Sqrt[b^2 - 4*a*c])/c]*Sqrt[1 - x^2]] - 2*Sqrt[2]*a*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^
2 - 4*a*c]))/c^2]*x*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 -
4*a*c])/c]*Sqrt[1 - x^2]] - Sqrt[2]*b*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))/c
^2]*x*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*Sqr
t[1 - x^2]] + Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[(-b^2 + c*(2*a + Sqrt[b^2 - 4*a*c]) + b*(-c + Sqrt[b^2 - 4*a*c]))
/c^2]*x*Log[2 + Sqrt[2]*Sqrt[-((b + Sqrt[b^2 - 4*a*c])/c)]*x + Sqrt[2]*Sqrt[(b + 2*c + Sqrt[b^2 - 4*a*c])/c]*S
qrt[1 - x^2]])/(2*a*Sqrt[b^2 - 4*a*c]*Sqrt[((b + 2*c - Sqrt[b^2 - 4*a*c])*(-b + Sqrt[b^2 - 4*a*c]))/c^2]*Sqrt[
-(((b + Sqrt[b^2 - 4*a*c])*(b + 2*c + Sqrt[b^2 - 4*a*c]))/c^2)]*x)

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Maple [C]  time = 0.023, size = 217, normalized size = 0.8 \begin{align*}{\frac{1}{4\,a}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a+8\,b+16\,c \right ){{\it \_Z}}^{4}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{2}+a \right ) }{\frac{ \left ( a+b \right ){{\it \_R}}^{6}+ \left ( 3\,a+3\,b+4\,c \right ){{\it \_R}}^{4}+ \left ( 3\,a+3\,b+4\,c \right ){{\it \_R}}^{2}+a+b}{{{\it \_R}}^{7}a+3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{3}b+8\,{{\it \_R}}^{3}c+{\it \_R}\,a+{\it \_R}\,b}\ln \left ({\frac{1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-{\it \_R} \right ) }}-2\,{\frac{1}{a}\arctan \left ({\frac{\sqrt{-{x}^{2}+1}-1}{x}} \right ) }-{\frac{1}{ax} \left ( -{x}^{2}+1 \right ) ^{{\frac{3}{2}}}}-{\frac{x}{a}\sqrt{-{x}^{2}+1}}-{\frac{\arcsin \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/a*sum(((a+b)*_R^6+(3*a+3*b+4*c)*_R^4+(3*a+3*b+4*c)*_R^2+a+b)/(_R^7*a+3*_R^5*a+3*_R^5*b+3*_R^3*a+4*_R^3*b+8
*_R^3*c+_R*a+_R*b)*ln(((-x^2+1)^(1/2)-1)/x-_R),_R=RootOf(a*_Z^8+(4*a+4*b)*_Z^6+(6*a+8*b+16*c)*_Z^4+(4*a+4*b)*_
Z^2+a))-2/a*arctan(((-x^2+1)^(1/2)-1)/x)-1/a/x*(-x^2+1)^(3/2)-1/a*x*(-x^2+1)^(1/2)-1/a*arcsin(x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-x^{2} + 1}}{{\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((-x^2+1)^(1/2)/x^2/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(-x^2 + 1)/((c*x^4 + b*x^2 + a)*x^2), x)

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Fricas [B]  time = 3.45911, size = 4091, normalized size = 15.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(1/2)*a*x*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4
+ a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b + b
^2)*c)*x^2 - 2*(a*b + b^2)*c + sqrt(1/2)*((a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*sqrt(-x^2 + 1)*x -
(a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*x - ((a^3*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*
a^4*b*c)*x)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(a*b^2
+ b^3 - (2*a^2 + 3*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)
/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) - 2*(a*c^2 - (a*b + b^2)*c)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*a*x*s
qrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b
+ a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b + b^2)*c)*x^2 - 2*(a*b
+ b^2)*c - sqrt(1/2)*((a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*sqrt(-x^2 + 1)*x - (a*b^3 + b^4 + 4*a^
2*c^2 - (4*a^2*b + 5*a*b^2)*c)*x - ((a^3*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*a^4*b*c)*x)*sqrt((a^
2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a
*b)*c + (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)
))/(a^3*b^2 - 4*a^4*c)) - 2*(a*c^2 - (a*b + b^2)*c)*sqrt(-x^2 + 1))/x^2) + sqrt(1/2)*a*x*sqrt(-(a*b^2 + b^3 -
(2*a^2 + 3*a*b)*c - (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^
2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b + b^2)*c)*x^2 - 2*(a*b + b^2)*c + sqrt(1/2)
*((a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*sqrt(-x^2 + 1)*x - (a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5
*a*b^2)*c)*x + ((a^3*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*a^4*b*c)*x)*sqrt((a^2*b^2 + 2*a*b^3 + b^
4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c - (a^3*b^2 - 4
*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*
c)) - 2*(a*c^2 - (a*b + b^2)*c)*sqrt(-x^2 + 1))/x^2) - sqrt(1/2)*a*x*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c -
(a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*
b^2 - 4*a^4*c))*log((2*a*c^2 - 2*(a*c^2 - (a*b + b^2)*c)*x^2 - 2*(a*b + b^2)*c - sqrt(1/2)*((a*b^3 + b^4 + 4*a
^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*sqrt(-x^2 + 1)*x - (a*b^3 + b^4 + 4*a^2*c^2 - (4*a^2*b + 5*a*b^2)*c)*x + ((a^3
*b^3 - 4*a^4*b*c)*sqrt(-x^2 + 1)*x - (a^3*b^3 - 4*a^4*b*c)*x)*sqrt((a^2*b^2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2
*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))*sqrt(-(a*b^2 + b^3 - (2*a^2 + 3*a*b)*c - (a^3*b^2 - 4*a^4*c)*sqrt((a^2*b^
2 + 2*a*b^3 + b^4 + a^2*c^2 - 2*(a^2*b + a*b^2)*c)/(a^6*b^2 - 4*a^7*c)))/(a^3*b^2 - 4*a^4*c)) - 2*(a*c^2 - (a*
b + b^2)*c)*sqrt(-x^2 + 1))/x^2) - 2*sqrt(-x^2 + 1))/(a*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{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((-x**2+1)**(1/2)/x**2/(c*x**4+b*x**2+a),x)

[Out]

Integral(sqrt(-(x - 1)*(x + 1))/(x**2*(a + b*x**2 + c*x**4)), x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out