Optimal. Leaf size=60 \[ -\frac {481 \log \left (x^2+x+1\right )}{5586}-\frac {79}{273 (x+5)}+\frac {200 \log (3-2 x)}{3211}+\frac {2731 \log (x+5)}{24843}+\frac {451 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2793 \sqrt {3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {6728, 634, 618, 204, 628} \[ -\frac {481 \log \left (x^2+x+1\right )}{5586}-\frac {79}{273 (x+5)}+\frac {200 \log (3-2 x)}{3211}+\frac {2731 \log (x+5)}{24843}+\frac {451 \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2793 \sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 618
Rule 628
Rule 634
Rule 6728
Rubi steps
\begin {align*} \int \frac {1+16 x}{(5+x)^2 (-3+2 x) \left (1+x+x^2\right )} \, dx &=\int \left (\frac {79}{273 (5+x)^2}+\frac {2731}{24843 (5+x)}+\frac {400}{3211 (-3+2 x)}+\frac {-15-481 x}{2793 \left (1+x+x^2\right )}\right ) \, dx\\ &=-\frac {79}{273 (5+x)}+\frac {200 \log (3-2 x)}{3211}+\frac {2731 \log (5+x)}{24843}+\frac {\int \frac {-15-481 x}{1+x+x^2} \, dx}{2793}\\ &=-\frac {79}{273 (5+x)}+\frac {200 \log (3-2 x)}{3211}+\frac {2731 \log (5+x)}{24843}+\frac {451 \int \frac {1}{1+x+x^2} \, dx}{5586}-\frac {481 \int \frac {1+2 x}{1+x+x^2} \, dx}{5586}\\ &=-\frac {79}{273 (5+x)}+\frac {200 \log (3-2 x)}{3211}+\frac {2731 \log (5+x)}{24843}-\frac {481 \log \left (1+x+x^2\right )}{5586}-\frac {451 \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 x\right )}{2793}\\ &=-\frac {79}{273 (5+x)}+\frac {451 \tan ^{-1}\left (\frac {1+2 x}{\sqrt {3}}\right )}{2793 \sqrt {3}}+\frac {200 \log (3-2 x)}{3211}+\frac {2731 \log (5+x)}{24843}-\frac {481 \log \left (1+x+x^2\right )}{5586}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 54, normalized size = 0.90 \[ \frac {-243867 \log \left (x^2+x+1\right )-\frac {819546}{x+5}+176400 \log (3-2 x)+311334 \log (x+5)+152438 \sqrt {3} \tan ^{-1}\left (\frac {2 x+1}{\sqrt {3}}\right )}{2832102} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 60, normalized size = 1.00 \[ \frac {152438 \, \sqrt {3} {\left (x + 5\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - 243867 \, {\left (x + 5\right )} \log \left (x^{2} + x + 1\right ) + 176400 \, {\left (x + 5\right )} \log \left (2 \, x - 3\right ) + 311334 \, {\left (x + 5\right )} \log \left (x + 5\right ) - 819546}{2832102 \, {\left (x + 5\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.90, size = 60, normalized size = 1.00 \[ \frac {451}{8379} \, \sqrt {3} \arctan \left (-\sqrt {3} {\left (\frac {14}{x + 5} - 3\right )}\right ) - \frac {79}{273 \, {\left (x + 5\right )}} - \frac {481}{5586} \, \log \left (-\frac {9}{x + 5} + \frac {21}{{\left (x + 5\right )}^{2}} + 1\right ) + \frac {200}{3211} \, \log \left ({\left | -\frac {13}{x + 5} + 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 48, normalized size = 0.80 \[ \frac {451 \sqrt {3}\, \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )}{8379}+\frac {200 \ln \left (2 x -3\right )}{3211}+\frac {2731 \ln \left (x +5\right )}{24843}-\frac {481 \ln \left (x^{2}+x +1\right )}{5586}-\frac {79}{273 \left (x +5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.06, size = 47, normalized size = 0.78 \[ \frac {451}{8379} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x + 1\right )}\right ) - \frac {79}{273 \, {\left (x + 5\right )}} - \frac {481}{5586} \, \log \left (x^{2} + x + 1\right ) + \frac {200}{3211} \, \log \left (2 \, x - 3\right ) + \frac {2731}{24843} \, \log \left (x + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.12, size = 61, normalized size = 1.02 \[ \frac {200\,\ln \left (x-\frac {3}{2}\right )}{3211}+\frac {2731\,\ln \left (x+5\right )}{24843}-\frac {79}{273\,\left (x+5\right )}-\ln \left (x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {481}{5586}+\frac {\sqrt {3}\,451{}\mathrm {i}}{16758}\right )+\ln \left (x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (-\frac {481}{5586}+\frac {\sqrt {3}\,451{}\mathrm {i}}{16758}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.27, size = 63, normalized size = 1.05 \[ \frac {200 \log {\left (x - \frac {3}{2} \right )}}{3211} + \frac {2731 \log {\left (x + 5 \right )}}{24843} - \frac {481 \log {\left (x^{2} + x + 1 \right )}}{5586} + \frac {451 \sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x}{3} + \frac {\sqrt {3}}{3} \right )}}{8379} - \frac {79}{273 x + 1365} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________