3.459 \(\int \frac{1}{x^6 (5+x^2)} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{75 x^3}-\frac{1}{25 x^5}-\frac{1}{125 x}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )}{125 \sqrt{5}} \]

[Out]

-1/(25*x^5) + 1/(75*x^3) - 1/(125*x) - ArcTan[x/Sqrt[5]]/(125*Sqrt[5])

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Rubi [A]  time = 0.0139152, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {325, 203} \[ \frac{1}{75 x^3}-\frac{1}{25 x^5}-\frac{1}{125 x}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )}{125 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(5 + x^2)),x]

[Out]

-1/(25*x^5) + 1/(75*x^3) - 1/(125*x) - ArcTan[x/Sqrt[5]]/(125*Sqrt[5])

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, 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^6 \left (5+x^2\right )} \, dx &=-\frac{1}{25 x^5}-\frac{1}{5} \int \frac{1}{x^4 \left (5+x^2\right )} \, dx\\ &=-\frac{1}{25 x^5}+\frac{1}{75 x^3}+\frac{1}{25} \int \frac{1}{x^2 \left (5+x^2\right )} \, dx\\ &=-\frac{1}{25 x^5}+\frac{1}{75 x^3}-\frac{1}{125 x}-\frac{1}{125} \int \frac{1}{5+x^2} \, dx\\ &=-\frac{1}{25 x^5}+\frac{1}{75 x^3}-\frac{1}{125 x}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )}{125 \sqrt{5}}\\ \end{align*}

Mathematica [A]  time = 0.0137162, size = 39, normalized size = 1. \[ \frac{1}{75 x^3}-\frac{1}{25 x^5}-\frac{1}{125 x}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{5}}\right )}{125 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(5 + x^2)),x]

[Out]

-1/(25*x^5) + 1/(75*x^3) - 1/(125*x) - ArcTan[x/Sqrt[5]]/(125*Sqrt[5])

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Maple [A]  time = 0.006, size = 29, normalized size = 0.7 \begin{align*} -{\frac{1}{25\,{x}^{5}}}+{\frac{1}{75\,{x}^{3}}}-{\frac{1}{125\,x}}-{\frac{\sqrt{5}}{625}\arctan \left ({\frac{x\sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^2+5),x)

[Out]

-1/25/x^5+1/75/x^3-1/125/x-1/625*arctan(1/5*x*5^(1/2))*5^(1/2)

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Maxima [A]  time = 1.44339, size = 41, normalized size = 1.05 \begin{align*} -\frac{1}{625} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) - \frac{3 \, x^{4} - 5 \, x^{2} + 15}{375 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^2+5),x, algorithm="maxima")

[Out]

-1/625*sqrt(5)*arctan(1/5*sqrt(5)*x) - 1/375*(3*x^4 - 5*x^2 + 15)/x^5

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Fricas [A]  time = 2.08892, size = 100, normalized size = 2.56 \begin{align*} -\frac{3 \, \sqrt{5} x^{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) + 15 \, x^{4} - 25 \, x^{2} + 75}{1875 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^2+5),x, algorithm="fricas")

[Out]

-1/1875*(3*sqrt(5)*x^5*arctan(1/5*sqrt(5)*x) + 15*x^4 - 25*x^2 + 75)/x^5

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Sympy [A]  time = 0.125101, size = 34, normalized size = 0.87 \begin{align*} - \frac{\sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} x}{5} \right )}}{625} - \frac{3 x^{4} - 5 x^{2} + 15}{375 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(x**2+5),x)

[Out]

-sqrt(5)*atan(sqrt(5)*x/5)/625 - (3*x**4 - 5*x**2 + 15)/(375*x**5)

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Giac [A]  time = 1.11461, size = 41, normalized size = 1.05 \begin{align*} -\frac{1}{625} \, \sqrt{5} \arctan \left (\frac{1}{5} \, \sqrt{5} x\right ) - \frac{3 \, x^{4} - 5 \, x^{2} + 15}{375 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^2+5),x, algorithm="giac")

[Out]

-1/625*sqrt(5)*arctan(1/5*sqrt(5)*x) - 1/375*(3*x^4 - 5*x^2 + 15)/x^5