3.461 \(\int \frac{1}{x (-2+x^2)^{5/2}} \, dx\)

Optimal. Leaf size=52 \[ \frac{1}{4 \sqrt{x^2-2}}-\frac{1}{6 \left (x^2-2\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{x^2-2}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

[Out]

-1/(6*(-2 + x^2)^(3/2)) + 1/(4*Sqrt[-2 + x^2]) + ArcTan[Sqrt[-2 + x^2]/Sqrt[2]]/(4*Sqrt[2])

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Rubi [A]  time = 0.023171, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {266, 51, 63, 203} \[ \frac{1}{4 \sqrt{x^2-2}}-\frac{1}{6 \left (x^2-2\right )^{3/2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{x^2-2}}{\sqrt{2}}\right )}{4 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*(-2 + x^2)^(5/2)),x]

[Out]

-1/(6*(-2 + x^2)^(3/2)) + 1/(4*Sqrt[-2 + x^2]) + ArcTan[Sqrt[-2 + x^2]/Sqrt[2]]/(4*Sqrt[2])

Rule 266

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

Rule 51

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{x \left (-2+x^2\right )^{5/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(-2+x)^{5/2} x} \, dx,x,x^2\right )\\ &=-\frac{1}{6 \left (-2+x^2\right )^{3/2}}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{(-2+x)^{3/2} x} \, dx,x,x^2\right )\\ &=-\frac{1}{6 \left (-2+x^2\right )^{3/2}}+\frac{1}{4 \sqrt{-2+x^2}}+\frac{1}{8} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-2+x} x} \, dx,x,x^2\right )\\ &=-\frac{1}{6 \left (-2+x^2\right )^{3/2}}+\frac{1}{4 \sqrt{-2+x^2}}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{2+x^2} \, dx,x,\sqrt{-2+x^2}\right )\\ &=-\frac{1}{6 \left (-2+x^2\right )^{3/2}}+\frac{1}{4 \sqrt{-2+x^2}}+\frac{\tan ^{-1}\left (\frac{\sqrt{-2+x^2}}{\sqrt{2}}\right )}{4 \sqrt{2}}\\ \end{align*}

Mathematica [C]  time = 0.0062441, size = 30, normalized size = 0.58 \[ -\frac{\, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};1-\frac{x^2}{2}\right )}{6 \left (x^2-2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-2 + x^2)^(5/2)),x]

[Out]

-Hypergeometric2F1[-3/2, 1, -1/2, 1 - x^2/2]/(6*(-2 + x^2)^(3/2))

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Maple [A]  time = 0.005, size = 37, normalized size = 0.7 \begin{align*} -{\frac{1}{6} \left ({x}^{2}-2 \right ) ^{-{\frac{3}{2}}}}+{\frac{1}{4}{\frac{1}{\sqrt{{x}^{2}-2}}}}-{\frac{\sqrt{2}}{8}\arctan \left ({\sqrt{2}{\frac{1}{\sqrt{{x}^{2}-2}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(x^2-2)^(5/2),x)

[Out]

-1/6/(x^2-2)^(3/2)+1/4/(x^2-2)^(1/2)-1/8*2^(1/2)*arctan(2^(1/2)/(x^2-2)^(1/2))

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Maxima [A]  time = 1.40477, size = 45, normalized size = 0.87 \begin{align*} -\frac{1}{8} \, \sqrt{2} \arcsin \left (\frac{\sqrt{2}}{{\left | x \right |}}\right ) + \frac{1}{4 \, \sqrt{x^{2} - 2}} - \frac{1}{6 \,{\left (x^{2} - 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-2)^(5/2),x, algorithm="maxima")

[Out]

-1/8*sqrt(2)*arcsin(sqrt(2)/abs(x)) + 1/4/sqrt(x^2 - 2) - 1/6/(x^2 - 2)^(3/2)

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Fricas [A]  time = 2.06344, size = 180, normalized size = 3.46 \begin{align*} \frac{3 \, \sqrt{2}{\left (x^{4} - 4 \, x^{2} + 4\right )} \arctan \left (-\frac{1}{2} \, \sqrt{2} x + \frac{1}{2} \, \sqrt{2} \sqrt{x^{2} - 2}\right ) +{\left (3 \, x^{2} - 8\right )} \sqrt{x^{2} - 2}}{12 \,{\left (x^{4} - 4 \, x^{2} + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-2)^(5/2),x, algorithm="fricas")

[Out]

1/12*(3*sqrt(2)*(x^4 - 4*x^2 + 4)*arctan(-1/2*sqrt(2)*x + 1/2*sqrt(2)*sqrt(x^2 - 2)) + (3*x^2 - 8)*sqrt(x^2 -
2))/(x^4 - 4*x^2 + 4)

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Sympy [C]  time = 5.23291, size = 986, normalized size = 18.96 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x**2-2)**(5/2),x)

[Out]

Piecewise((6*I*x**4*log(x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 3*I*x**4*log(x**2)/(24*sqrt(2)*x
**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 6*x**4*asin(sqrt(2)/x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2))
+ 6*sqrt(2)*x**2*sqrt(x**2 - 2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 24*I*x**2*log(x)/(24*sqrt(2
)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 12*I*x**2*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2))
+ 24*x**2*asin(sqrt(2)/x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 16*sqrt(2)*sqrt(x**2 - 2)/(24*sqr
t(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 24*I*log(x)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 12*
I*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 24*asin(sqrt(2)/x)/(24*sqrt(2)*x**4 - 96*sqrt(2
)*x**2 + 96*sqrt(2)), Abs(x**2)/2 > 1), (-3*I*x**4*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2))
+ 6*I*x**4*log(sqrt(1 - x**2/2) + 1)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 3*pi*x**4/(24*sqrt(2)*
x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 3*I*x**4*log(2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 6*sq
rt(2)*I*x**2*sqrt(2 - x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 12*I*x**2*log(x**2)/(24*sqrt(2)
*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 24*I*x**2*log(sqrt(1 - x**2/2) + 1)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2
+ 96*sqrt(2)) + 12*pi*x**2/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 12*I*x**2*log(2)/(24*sqrt(2)*x*
*4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 16*sqrt(2)*I*sqrt(2 - x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(
2)) - 12*I*log(x**2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 24*I*log(sqrt(1 - x**2/2) + 1)/(24*sqr
t(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) - 12*pi/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)) + 12*I*log(
2)/(24*sqrt(2)*x**4 - 96*sqrt(2)*x**2 + 96*sqrt(2)), True))

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Giac [A]  time = 1.0771, size = 47, normalized size = 0.9 \begin{align*} \frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} \sqrt{x^{2} - 2}\right ) + \frac{3 \, x^{2} - 8}{12 \,{\left (x^{2} - 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(x^2-2)^(5/2),x, algorithm="giac")

[Out]

1/8*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(x^2 - 2)) + 1/12*(3*x^2 - 8)/(x^2 - 2)^(3/2)