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

Optimal. Leaf size=65 \[ -\frac {1}{36} \sqrt {2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {3 x^2-1}-\frac {7}{144} \sin ^{-1}\left (3-6 x^2\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {446, 80, 50, 53, 619, 216} \begin {gather*} -\frac {1}{36} \sqrt {2-3 x^2} \left (3 x^2-1\right )^{3/2}-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {3 x^2-1}-\frac {7}{144} \sin ^{-1}\left (3-6 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[2 - 3*x^2]*Sqrt[-1 + 3*x^2])/72 - (Sqrt[2 - 3*x^2]*(-1 + 3*x^2)^(3/2))/36 - (7*ArcSin[3 - 6*x^2])/144

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 53

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

Rule 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 216

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

Rule 446

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

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {x^3 \sqrt {-1+3 x^2}}{\sqrt {2-3 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x \sqrt {-1+3 x}}{\sqrt {2-3 x}} \, dx,x,x^2\right )\\ &=-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac {7}{24} \operatorname {Subst}\left (\int \frac {\sqrt {-1+3 x}}{\sqrt {2-3 x}} \, dx,x,x^2\right )\\ &=-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac {7}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2-3 x} \sqrt {-1+3 x}} \, dx,x,x^2\right )\\ &=-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}+\frac {7}{48} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-2+9 x-9 x^2}} \, dx,x,x^2\right )\\ &=-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac {7}{432} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{9}}} \, dx,x,9 \left (1-2 x^2\right )\right )\\ &=-\frac {7}{72} \sqrt {2-3 x^2} \sqrt {-1+3 x^2}-\frac {1}{36} \sqrt {2-3 x^2} \left (-1+3 x^2\right )^{3/2}-\frac {7}{144} \sin ^{-1}\left (3-6 x^2\right )\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 44, normalized size = 0.68 \begin {gather*} \frac {1}{72} \left (-7 \sin ^{-1}\left (\sqrt {2-3 x^2}\right )-\sqrt {-9 x^4+9 x^2-2} \left (6 x^2+5\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-((5 + 6*x^2)*Sqrt[-2 + 9*x^2 - 9*x^4]) - 7*ArcSin[Sqrt[2 - 3*x^2]])/72

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IntegrateAlgebraic [A]  time = 0.53, size = 96, normalized size = 1.48 \begin {gather*} -\frac {\sqrt {2-3 x^2} \left (\frac {7 \left (2-3 x^2\right )}{3 x^2-1}+9\right )}{72 \sqrt {3 x^2-1} \left (\frac {2-3 x^2}{3 x^2-1}+1\right )^2}-\frac {7}{72} \tan ^{-1}\left (\frac {\sqrt {2-3 x^2}}{\sqrt {3 x^2-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/72*(Sqrt[2 - 3*x^2]*(9 + (7*(2 - 3*x^2))/(-1 + 3*x^2)))/(Sqrt[-1 + 3*x^2]*(1 + (2 - 3*x^2)/(-1 + 3*x^2))^2)
- (7*ArcTan[Sqrt[2 - 3*x^2]/Sqrt[-1 + 3*x^2]])/72

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fricas [A]  time = 0.71, size = 72, normalized size = 1.11 \begin {gather*} -\frac {1}{72} \, {\left (6 \, x^{2} + 5\right )} \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} - \frac {7}{144} \, \arctan \left (\frac {3 \, \sqrt {3 \, x^{2} - 1} {\left (2 \, x^{2} - 1\right )} \sqrt {-3 \, x^{2} + 2}}{2 \, {\left (9 \, x^{4} - 9 \, x^{2} + 2\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(6*x^2 + 5)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) - 7/144*arctan(3/2*sqrt(3*x^2 - 1)*(2*x^2 - 1)*sqrt(-3*x^2
+ 2)/(9*x^4 - 9*x^2 + 2))

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giac [A]  time = 0.38, size = 40, normalized size = 0.62 \begin {gather*} -\frac {1}{72} \, {\left (6 \, x^{2} + 5\right )} \sqrt {3 \, x^{2} - 1} \sqrt {-3 \, x^{2} + 2} + \frac {7}{72} \, \arcsin \left (\sqrt {3 \, x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(6*x^2 + 5)*sqrt(3*x^2 - 1)*sqrt(-3*x^2 + 2) + 7/72*arcsin(sqrt(3*x^2 - 1))

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maple [A]  time = 0.03, size = 81, normalized size = 1.25 \begin {gather*} \frac {\sqrt {3 x^{2}-1}\, \sqrt {-3 x^{2}+2}\, \left (-12 \sqrt {-9 x^{4}+9 x^{2}-2}\, x^{2}+7 \arcsin \left (6 x^{2}-3\right )-10 \sqrt {-9 x^{4}+9 x^{2}-2}\right )}{144 \sqrt {-9 x^{4}+9 x^{2}-2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(3*x^2-1)^(1/2)*(-3*x^2+2)^(1/2)*(-12*x^2*(-9*x^4+9*x^2-2)^(1/2)+7*arcsin(6*x^2-3)-10*(-9*x^4+9*x^2-2)^(
1/2))/(-9*x^4+9*x^2-2)^(1/2)

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maxima [A]  time = 2.01, size = 46, normalized size = 0.71 \begin {gather*} -\frac {1}{12} \, \sqrt {-9 \, x^{4} + 9 \, x^{2} - 2} x^{2} - \frac {5}{72} \, \sqrt {-9 \, x^{4} + 9 \, x^{2} - 2} + \frac {7}{144} \, \arcsin \left (6 \, x^{2} - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*sqrt(-9*x^4 + 9*x^2 - 2)*x^2 - 5/72*sqrt(-9*x^4 + 9*x^2 - 2) + 7/144*arcsin(6*x^2 - 3)

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mupad [B]  time = 12.04, size = 414, normalized size = 6.37 \begin {gather*} -\frac {7\,\mathrm {atan}\left (\frac {\sqrt {3\,x^2-1}-\mathrm {i}}{\sqrt {2}-\sqrt {2-3\,x^2}}\right )}{36}+\frac {\frac {7\,\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}{36\,\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}+\frac {143\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^3}{36\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^3}-\frac {143\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^5}{36\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^5}-\frac {7\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^7}{36\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^7}+\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^2\,4{}\mathrm {i}}{9\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^2}-\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^4\,40{}\mathrm {i}}{9\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^4}+\frac {\sqrt {2}\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^6\,4{}\mathrm {i}}{9\,{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^6}}{\frac {4\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^2}+\frac {6\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^4}+\frac {4\,{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^6}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^6}+\frac {{\left (\sqrt {3\,x^2-1}-\mathrm {i}\right )}^8}{{\left (\sqrt {2}-\sqrt {2-3\,x^2}\right )}^8}+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((7*((3*x^2 - 1)^(1/2) - 1i))/(36*(2^(1/2) - (2 - 3*x^2)^(1/2))) + (143*((3*x^2 - 1)^(1/2) - 1i)^3)/(36*(2^(1/
2) - (2 - 3*x^2)^(1/2))^3) - (143*((3*x^2 - 1)^(1/2) - 1i)^5)/(36*(2^(1/2) - (2 - 3*x^2)^(1/2))^5) - (7*((3*x^
2 - 1)^(1/2) - 1i)^7)/(36*(2^(1/2) - (2 - 3*x^2)^(1/2))^7) + (2^(1/2)*((3*x^2 - 1)^(1/2) - 1i)^2*4i)/(9*(2^(1/
2) - (2 - 3*x^2)^(1/2))^2) - (2^(1/2)*((3*x^2 - 1)^(1/2) - 1i)^4*40i)/(9*(2^(1/2) - (2 - 3*x^2)^(1/2))^4) + (2
^(1/2)*((3*x^2 - 1)^(1/2) - 1i)^6*4i)/(9*(2^(1/2) - (2 - 3*x^2)^(1/2))^6))/((4*((3*x^2 - 1)^(1/2) - 1i)^2)/(2^
(1/2) - (2 - 3*x^2)^(1/2))^2 + (6*((3*x^2 - 1)^(1/2) - 1i)^4)/(2^(1/2) - (2 - 3*x^2)^(1/2))^4 + (4*((3*x^2 - 1
)^(1/2) - 1i)^6)/(2^(1/2) - (2 - 3*x^2)^(1/2))^6 + ((3*x^2 - 1)^(1/2) - 1i)^8/(2^(1/2) - (2 - 3*x^2)^(1/2))^8
+ 1) - (7*atan(((3*x^2 - 1)^(1/2) - 1i)/(2^(1/2) - (2 - 3*x^2)^(1/2))))/36

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3} \sqrt {3 x^{2} - 1}}{\sqrt {2 - 3 x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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