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

Optimal. Leaf size=23 \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]

[Out]

Sqrt[-2 + x^2] + (-2 + x^2)^(3/2)/3

________________________________________________________________________________________

Rubi [A]  time = 0.0211766, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1593, 444, 43} \[ \frac{1}{3} \left (x^2-2\right )^{3/2}+\sqrt{x^2-2} \]

Antiderivative was successfully verified.

[In]

Int[(-x + x^3)/Sqrt[-2 + x^2],x]

[Out]

Sqrt[-2 + x^2] + (-2 + x^2)^(3/2)/3

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(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] && EqQ[m
- n + 1, 0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{-x+x^3}{\sqrt{-2+x^2}} \, dx &=\int \frac{x \left (-1+x^2\right )}{\sqrt{-2+x^2}} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+x}{\sqrt{-2+x}} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{\sqrt{-2+x}}+\sqrt{-2+x}\right ) \, dx,x,x^2\right )\\ &=\sqrt{-2+x^2}+\frac{1}{3} \left (-2+x^2\right )^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0081222, size = 18, normalized size = 0.78 \[ \frac{1}{3} \sqrt{x^2-2} \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-2 + x^2]*(1 + x^2))/3

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 15, normalized size = 0.7 \begin{align*}{\frac{{x}^{2}+1}{3}\sqrt{{x}^{2}-2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.970398, size = 30, normalized size = 1.3 \begin{align*} \frac{1}{3} \, \sqrt{x^{2} - 2} x^{2} + \frac{1}{3} \, \sqrt{x^{2} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.42835, size = 39, normalized size = 1.7 \begin{align*} \frac{1}{3} \,{\left (x^{2} + 1\right )} \sqrt{x^{2} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^2 + 1)*sqrt(x^2 - 2)

________________________________________________________________________________________

Sympy [A]  time = 0.343196, size = 22, normalized size = 0.96 \begin{align*} \frac{x^{2} \sqrt{x^{2} - 2}}{3} + \frac{\sqrt{x^{2} - 2}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**2*sqrt(x**2 - 2)/3 + sqrt(x**2 - 2)/3

________________________________________________________________________________________

Giac [A]  time = 1.14376, size = 23, normalized size = 1. \begin{align*} \frac{1}{3} \,{\left (x^{2} - 2\right )}^{\frac{3}{2}} + \sqrt{x^{2} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(x^2 - 2)^(3/2) + sqrt(x^2 - 2)