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

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.00, size = 31, normalized size = 1.00 \[ -\frac {1}{20 x^4}+\frac {1}{50 x^2}-\frac {1}{250} \log \left (x^2+5\right )+\frac {\log (x)}{125} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(5 + x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.01, size = 31, normalized size = 1.00 \[ -\frac {1}{250} \log \left (x^2+5\right )+\frac {2 x^2-5}{100 x^4}+\frac {\log (x)}{125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-5 + 2*x^2)/(100*x^4) + Log[x]/125 - Log[5 + x^2]/250

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fricas [A]  time = 1.28, size = 30, normalized size = 0.97 \[ -\frac {2 \, x^{4} \log \left (x^{2} + 5\right ) - 4 \, x^{4} \log \relax (x) - 10 \, x^{2} + 25}{500 \, x^{4}} \]

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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giac [A]  time = 0.59, size = 32, normalized size = 1.03 \[ -\frac {3 \, x^{4} - 10 \, x^{2} + 25}{500 \, x^{4}} - \frac {1}{250} \, \log \left (x^{2} + 5\right ) + \frac {1}{250} \, \log \left (x^{2}\right ) \]

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)

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maple [A]  time = 0.32, size = 24, normalized size = 0.77

method	result	size
default	\(-\frac {1}{20 x^{4}}+\frac {1}{50 x^{2}}+\frac {\ln \relax (x )}{125}-\frac {\ln \left (x^{2}+5\right )}{250}\)	\(24\)
norman	\(\frac {-\frac {1}{20}+\frac {x^{2}}{50}}{x^{4}}+\frac {\ln \relax (x )}{125}-\frac {\ln \left (x^{2}+5\right )}{250}\)	\(25\)
risch	\(\frac {-\frac {1}{20}+\frac {x^{2}}{50}}{x^{4}}+\frac {\ln \relax (x )}{125}-\frac {\ln \left (x^{2}+5\right )}{250}\)	\(25\)
meijerg	\(-\frac {\ln \left (1+\frac {x^{2}}{5}\right )}{250}+\frac {\ln \relax (x )}{125}-\frac {\ln \relax (5)}{250}-\frac {1}{20 x^{4}}+\frac {1}{50 x^{2}}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.41, size = 27, normalized size = 0.87 \[ \frac {2 \, x^{2} - 5}{100 \, x^{4}} - \frac {1}{250} \, \log \left (x^{2} + 5\right ) + \frac {1}{250} \, \log \left (x^{2}\right ) \]

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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mupad [B]  time = 0.31, size = 24, normalized size = 0.77 \[ \frac {\ln \relax (x)}{125}-\frac {\ln \left (x^2+5\right )}{250}+\frac {\frac {x^2}{50}-\frac {1}{20}}{x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(x^2 + 5)),x)

[Out]

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

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sympy [A]  time = 0.12, size = 24, normalized size = 0.77 \[ \frac {\log {\relax (x )}}{125} - \frac {\log {\left (x^{2} + 5 \right )}}{250} + \frac {2 x^{2} - 5}{100 x^{4}} \]

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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