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

Optimal. Leaf size=31 \[ \frac{1}{50 x^2}-\frac{1}{20 x^4}-\frac{1}{250} \log \left (x^2+5\right )+\frac{\log (x)}{125} \]

[Out]

-1/(20*x^4) + 1/(50*x^2) + Log[x]/125 - Log[5 + x^2]/250

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Rubi [A]  time = 0.0148877, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {266, 44} \[ \frac{1}{50 x^2}-\frac{1}{20 x^4}-\frac{1}{250} \log \left (x^2+5\right )+\frac{\log (x)}{125} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(5 + x^2)),x]

[Out]

-1/(20*x^4) + 1/(50*x^2) + Log[x]/125 - Log[5 + x^2]/250

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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (5+x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^3 (5+x)} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{5 x^3}-\frac{1}{25 x^2}+\frac{1}{125 x}-\frac{1}{125 (5+x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{20 x^4}+\frac{1}{50 x^2}+\frac{\log (x)}{125}-\frac{1}{250} \log \left (5+x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0036833, size = 31, normalized size = 1. \[ \frac{1}{50 x^2}-\frac{1}{20 x^4}-\frac{1}{250} \log \left (x^2+5\right )+\frac{\log (x)}{125} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(5 + x^2)),x]

[Out]

-1/(20*x^4) + 1/(50*x^2) + Log[x]/125 - Log[5 + x^2]/250

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Maple [A]  time = 0.006, size = 24, normalized size = 0.8 \begin{align*} -{\frac{1}{20\,{x}^{4}}}+{\frac{1}{50\,{x}^{2}}}+{\frac{\ln \left ( x \right ) }{125}}-{\frac{\ln \left ({x}^{2}+5 \right ) }{250}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/20/x^4+1/50/x^2+1/125*ln(x)-1/250*ln(x^2+5)

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Maxima [A]  time = 0.931927, size = 36, normalized size = 1.16 \begin{align*} \frac{2 \, x^{2} - 5}{100 \, x^{4}} - \frac{1}{250} \, \log \left (x^{2} + 5\right ) + \frac{1}{250} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/100*(2*x^2 - 5)/x^4 - 1/250*log(x^2 + 5) + 1/250*log(x^2)

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Fricas [A]  time = 2.29247, size = 84, normalized size = 2.71 \begin{align*} -\frac{2 \, x^{4} \log \left (x^{2} + 5\right ) - 4 \, x^{4} \log \left (x\right ) - 10 \, x^{2} + 25}{500 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/500*(2*x^4*log(x^2 + 5) - 4*x^4*log(x) - 10*x^2 + 25)/x^4

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Sympy [A]  time = 0.112281, size = 24, normalized size = 0.77 \begin{align*} \frac{\log{\left (x \right )}}{125} - \frac{\log{\left (x^{2} + 5 \right )}}{250} + \frac{2 x^{2} - 5}{100 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x)/125 - log(x**2 + 5)/250 + (2*x**2 - 5)/(100*x**4)

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Giac [A]  time = 1.14959, size = 43, normalized size = 1.39 \begin{align*} -\frac{3 \, x^{4} - 10 \, x^{2} + 25}{500 \, x^{4}} - \frac{1}{250} \, \log \left (x^{2} + 5\right ) + \frac{1}{250} \, \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/500*(3*x^4 - 10*x^2 + 25)/x^4 - 1/250*log(x^2 + 5) + 1/250*log(x^2)