3.1001 \(\int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}}-\frac{\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3}} \]

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3] - EllipticF[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3]

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Rubi [A]  time = 0.0282783, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {493, 424, 419} \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]

[Out]

EllipticE[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3] - EllipticF[ArcSin[Sqrt[3/2]*x], -2/3]/Sqrt[3]

Rule 493

Int[(x_)^(n_)/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[1/b, Int[Sqrt[a +
b*x^n]/Sqrt[c + d*x^n], x], x] - Dist[a/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b, c,
d}, x] && NeQ[b*c - a*d, 0] && (EqQ[n, 2] || EqQ[n, 4]) &&  !(EqQ[n, 2] && SimplerSqrtQ[-(b/a), -(d/c)])

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

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

Rubi steps

\begin{align*} \int \frac{x^2}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx &=-\int \frac{1}{\sqrt{2-3 x^2} \sqrt{1+x^2}} \, dx+\int \frac{\sqrt{1+x^2}}{\sqrt{2-3 x^2}} \, dx\\ &=\frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}}-\frac{F\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )}{\sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.029042, size = 37, normalized size = 0.88 \[ \frac{E\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )|-\frac{2}{3}\right )-\text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),-\frac{2}{3}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(Sqrt[2 - 3*x^2]*Sqrt[1 + x^2]),x]

[Out]

(EllipticE[ArcSin[Sqrt[3/2]*x], -2/3] - EllipticF[ArcSin[Sqrt[3/2]*x], -2/3])/Sqrt[3]

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Maple [A]  time = 0.017, size = 35, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{3}}{3} \left ({\it EllipticF} \left ({\frac{x\sqrt{6}}{2}},{\frac{i}{3}}\sqrt{6} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{6}}{2}},{\frac{i}{3}}\sqrt{6} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*3^(1/2)*(EllipticF(1/2*x*6^(1/2),1/3*I*6^(1/2))-EllipticE(1/2*x*6^(1/2),1/3*I*6^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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