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

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

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{3} \left (x^2-2\right )^{3/2}+\sqrt {x^2-2} \end {gather*}

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 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])

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 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]

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*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{3} \sqrt {x^2-2} \left (x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-2 + x^2]*(1 + x^2))/3

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IntegrateAlgebraic [A]  time = 0.02, size = 18, normalized size = 0.78 \begin {gather*} \frac {1}{3} \sqrt {x^2-2} \left (x^2+1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[-2 + x^2]*(1 + x^2))/3

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fricas [A]  time = 0.82, size = 14, normalized size = 0.61 \begin {gather*} \frac {1}{3} \, {\left (x^{2} + 1\right )} \sqrt {x^{2} - 2} \end {gather*}

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)

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giac [A]  time = 0.40, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, {\left (x^{2} - 2\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 2} \end {gather*}

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)

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maple [A]  time = 0.01, size = 15, normalized size = 0.65 \begin {gather*} \frac {\left (x^{2}+1\right ) \sqrt {x^{2}-2}}{3} \end {gather*}

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)

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maxima [A]  time = 1.34, size = 22, normalized size = 0.96 \begin {gather*} \frac {1}{3} \, \sqrt {x^{2} - 2} x^{2} + \frac {1}{3} \, \sqrt {x^{2} - 2} \end {gather*}

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)

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mupad [B]  time = 0.10, size = 14, normalized size = 0.61 \begin {gather*} \frac {\left (x^2+1\right )\,\sqrt {x^2-2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - x^3)/(x^2 - 2)^(1/2),x)

[Out]

((x^2 + 1)*(x^2 - 2)^(1/2))/3

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sympy [A]  time = 0.69, size = 22, normalized size = 0.96 \begin {gather*} \frac {x^{2} \sqrt {x^{2} - 2}}{3} + \frac {\sqrt {x^{2} - 2}}{3} \end {gather*}

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

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