Optimal. Leaf size=51 \[ \frac {1}{2} \sqrt {x^4+5} x^4-\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )-\frac {1}{2} \left (10-x^2\right ) \sqrt {x^4+5} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1252, 833, 780, 215} \[ \frac {1}{2} \sqrt {x^4+5} x^4-\frac {1}{2} \left (10-x^2\right ) \sqrt {x^4+5}-\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 780
Rule 833
Rule 1252
Rubi steps
\begin {align*} \int \frac {x^5 \left (2+3 x^2\right )}{\sqrt {5+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2 (2+3 x)}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {5+x^4}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {x (-30+6 x)}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {5+x^4}-\frac {1}{2} \left (10-x^2\right ) \sqrt {5+x^4}-\frac {5}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {5+x^2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} x^4 \sqrt {5+x^4}-\frac {1}{2} \left (10-x^2\right ) \sqrt {5+x^4}-\frac {5}{2} \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 35, normalized size = 0.69 \[ \frac {1}{2} \left (\sqrt {x^4+5} \left (x^4+x^2-10\right )-5 \sinh ^{-1}\left (\frac {x^2}{\sqrt {5}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 34, normalized size = 0.67 \[ \frac {1}{2} \, {\left (x^{4} + x^{2} - 10\right )} \sqrt {x^{4} + 5} + \frac {5}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 37, normalized size = 0.73 \[ \frac {1}{2} \, \sqrt {x^{4} + 5} {\left ({\left (x^{2} + 1\right )} x^{2} - 10\right )} + \frac {5}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 39, normalized size = 0.76 \[ \frac {\sqrt {x^{4}+5}\, x^{2}}{2}-\frac {5 \arcsinh \left (\frac {\sqrt {5}\, x^{2}}{5}\right )}{2}+\frac {\sqrt {x^{4}+5}\, \left (x^{4}-10\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.22, size = 76, normalized size = 1.49 \[ \frac {1}{2} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} - \frac {15}{2} \, \sqrt {x^{4} + 5} + \frac {5 \, \sqrt {x^{4} + 5}}{2 \, x^{2} {\left (\frac {x^{4} + 5}{x^{4}} - 1\right )}} - \frac {5}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} + 1\right ) + \frac {5}{4} \, \log \left (\frac {\sqrt {x^{4} + 5}}{x^{2}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.31, size = 32, normalized size = 0.63 \[ \sqrt {x^4+5}\,\left (\frac {x^4}{2}+\frac {x^2}{2}-5\right )-\frac {5\,\mathrm {asinh}\left (\frac {\sqrt {5}\,x^2}{5}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.52, size = 66, normalized size = 1.29 \[ \frac {x^{6}}{2 \sqrt {x^{4} + 5}} + \frac {x^{4} \sqrt {x^{4} + 5}}{2} + \frac {5 x^{2}}{2 \sqrt {x^{4} + 5}} - 5 \sqrt {x^{4} + 5} - \frac {5 \operatorname {asinh}{\left (\frac {\sqrt {5} x^{2}}{5} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________