\(\int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx\) [964]

Optimal result

Integrand size = 26, antiderivative size = 99 \[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {7}{135} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{15} x^3 \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {8 E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{45 \sqrt {3}}-\frac {2 \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{27 \sqrt {3}} \]

[Out]

-8/135*(x^2)^(1/2)/x*EllipticE(1/2*(-6*x^2+4)^(1/2),2^(1/2))*3^(1/2)-2/81*(x^2)^(1/2)/x*EllipticF(1/2*(-6*x^2+
4)^(1/2),2^(1/2))*3^(1/2)-7/135*x*(-3*x^2+2)^(1/2)*(3*x^2-1)^(1/2)-1/15*x^3*(-3*x^2+2)^(1/2)*(3*x^2-1)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {489, 596, 538, 436, 431} \[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=-\frac {2 \operatorname {EllipticF}\left (\arccos \left (\sqrt {\frac {3}{2}} x\right ),2\right )}{27 \sqrt {3}}-\frac {8 E\left (\left .\arccos \left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{45 \sqrt {3}}-\frac {7}{135} \sqrt {2-3 x^2} \sqrt {3 x^2-1} x-\frac {1}{15} \sqrt {2-3 x^2} \sqrt {3 x^2-1} x^3 \]

[In]

Int[(x^4*Sqrt[-1 + 3*x^2])/Sqrt[2 - 3*x^2],x]

[Out]

(-7*x*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/135 - (x^3*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/15 - (8*EllipticE[ArcCos[
Sqrt[3/2]*x], 2])/(45*Sqrt[3]) - (2*EllipticF[ArcCos[Sqrt[3/2]*x], 2])/(27*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]))))))

Rule 596

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

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{15} x^3 \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{15} \int \frac {x^2 \left (-6+21 x^2\right )}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx \\ & = -\frac {7}{135} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{15} x^3 \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {1}{405} \int \frac {-42+216 x^2}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx \\ & = -\frac {7}{135} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{15} x^3 \sqrt {2-3 x^2} \sqrt {-1+3 x^2}+\frac {2}{27} \int \frac {1}{\sqrt {2-3 x^2} \sqrt {-1+3 x^2}} \, dx+\frac {8}{45} \int \frac {\sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx \\ & = -\frac {7}{135} x \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{15} x^3 \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {8 E\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{45 \sqrt {3}}-\frac {2 F\left (\left .\cos ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{27 \sqrt {3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93 \[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {-3 x \sqrt {2-3 x^2} \left (-7+12 x^2+27 x^4\right )-24 \sqrt {6-18 x^2} E\left (\arcsin \left (\sqrt {3} x\right )|\frac {1}{2}\right )+17 \sqrt {6-18 x^2} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {3} x\right ),\frac {1}{2}\right )}{405 \sqrt {-1+3 x^2}} \]

[In]

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

[Out]

(-3*x*Sqrt[2 - 3*x^2]*(-7 + 12*x^2 + 27*x^4) - 24*Sqrt[6 - 18*x^2]*EllipticE[ArcSin[Sqrt[3]*x], 1/2] + 17*Sqrt
[6 - 18*x^2]*EllipticF[ArcSin[Sqrt[3]*x], 1/2])/(405*Sqrt[-1 + 3*x^2])

Maple [A] (verified)

Time = 4.80 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {\sqrt {3 x^{2}-1}\, \sqrt {2}\, \sqrt {-6 x^{2}+4}\, \left (243 x^{7}-54 x^{5}+5 \sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}, \sqrt {2}\right )-12 \sqrt {2}\, \sqrt {3}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, E\left (\frac {x \sqrt {2}\, \sqrt {3}}{2}, \sqrt {2}\right )-135 x^{3}+42 x \right )}{810 \left (9 x^{4}-9 x^{2}+2\right )}\)	\(135\)
elliptic	\(\frac {\sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \left (-\frac {7 x \sqrt {-9 x^{4}+9 x^{2}-2}}{135}-\frac {7 \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )}{405 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {4 \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )-E\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )\right )}{135 \sqrt {-9 x^{4}+9 x^{2}-2}}-\frac {x^{3} \sqrt {-9 x^{4}+9 x^{2}-2}}{15}\right )}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\)	\(182\)
risch	\(\frac {x \left (9 x^{2}+7\right ) \left (3 x^{2}-2\right ) \sqrt {3 x^{2}-1}\, \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{135 \sqrt {-\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )}\, \sqrt {-3 x^{2}+2}}+\frac {\left (-\frac {7 \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )}{405 \sqrt {-9 x^{4}+9 x^{2}-2}}+\frac {4 \sqrt {6}\, \sqrt {-6 x^{2}+4}\, \sqrt {-3 x^{2}+1}\, \left (F\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )-E\left (\frac {x \sqrt {6}}{2}, \sqrt {2}\right )\right )}{135 \sqrt {-9 x^{4}+9 x^{2}-2}}\right ) \sqrt {\left (3 x^{2}-1\right ) \left (-3 x^{2}+2\right )}}{\sqrt {-3 x^{2}+2}\, \sqrt {3 x^{2}-1}}\)	\(216\)

[In]

int(x^4*(3*x^2-1)^(1/2)/(-3*x^2+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/810*(3*x^2-1)^(1/2)*2^(1/2)*(-6*x^2+4)^(1/2)*(243*x^7-54*x^5+5*2^(1/2)*3^(1/2)*(-6*x^2+4)^(1/2)*(-3*x^2+1)^
(1/2)*EllipticF(1/2*x*2^(1/2)*3^(1/2),2^(1/2))-12*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))-135*x^3+42*x)/(9*x^4-9*x^2+2)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.85 \[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\frac {-16 i \, \sqrt {3} \sqrt {2} x E(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {1}{2}) + 9 i \, \sqrt {3} \sqrt {2} x F(\arcsin \left (\frac {\sqrt {3} \sqrt {2}}{3 \, x}\right )\,|\,\frac {1}{2}) - 3 \, {\left (9 \, x^{4} + 7 \, x^{2} + 8\right )} \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2}}{405 \, x} \]

[In]

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

[Out]

1/405*(-16*I*sqrt(3)*sqrt(2)*x*elliptic_e(arcsin(1/3*sqrt(3)*sqrt(2)/x), 1/2) + 9*I*sqrt(3)*sqrt(2)*x*elliptic
_f(arcsin(1/3*sqrt(3)*sqrt(2)/x), 1/2) - 3*(9*x^4 + 7*x^2 + 8)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2))/x

Sympy [F]

\[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {x^{4} \sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {\sqrt {3 \, x^{2} - 1} x^{4}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int { \frac {\sqrt {3 \, x^{2} - 1} x^{4}}{\sqrt {-3 \, x^{2} + 2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx=\int \frac {x^4\,\sqrt {3\,x^2-1}}{\sqrt {2-3\,x^2}} \,d x \]

[In]

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

[Out]

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