\(\int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx\) [966]

Optimal result

Integrand size = 26, antiderivative size = 70 \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{9 \sqrt {3}} \]

[Out]

-1/9*(x^2)^(1/2)/x*EllipticE(1/2*(-6*x^2+4)^(1/2),2^(1/2))*3^(1/2)-1/27*(x^2)^(1/2)/x*EllipticF(1/2*(-6*x^2+4)
^(1/2),2^(1/2))*3^(1/2)-1/9*x*(-3*x^2+2)^(1/2)*(3*x^2-1)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {489, 538, 436, 431} \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {\operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{9 \sqrt {3}}-\frac {E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}-\frac {1}{9} \sqrt {2-3 x^2} \sqrt {3 x^2-1} x \]

[In]

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

[Out]

-1/9*(x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2]) - EllipticE[ArcCos[Sqrt[3/2]*x], 2]/(3*Sqrt[3]) - EllipticF[ArcCos[S
qrt[3/2]*x], 2]/(9*Sqrt[3])

Rule 431

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

Rule 436

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

Rule 489

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[e^(n - 1
)*(e*x)^(m - n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(b*(m + n*(p + q) + 1))), x] - Dist[e^n/(b*(m + n*(p +
q) + 1)), Int[(e*x)^(m - n)*(a + b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[a*c*(m - n + 1) + (a*d*(m - n + 1) - n*q*(b
*c - a*d))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] &&
GtQ[m - n + 1, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 538

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{9} \int \frac {-2+9 x^2}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx \\ & = -\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{9} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx+\frac {1}{3} \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx \\ & = -\frac {1}{9} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {E\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{3 \sqrt {3}}-\frac {F\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{9 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.24 \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {3 x \left (1-3 x^2\right ) \sqrt {2-3 x^2}-3 \sqrt {6-18 x^2} E\left (\arcsin \left (\sqrt {3} x\right )|\frac {1}{2}\right )+2 \sqrt {6-18 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} x\right ),\frac {1}{2}\right )}{27 \sqrt {-1+3 x^2}} \]

[In]

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

[Out]

(3*x*(1 - 3*x^2)*Sqrt[2 - 3*x^2] - 3*Sqrt[6 - 18*x^2]*EllipticE[ArcSin[Sqrt[3]*x], 1/2] + 2*Sqrt[6 - 18*x^2]*E
llipticF[ArcSin[Sqrt[3]*x], 1/2])/(27*Sqrt[-1 + 3*x^2])

Maple [A] (verified)

Time = 3.96 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.84

method	result	size
default	\(-\frac {\sqrt {3 x^{2}-1}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \left (54 x^{5}+\sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}, \sqrt {2}\right )-3 \sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, E\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}, \sqrt {2}\right )-54 x^{3}+12 x \right )}{108 \left (9 x^{4}-9 x^{2}+2\right )}\)	\(129\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (-\frac {x \sqrt {-9 x^{4}+9 x^{2}-2}}{9}-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )}{27 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )-E\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )\right )}{18 \sqrt {-9 x^{4}+9 x^{2}-2}}\right )}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\)	\(163\)
risch	\(\frac {x \left (3 x^{2}-2\right ) \sqrt {3 x^{2}-1}\, \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{9 \sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \sqrt {-3 x^{2}+2}}+\frac {\left (-\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )}{27 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {\sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )-E\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )\right )}{18 \sqrt {-9 x^{4}+9 x^{2}-2}}\right ) \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\)	\(209\)

[In]

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

[Out]

-1/108*(3*x^2-1)^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(54*x^5+2^(1/2)*3^(1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*Elli
pticF(1/2*x*2^(1/2)*3^(1/2),2^(1/2))-3*2^(1/2)*3^(1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^(1/2)*EllipticE(1/2*x*2^(1/
2)*3^(1/2),2^(1/2))-54*x^3+12*x)/(9*x^4-9*x^2+2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.10 \[ \int \frac {x^2 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {-2 i \, \sqrt {3} \sqrt {2} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {1}{2}) + i \, \sqrt {3} \sqrt {2} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {1}{2}) - 3 \, \sqrt {3 \, x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {-3 \, x^{2} + 2}}{27 \, x} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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