3.53 \(\int \frac{1}{\sqrt{-3+7 x^2+2 x^4}} \, dx\)

Optimal. Leaf size=148 \[ \frac{\sqrt{\frac{6-\left (7-\sqrt{73}\right ) x^2}{6-\left (7+\sqrt{73}\right ) x^2}} \sqrt{\left (7+\sqrt{73}\right ) x^2-6} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{73} x}{\sqrt{\left (7+\sqrt{73}\right ) x^2-6}}\right ),\frac{1}{146} \left (73+7 \sqrt{73}\right )\right )}{2 \sqrt{3} \sqrt [4]{73} \sqrt{\frac{1}{6-\left (7+\sqrt{73}\right ) x^2}} \sqrt{2 x^4+7 x^2-3}} \]

[Out]

(Sqrt[(6 - (7 - Sqrt[73])*x^2)/(6 - (7 + Sqrt[73])*x^2)]*Sqrt[-6 + (7 + Sqrt[73])*x^2]*EllipticF[ArcSin[(Sqrt[
2]*73^(1/4)*x)/Sqrt[-6 + (7 + Sqrt[73])*x^2]], (73 + 7*Sqrt[73])/146])/(2*Sqrt[3]*73^(1/4)*Sqrt[(6 - (7 + Sqrt
[73])*x^2)^(-1)]*Sqrt[-3 + 7*x^2 + 2*x^4])

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Rubi [A]  time = 0.02914, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {1098} \[ \frac{\sqrt{\frac{6-\left (7-\sqrt{73}\right ) x^2}{6-\left (7+\sqrt{73}\right ) x^2}} \sqrt{\left (7+\sqrt{73}\right ) x^2-6} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{73} x}{\sqrt{\left (7+\sqrt{73}\right ) x^2-6}}\right )|\frac{1}{146} \left (73+7 \sqrt{73}\right )\right )}{2 \sqrt{3} \sqrt [4]{73} \sqrt{\frac{1}{6-\left (7+\sqrt{73}\right ) x^2}} \sqrt{2 x^4+7 x^2-3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[-3 + 7*x^2 + 2*x^4],x]

[Out]

(Sqrt[(6 - (7 - Sqrt[73])*x^2)/(6 - (7 + Sqrt[73])*x^2)]*Sqrt[-6 + (7 + Sqrt[73])*x^2]*EllipticF[ArcSin[(Sqrt[
2]*73^(1/4)*x)/Sqrt[-6 + (7 + Sqrt[73])*x^2]], (73 + 7*Sqrt[73])/146])/(2*Sqrt[3]*73^(1/4)*Sqrt[(6 - (7 + Sqrt
[73])*x^2)^(-1)]*Sqrt[-3 + 7*x^2 + 2*x^4])

Rule 1098

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[(2*a +
(b - q)*x^2)/(2*a + (b + q)*x^2)]*Sqrt[(2*a + (b + q)*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q
)]], (b + q)/(2*q)])/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{-3+7 x^2+2 x^4}} \, dx &=\frac{\sqrt{\frac{6-\left (7-\sqrt{73}\right ) x^2}{6-\left (7+\sqrt{73}\right ) x^2}} \sqrt{-6+\left (7+\sqrt{73}\right ) x^2} F\left (\sin ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{73} x}{\sqrt{-6+\left (7+\sqrt{73}\right ) x^2}}\right )|\frac{1}{146} \left (73+7 \sqrt{73}\right )\right )}{2 \sqrt{3} \sqrt [4]{73} \sqrt{\frac{1}{6-\left (7+\sqrt{73}\right ) x^2}} \sqrt{-3+7 x^2+2 x^4}}\\ \end{align*}

Mathematica [C]  time = 0.052869, size = 80, normalized size = 0.54 \[ -\frac{i \sqrt{-4 x^4-14 x^2+6} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{2 x}{\sqrt{7+\sqrt{73}}}\right ),\frac{1}{12} \left (-61-7 \sqrt{73}\right )\right )}{\sqrt{\sqrt{73}-7} \sqrt{2 x^4+7 x^2-3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[-3 + 7*x^2 + 2*x^4],x]

[Out]

((-I)*Sqrt[6 - 14*x^2 - 4*x^4]*EllipticF[I*ArcSinh[(2*x)/Sqrt[7 + Sqrt[73]]], (-61 - 7*Sqrt[73])/12])/(Sqrt[-7
+ Sqrt[73]]*Sqrt[-3 + 7*x^2 + 2*x^4])

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Maple [C]  time = 0.18, size = 84, normalized size = 0.6 \begin{align*} 6\,{\frac{\sqrt{1- \left ( 7/6-1/6\,\sqrt{73} \right ){x}^{2}}\sqrt{1- \left ( 1/6\,\sqrt{73}+7/6 \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,\sqrt{42-6\,\sqrt{73}}x,{\frac{7\,i}{12}}\sqrt{6}+i/12\sqrt{438} \right ) }{\sqrt{42-6\,\sqrt{73}}\sqrt{2\,{x}^{4}+7\,{x}^{2}-3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2*x^4+7*x^2-3)^(1/2),x)

[Out]

6/(42-6*73^(1/2))^(1/2)*(1-(7/6-1/6*73^(1/2))*x^2)^(1/2)*(1-(1/6*73^(1/2)+7/6)*x^2)^(1/2)/(2*x^4+7*x^2-3)^(1/2
)*EllipticF(1/6*(42-6*73^(1/2))^(1/2)*x,7/12*I*6^(1/2)+1/12*I*438^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+7*x^2-3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(2*x^4 + 7*x^2 - 3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\sqrt{2 \, x^{4} + 7 \, x^{2} - 3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+7*x^2-3)^(1/2),x, algorithm="fricas")

[Out]

integral(1/sqrt(2*x^4 + 7*x^2 - 3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 x^{4} + 7 x^{2} - 3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x**4+7*x**2-3)**(1/2),x)

[Out]

Integral(1/sqrt(2*x**4 + 7*x**2 - 3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{2 \, x^{4} + 7 \, x^{2} - 3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2*x^4+7*x^2-3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(2*x^4 + 7*x^2 - 3), x)