Optimal. Leaf size=31 \[ -\frac {1}{20 x^4}+\frac {1}{50 x^2}-\frac {1}{250} \log \left (x^2+5\right )+\frac {\log (x)}{125} \]
________________________________________________________________________________________
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]
[Out]
Rule 44
Rule 266
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 = 0.00, 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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________