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

Optimal. Leaf size=99 \[ -\frac{2 \text{EllipticF}\left (\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{27 \sqrt{3}}-\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{8 E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{45 \sqrt{3}} \]

[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])

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Rubi [A]  time = 0.0934089, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {478, 582, 524, 425, 420} \[ -\frac{1}{15} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x^3-\frac{7}{135} \sqrt{2-3 x^2} \sqrt{3 x^2-1} x-\frac{2 F\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{27 \sqrt{3}}-\frac{8 E\left (\left .\cos ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{45 \sqrt{3}} \]

Antiderivative was successfully verified.

[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 478

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 582

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]

Rule 524

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 425

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> -Simp[(Sqrt[a - (b*c)/d]*EllipticE[ArcCo
s[Rt[-(d/c), 2]*x], (b*c)/(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 - (b*c)/d, 0]

Rule 420

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

Rubi steps

\begin{align*} \int \frac{x^4 \sqrt{-1+3 x^2}}{\sqrt{2-3 x^2}} \, dx &=-\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]  time = 0.0918039, size = 92, normalized size = 0.93 \[ \frac{10 \sqrt{3-9 x^2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )-3 x \sqrt{2-3 x^2} \left (27 x^4+12 x^2-7\right )-24 \sqrt{3-9 x^2} E\left (\left .\sin ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{405 \sqrt{3 x^2-1}} \]

Antiderivative was successfully verified.

[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[3 - 9*x^2]*EllipticE[ArcSin[Sqrt[3/2]*x], 2] + 10*Sqrt[
3 - 9*x^2]*EllipticF[ArcSin[Sqrt[3/2]*x], 2])/(405*Sqrt[-1 + 3*x^2])

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Maple [A]  time = 0.02, size = 135, normalized size = 1.4 \begin{align*} -{\frac{\sqrt{2}}{7290\,{x}^{4}-7290\,{x}^{2}+1620}\sqrt{3\,{x}^{2}-1}\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}{\it EllipticF} \left ( 1/2\,x\sqrt{2}\sqrt{3},\sqrt{2} \right ) -12\,\sqrt{2}\sqrt{3}\sqrt{-6\,{x}^{2}+4}\sqrt{-3\,{x}^{2}+1}{\it EllipticE} \left ( 1/2\,x\sqrt{2}\sqrt{3},\sqrt{2} \right ) -135\,{x}^{3}+42\,x \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{3 \, x^{2} - 1} x^{4}}{\sqrt{-3 \, x^{2} + 2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)