\(\int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx\) [1001]

Optimal result

Integrand size = 24, antiderivative size = 42 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {507, 435, 430} \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )}{\sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \]

[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 430

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

Rule 435

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

Rule 507

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])

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.88 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\frac {E\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right )|-\frac {2}{3}\right )-\operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{2}} x\right ),-\frac {2}{3}\right )}{\sqrt {3}} \]

[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]

Maple [A] (verified)

Time = 3.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83

method	result	size
default	\(-\frac {\sqrt {3}\, \left (F\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )-E\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{3}\)	\(35\)
elliptic	\(-\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \left (F\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )-E\left (\frac {x \sqrt {6}}{2}, \frac {i \sqrt {6}}{3}\right )\right )}{6 \sqrt {-3 x^{2}+2}\, \sqrt {-3 x^{4}-x^{2}+2}}\)	\(83\)

[In]

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

[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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (35) = 70\).

Time = 0.10 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.81 \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=-\frac {2 \, \sqrt {3} \sqrt {2} \sqrt {-3} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,-\frac {3}{2}) - 2 \, \sqrt {3} \sqrt {2} \sqrt {-3} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,-\frac {3}{2}) + 9 \, \sqrt {x^{2} + 1} \sqrt {-3 \, x^{2} + 2}}{27 \, x} \]

[In]

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

[Out]

-1/27*(2*sqrt(3)*sqrt(2)*sqrt(-3)*x*elliptic_e(arcsin(1/3*sqrt(3)*sqrt(2)/x), -3/2) - 2*sqrt(3)*sqrt(2)*sqrt(-
3)*x*elliptic_f(arcsin(1/3*sqrt(3)*sqrt(2)/x), -3/2) + 9*sqrt(x^2 + 1)*sqrt(-3*x^2 + 2))/x

Sympy [F]

\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\int \frac {x^{2}}{\sqrt {2 - 3 x^{2}} \sqrt {x^{2} + 1}}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\int { \frac {x^{2}}{\sqrt {x^{2} + 1} \sqrt {-3 \, x^{2} + 2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\sqrt {2-3 x^2} \sqrt {1+x^2}} \, dx=\int \frac {x^2}{\sqrt {x^2+1}\,\sqrt {2-3\,x^2}} \,d x \]

[In]

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

[Out]

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