Optimal. Leaf size=45 \[ -\sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )-\frac {x^9}{3 \sqrt {x^6+2}}+\frac {1}{2} \sqrt {x^6+2} x^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {275, 288, 321, 215} \[ -\frac {x^9}{3 \sqrt {x^6+2}}+\frac {1}{2} \sqrt {x^6+2} x^3-\sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 215
Rule 275
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^{14}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^4}{\left (2+x^2\right )^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {x^9}{3 \sqrt {2+x^6}}+\operatorname {Subst}\left (\int \frac {x^2}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {x^9}{3 \sqrt {2+x^6}}+\frac {1}{2} x^3 \sqrt {2+x^6}-\operatorname {Subst}\left (\int \frac {1}{\sqrt {2+x^2}} \, dx,x,x^3\right )\\ &=-\frac {x^9}{3 \sqrt {2+x^6}}+\frac {1}{2} x^3 \sqrt {2+x^6}-\sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 43, normalized size = 0.96 \[ \frac {x^9+6 x^3-6 \sqrt {x^6+2} \sinh ^{-1}\left (\frac {x^3}{\sqrt {2}}\right )}{6 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.84, size = 54, normalized size = 1.20 \[ \frac {4 \, x^{6} + 6 \, {\left (x^{6} + 2\right )} \log \left (-x^{3} + \sqrt {x^{6} + 2}\right ) + {\left (x^{9} + 6 \, x^{3}\right )} \sqrt {x^{6} + 2} + 8}{6 \, {\left (x^{6} + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{14}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.14, size = 30, normalized size = 0.67 \[ \frac {\left (x^{6}+6\right ) x^{3}}{6 \sqrt {x^{6}+2}}-\arcsinh \left (\frac {\sqrt {2}\, x^{3}}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.01, size = 73, normalized size = 1.62 \[ -\frac {\frac {3 \, {\left (x^{6} + 2\right )}}{x^{6}} - 2}{3 \, {\left (\frac {\sqrt {x^{6} + 2}}{x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{x^{9}}\right )}} - \frac {1}{2} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} + 1\right ) + \frac {1}{2} \, \log \left (\frac {\sqrt {x^{6} + 2}}{x^{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^{14}}{{\left (x^6+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.51, size = 36, normalized size = 0.80 \[ \frac {x^{9}}{6 \sqrt {x^{6} + 2}} + \frac {x^{3}}{\sqrt {x^{6} + 2}} - \operatorname {asinh}{\left (\frac {\sqrt {2} x^{3}}{2} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________