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

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

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Rubi [A]  time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.02, size = 39, normalized size = 1.00 \[ -\frac {1}{25 x^5}+\frac {1}{75 x^3}-\frac {1}{125 x}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{125 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(5 + x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 37, normalized size = 0.95 \[ \frac {-3 x^4+5 x^2-15}{375 x^5}-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {5}}\right )}{125 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x^6*(5 + x^2)),x]

[Out]

(-15 + 5*x^2 - 3*x^4)/(375*x^5) - ArcTan[x/Sqrt[5]]/(125*Sqrt[5])

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fricas [A]  time = 1.36, size = 32, normalized size = 0.82 \[ -\frac {3 \, \sqrt {5} x^{5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) + 15 \, x^{4} - 25 \, x^{2} + 75}{1875 \, x^{5}} \]

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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giac [A]  time = 0.58, size = 30, normalized size = 0.77 \[ -\frac {1}{625} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - \frac {3 \, x^{4} - 5 \, x^{2} + 15}{375 \, x^{5}} \]

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

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maple [A]  time = 0.32, size = 29, normalized size = 0.74

method	result	size
default	\(-\frac {1}{25 x^{5}}+\frac {1}{75 x^{3}}-\frac {1}{125 x}-\frac {\arctan \left (\frac {x \sqrt {5}}{5}\right ) \sqrt {5}}{625}\)	\(29\)
risch	\(\frac {-\frac {1}{125} x^{4}+\frac {1}{75} x^{2}-\frac {1}{25}}{x^{5}}-\frac {\arctan \left (\frac {x \sqrt {5}}{5}\right ) \sqrt {5}}{625}\)	\(30\)
meijerg	\(\frac {\sqrt {5}\, \left (-\frac {2 \sqrt {5}}{x}+\frac {10 \sqrt {5}}{3 x^{3}}-\frac {10 \sqrt {5}}{x^{5}}-2 \arctan \left (\frac {x \sqrt {5}}{5}\right )\right )}{1250}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.95, size = 30, normalized size = 0.77 \[ -\frac {1}{625} \, \sqrt {5} \arctan \left (\frac {1}{5} \, \sqrt {5} x\right ) - \frac {3 \, x^{4} - 5 \, x^{2} + 15}{375 \, x^{5}} \]

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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mupad [B]  time = 0.03, size = 30, normalized size = 0.77 \[ -\frac {\frac {x^4}{125}-\frac {x^2}{75}+\frac {1}{25}}{x^5}-\frac {\sqrt {5}\,\mathrm {atan}\left (\frac {\sqrt {5}\,x}{5}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^6*(x^2 + 5)),x)

[Out]

- (x^4/125 - x^2/75 + 1/25)/x^5 - (5^(1/2)*atan((5^(1/2)*x)/5))/625

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sympy [A]  time = 0.13, size = 32, normalized size = 0.82 \[ - \frac {\sqrt {5} \operatorname {atan}{\left (\frac {\sqrt {5} x}{5} \right )}}{625} + \frac {- 3 x^{4} + 5 x^{2} - 15}{375 x^{5}} \]

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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