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

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])/Sqrt[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 = 0.267961, 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, Rules used = {1247, 699, 1130, 208} \[ \frac{\sqrt{\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{\sqrt{b^2-4 a c}+b+2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}}-\frac{\sqrt{-\sqrt{b^2-4 a c}+b+2 c} \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{1-x^2}}{\sqrt{-\sqrt{b^2-4 a c}+b+2 c}}\right )}{\sqrt{2} \sqrt{c} \sqrt{b^2-4 a c}} \]

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])/Sqrt[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])

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 699

Int[Sqrt[(d_.) + (e_.)*(x_)]/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[2*e, Subst[Int[x^2/(c*d^2
- b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^
2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 1130

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(
d^2*(b/q + 1))/2, Int[(d*x)^(m - 2)/(b/2 + q/2 + c*x^2), x], x] - Dist[(d^2*(b/q - 1))/2, Int[(d*x)^(m - 2)/(b
/2 - q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && GeQ[m, 2]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.245174, 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 = 0.036, size = 1167, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

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. \begin{align*} \int \frac{\sqrt{-x^{2} + 1} x}{c x^{4} + b x^{2} + a}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 5.87307, size = 1868, normalized size = 10.26 \begin{align*} \text{result too large to display} \end{align*}

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, 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))*l
og((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. \begin{align*} \int \frac{x \sqrt{- \left (x - 1\right ) \left (x + 1\right )}}{a + b x^{2} + c x^{4}}\, dx \end{align*}

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

[Out]

Timed out