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

Optimal. Leaf size=220 \[ \frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

(Sqrt[b + 2*c - 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])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b
^2 - 4*a*c]]) - (Sqrt[b + 2*c + 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])])/(Sqrt[b^2 - 4*a*c]
*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.725753, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{-\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \sqrt{b-\sqrt{b^2-4 a c}}}-\frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tan ^{-1}\left (\frac{x \sqrt{\sqrt{b^2-4 a c}+b+2 c}}{\sqrt{1-x^2} \sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{b^2-4 a c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[b + 2*c - 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])])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b
^2 - 4*a*c]]) - (Sqrt[b + 2*c + 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])])/(Sqrt[b^2 - 4*a*c]
*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 78.4928, size = 196, normalized size = 0.89 \[ - \frac{\sqrt{b + 2 c + \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{x \sqrt{b + 2 c + \sqrt{- 4 a c + b^{2}}}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- x^{2} + 1}} \right )}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{b + 2 c - \sqrt{- 4 a c + b^{2}}} \operatorname{atan}{\left (\frac{x \sqrt{b + 2 c - \sqrt{- 4 a c + b^{2}}}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- x^{2} + 1}} \right )}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.100479, size = 0, normalized size = 0. \[ \int \frac{\sqrt{1-x^2}}{a+b x^2+c x^4} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [C]  time = 0.016, size = 130, normalized size = 0.6 \[ -{\frac{1}{4}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{8}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{6}+ \left ( 6\,a+8\,b+16\,c \right ){{\it \_Z}}^{4}+ \left ( 4\,a+4\,b \right ){{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{6}-{{\it \_R}}^{4}-{{\it \_R}}^{2}+1}{{{\it \_R}}^{7}a+3\,{{\it \_R}}^{5}a+3\,{{\it \_R}}^{5}b+3\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{3}b+8\,{{\it \_R}}^{3}c+{\it \_R}\,a+{\it \_R}\,b}\ln \left ({\frac{1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-{\it \_R} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sum((_R^6-_R^4-_R^2+1)/(_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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-x^{2} + 1}}{c x^{4} + b x^{2} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-x^2 + 1)/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.378485, size = 1025, normalized size = 4.66 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(1/2)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2
- 4*a^2*c))*log(-(x^2 + sqrt(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2
- 4*a^2*c)*x)*sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2
- 4*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) - 1/2*sqrt(1/2)*
sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*l
og(-(x^2 - sqrt(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*
sqrt(-(2*a + b + (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/s
qrt(a^2*b^2 - 4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) - 1/2*sqrt(1/2)*sqrt(-(2*a + b
- (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 + sqr
t(1/2)*((a*b^2 - 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2*a + b
- (a*b^2 - 4*a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 -
4*a^3*c) + sqrt(-x^2 + 1) - 1)/x^2) + 1/2*sqrt(1/2)*sqrt(-(2*a + b - (a*b^2 - 4*
a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))*log(-(x^2 - sqrt(1/2)*((a*b^2
- 4*a^2*c)*sqrt(-x^2 + 1)*x - (a*b^2 - 4*a^2*c)*x)*sqrt(-(2*a + b - (a*b^2 - 4*
a^2*c)/sqrt(a^2*b^2 - 4*a^3*c))/(a*b^2 - 4*a^2*c))/sqrt(a^2*b^2 - 4*a^3*c) + sqr
t(-x^2 + 1) - 1)/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)**(1/2)/(c*x**4+b*x**2+a),x)

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out