3.336 \(\int \frac {1}{(7+5 x^2) \sqrt {2+x^2-x^4}} \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{7} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]

[Out]

1/7*EllipticPi(1/2*x*2^(1/2),-10/7,I*2^(1/2))

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Rubi [A]  time = 0.03, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1212, 537} \[ \frac {1}{7} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]

[Out]

EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2]/7

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 1212

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

Rubi steps

\begin {align*} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx &=2 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx\\ &=\frac {1}{7} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}

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Mathematica [C]  time = 0.10, size = 24, normalized size = 1.41 \[ -\frac {i \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )}{7 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((7 + 5*x^2)*Sqrt[2 + x^2 - x^4]),x]

[Out]

((-1/7*I)*EllipticPi[5/7, I*ArcSinh[x], -1/2])/Sqrt[2]

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + x^{2} + 2}}{5 \, x^{6} + 2 \, x^{4} - 17 \, x^{2} - 14}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+7)/(-x^4+x^2+2)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-x^4 + x^2 + 2)/(5*x^6 + 2*x^4 - 17*x^2 - 14), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+7)/(-x^4+x^2+2)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)), x)

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maple [B]  time = 0.02, size = 48, normalized size = 2.82 \[ \frac {\sqrt {2}\, \sqrt {-\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, x}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{7 \sqrt {-x^{4}+x^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*x^2+7)/(-x^4+x^2+2)^(1/2),x)

[Out]

1/7*2^(1/2)*(-1/2*x^2+1)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticPi(1/2*2^(1/2)*x,-10/7,I*2^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x^2+7)/(-x^4+x^2+2)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-x^4 + x^2 + 2)*(5*x^2 + 7)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {1}{\left (5\,x^2+7\right )\,\sqrt {-x^4+x^2+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((5*x^2 + 7)*(x^2 - x^4 + 2)^(1/2)),x)

[Out]

int(1/((5*x^2 + 7)*(x^2 - x^4 + 2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(5*x**2+7)/(-x**4+x**2+2)**(1/2),x)

[Out]

Integral(1/(sqrt(-(x**2 - 2)*(x**2 + 1))*(5*x**2 + 7)), x)

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