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]

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

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Rubi [A]  time = 0.78, antiderivative size = 265, normalized size of antiderivative = 1.00, 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 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]

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 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 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 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]

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*}

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Mathematica [B]  time = 4.95, 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]

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

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fricas [B]  time = 1.86, size = 1998, normalized size = 7.54 \[ \text {result too large to display} \]

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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giac [B]  time = 5.04, size = 3965, normalized size = 14.96 \[ \text {result too large to display} \]

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]

1/8*(4*a^6*b^3 + 6*a^5*b^4 + 2*a^4*b^5 - 16*a^7*b*c - 32*a^6*b^2*c - 12*a^5*b^3*c + 32*a^7*c^2 + 16*a^6*b*c^2
+ 6*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^6*b + 13*sqrt(2)*sqrt(2*a^2 + a*b - sq
rt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^5*b^2 + 7*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*
a*c)*a^4*b^3 - sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*b^4 - sqrt(2)*sqrt(2*a^2
+ a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sq
rt(b^2 - 4*a*c)*a^6*c - 6*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^5*b*c + 12*sqrt(
2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^4*b^2*c + 6*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2
- 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*b^3*c - 16*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c
)*a^5*c^2 - 8*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^4*b*c^2 - 4*(b^2 - 4*a*c)*a^
6*b - 6*(b^2 - 4*a*c)*a^5*b^2 - 2*(b^2 - 4*a*c)*a^4*b^3 + 8*(b^2 - 4*a*c)*a^6*c + 4*(b^2 - 4*a*c)*a^5*b*c - (2
*a^3*b^4 + 2*a^2*b^5 - 16*a^4*b^2*c - 16*a^3*b^3*c + 32*a^5*c^2 + 32*a^4*b*c^2 + 3*sqrt(2)*sqrt(2*a^2 + a*b -
sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*b^2 + 5*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 -
4*a*c)*a^2*b^3 + sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^4 - sqrt(2)*sqrt(2*a^2
+ a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b
^2 - 4*a*c)*a^4*c - 20*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*b*c + 8*sqrt(2)*s
qrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^3*c - 16*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*
a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*c^2 - 16*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*b
*c^2 - 2*(b^2 - 4*a*c)*a^3*b^2 - 2*(b^2 - 4*a*c)*a^2*b^3 + 8*(b^2 - 4*a*c)*a^4*c + 8*(b^2 - 4*a*c)*a^3*b*c)*a^
2 + 2*(3*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^5*b^2 + 5*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a
*c)*a)*a^4*b^3 + sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^3*b^4 - 2*a^4*b^4 - sqrt(2)*sqrt(2*a^2 + a*
b - sqrt(b^2 - 4*a*c)*a)*a^2*b^5 - 2*a^3*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^6*c - 20*s
qrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^5*b*c + 3*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^4
*b^2*c + 16*a^5*b^2*c + 10*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^3*b^3*c + 16*a^4*b^3*c - sqrt(2)*
sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^2*b^4*c - 2*a^3*b^4*c - 28*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a
*c)*a)*a^5*c^2 - 32*a^6*c^2 - 24*sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^4*b*c^2 - 32*a^5*b*c^2 + 8*
sqrt(2)*sqrt(2*a^2 + a*b - sqrt(b^2 - 4*a*c)*a)*a^3*b^2*c^2 + 16*a^4*b^2*c^2 - 16*sqrt(2)*sqrt(2*a^2 + a*b - s
qrt(b^2 - 4*a*c)*a)*a^4*c^3 - 32*a^5*c^3 + 2*(b^2 - 4*a*c)*a^4*b^2 + 2*(b^2 - 4*a*c)*a^3*b^3 - 8*(b^2 - 4*a*c)
*a^5*c - 8*(b^2 - 4*a*c)*a^4*b*c + 2*(b^2 - 4*a*c)*a^3*b^2*c - 8*(b^2 - 4*a*c)*a^4*c^2)*abs(a))*arctan(-1/2*sq
rt(2)*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/sqrt((2*a^2 + a*b + sqrt(-4*(a^2 + a*b + a*c)*a^2 + (2
*a^2 + a*b)^2))/a^2))/(3*a^8*b^2 + 5*a^7*b^3 + a^6*b^4 - a^5*b^5 - 12*a^9*c - 20*a^8*b*c + 3*a^7*b^2*c + 10*a^
6*b^3*c - a^5*b^4*c - 28*a^8*c^2 - 24*a^7*b*c^2 + 8*a^6*b^2*c^2 - 16*a^7*c^3) - 1/8*(4*a^6*b^3 + 6*a^5*b^4 + 2
*a^4*b^5 - 16*a^7*b*c - 32*a^6*b^2*c - 12*a^5*b^3*c + 32*a^7*c^2 + 16*a^6*b*c^2 + 6*sqrt(2)*sqrt(2*a^2 + a*b +
sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^6*b + 13*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 -
4*a*c)*a^5*b^2 + 7*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^4*b^3 - sqrt(2)*sqrt(2*
a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*b^4 - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*s
qrt(b^2 - 4*a*c)*a^2*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^6*c - 6*sqrt
(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^5*b*c + 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2
- 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^4*b^2*c + 6*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)
*a^3*b^3*c - 16*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^5*c^2 - 8*sqrt(2)*sqrt(2*a
^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^4*b*c^2 - 4*(b^2 - 4*a*c)*a^6*b - 6*(b^2 - 4*a*c)*a^5*b^2
- 2*(b^2 - 4*a*c)*a^4*b^3 + 8*(b^2 - 4*a*c)*a^6*c + 4*(b^2 - 4*a*c)*a^5*b*c - (2*a^3*b^4 + 2*a^2*b^5 - 16*a^4*
b^2*c - 16*a^3*b^3*c + 32*a^5*c^2 + 32*a^4*b*c^2 + 3*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2
- 4*a*c)*a^3*b^2 + 5*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*b^3 + sqrt(2)*sqrt(
2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^4 - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*s
qrt(b^2 - 4*a*c)*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^4*c - 20*sqrt(2)
*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*b*c + 8*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4
*a*c)*a)*sqrt(b^2 - 4*a*c)*a*b^3*c - 16*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^3*
c^2 - 16*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*sqrt(b^2 - 4*a*c)*a^2*b*c^2 - 2*(b^2 - 4*a*c)*a^3*b^2
- 2*(b^2 - 4*a*c)*a^2*b^3 + 8*(b^2 - 4*a*c)*a^4*c + 8*(b^2 - 4*a*c)*a^3*b*c)*a^2 - 2*(3*sqrt(2)*sqrt(2*a^2 +
a*b + sqrt(b^2 - 4*a*c)*a)*a^5*b^2 + 5*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^4*b^3 + sqrt(2)*sqrt(
2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^3*b^4 + 2*a^4*b^4 - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^2*b
^5 + 2*a^3*b^5 - 12*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^6*c - 20*sqrt(2)*sqrt(2*a^2 + a*b + sqrt
(b^2 - 4*a*c)*a)*a^5*b*c + 3*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^4*b^2*c - 16*a^5*b^2*c + 10*sqr
t(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^3*b^3*c - 16*a^4*b^3*c - sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 -
4*a*c)*a)*a^2*b^4*c + 2*a^3*b^4*c - 28*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^5*c^2 + 32*a^6*c^2 -
24*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^4*b*c^2 + 32*a^5*b*c^2 + 8*sqrt(2)*sqrt(2*a^2 + a*b + sqr
t(b^2 - 4*a*c)*a)*a^3*b^2*c^2 - 16*a^4*b^2*c^2 - 16*sqrt(2)*sqrt(2*a^2 + a*b + sqrt(b^2 - 4*a*c)*a)*a^4*c^3 +
32*a^5*c^3 - 2*(b^2 - 4*a*c)*a^4*b^2 - 2*(b^2 - 4*a*c)*a^3*b^3 + 8*(b^2 - 4*a*c)*a^5*c + 8*(b^2 - 4*a*c)*a^4*b
*c - 2*(b^2 - 4*a*c)*a^3*b^2*c + 8*(b^2 - 4*a*c)*a^4*c^2)*abs(a))*arctan(-1/2*sqrt(2)*(x/(sqrt(-x^2 + 1) - 1)
- (sqrt(-x^2 + 1) - 1)/x)/sqrt((2*a^2 + a*b - sqrt(-4*(a^2 + a*b + a*c)*a^2 + (2*a^2 + a*b)^2))/a^2))/(3*a^8*b
^2 + 5*a^7*b^3 + a^6*b^4 - a^5*b^5 - 12*a^9*c - 20*a^8*b*c + 3*a^7*b^2*c + 10*a^6*b^3*c - a^5*b^4*c - 28*a^8*c
^2 - 24*a^7*b*c^2 + 8*a^6*b^2*c^2 - 16*a^7*c^3) + 1/2*(x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/a

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maple [C]  time = 0.03, size = 217, normalized size = 0.82 \[ -\frac {\sqrt {-x^{2}+1}\, x}{a}-\frac {\arcsin \relax (x )}{a}-\frac {2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )}{a}+\frac {\left (\left (a +b \right ) \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{6}+\left (3 a +3 b +4 c \right ) \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{4}+\left (3 a +3 b +4 c \right ) \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{2}+a +b \right ) \ln \left (-\RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )+\frac {\sqrt {-x^{2}+1}-1}{x}\right )}{4 a \left (\RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{7} a +3 \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{5} a +3 \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{5} b +3 \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{3} a +4 \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{3} b +8 \RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right )^{3} c +\RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right ) a +\RootOf \left (\textit {\_Z}^{8} a +\left (4 a +4 b \right ) \textit {\_Z}^{6}+\left (6 a +8 b +16 c \right ) \textit {\_Z}^{4}+\left (4 a +4 b \right ) \textit {\_Z}^{2}+a \right ) b \right )}-\frac {\left (-x^{2}+1\right )^{\frac {3}{2}}}{a x} \]

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/a/x*(-x^2+1)^(3/2)-1/a*x*(-x^2+1)^(1/2)-1/a*arcsin(x)+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(-_R+((-x^2+1)^(1/2)-1)/x),_R=Ro
otOf(_Z^8*a+(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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-x^{2} + 1}}{{\left (c x^{4} + b x^{2} + a\right )} x^{2}}\,{d x} \]

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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mupad [B]  time = 1.21, size = 1234, normalized size = 4.66 \[ -\frac {\sqrt {1-x^2}}{a\,x}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b^3\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b^2\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a^2\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c^2\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b^2\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}-3\,a\,b\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a^2\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b^3\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b^2\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a^2\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c^2\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b^2\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}-3\,a\,b\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a^2\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b^3\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b^2\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a^2\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c^2\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b^2\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}-3\,a\,b\,c\,\sqrt {-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,c\,{\left (-\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a^2\,\left (4\,a\,c-b^2\right )\,\sqrt {\frac {b+\sqrt {b^2-4\,a\,c}}{2\,c}+1}}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}}\right )\,\left (b^3\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b^2\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a^2\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}-2\,a\,c^2\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}+b^2\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}-3\,a\,b\,c\,\sqrt {-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}}+a\,b\,c\,{\left (-\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}\right )}^{3/2}\right )}{2\,a^2\,\sqrt {\frac {b-\sqrt {b^2-4\,a\,c}}{2\,c}+1}\,\left (4\,a\,c-b^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((((x*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 -
x^2)^(1/2)*1i)/(x - (-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b^3*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)
+ a*b^2*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a^2*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c^2*
(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b^2*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) - 3*a*b*c*(-(b + (b^
2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*a^2*(4*a*c - b^2)*((b + (
b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)) - (1 - x^2)^(1/2)/(a*x) + (log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(
1/2) - 1)*1i)/((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) - (1 - x^2)^(1/2)*1i)/(x - (-(b - (b^2 - 4*a*c)^(1/2
))/(2*c))^(1/2)))*(b^3*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/
2) - 2*a^2*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b^2
*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) - 3*a*b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b*c*(-(b -
(b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*a^2*((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(4*a*c - b^2)) - (log((
((x*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 1)*1i)/((b + (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^
(1/2)*1i)/(x + (-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b^3*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b^
2*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a^2*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a*c^2*(-(b +
(b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b^2*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) - 3*a*b*c*(-(b + (b^2 - 4*
a*c)^(1/2))/(2*c))^(1/2) + a*b*c*(-(b + (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2)))/(2*a^2*(4*a*c - b^2)*((b + (b^2 -
4*a*c)^(1/2))/(2*c) + 1)^(1/2)) - (log((((x*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + 1)*1i)/((b - (b^2 - 4*a
*c)^(1/2))/(2*c) + 1)^(1/2) + (1 - x^2)^(1/2)*1i)/(x + (-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2)))*(b^3*(-(b -
(b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) - 2*a^2*c*(-(b - (b^2 - 4*a
*c)^(1/2))/(2*c))^(1/2) - 2*a*c^2*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3/2) + b^2*c*(-(b - (b^2 - 4*a*c)^(1/2))
/(2*c))^(3/2) - 3*a*b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(1/2) + a*b*c*(-(b - (b^2 - 4*a*c)^(1/2))/(2*c))^(3
/2)))/(2*a^2*((b - (b^2 - 4*a*c)^(1/2))/(2*c) + 1)^(1/2)*(4*a*c - b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{x^{2} \left (a + b x^{2} + c x^{4}\right )}\, dx \]

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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