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

Optimal. Leaf size=182 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]

[Out]

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

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Rubi [A]  time = 0.619204, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 57.6882, size = 168, normalized size = 0.92 \[ - \frac{\sqrt{2} \sqrt{b + 2 c - \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{- x^{2} + 1}}{\sqrt{b + 2 c - \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \sqrt{b + 2 c + \sqrt{- 4 a c + b^{2}}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{- x^{2} + 1}}{\sqrt{b + 2 c + \sqrt{- 4 a c + b^{2}}}} \right )}}{2 \sqrt{c} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.260093, size = 169, normalized size = 0.93 \[ \frac{\sqrt{-\sqrt{b^2-4 a c}-b-2 c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}-b-2 c}}\right )-\sqrt{\sqrt{b^2-4 a c}-b-2 c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}-b-2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

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Maple [B]  time = 0.052, size = 1167, normalized size = 6.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.937263, size = 1176, normalized size = 6.46 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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