Optimal. Leaf size=49 \[ -\frac {1}{18 \sqrt {x^6+2} x^9}+\frac {x^3}{9 \sqrt {x^6+2}}+\frac {1}{9 \sqrt {x^6+2} x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {271, 264} \[ \frac {x^3}{9 \sqrt {x^6+2}}+\frac {1}{9 \sqrt {x^6+2} x^3}-\frac {1}{18 \sqrt {x^6+2} x^9} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 271
Rubi steps
\begin {align*} \int \frac {1}{x^{10} \left (2+x^6\right )^{3/2}} \, dx &=-\frac {1}{18 x^9 \sqrt {2+x^6}}-\frac {2}{3} \int \frac {1}{x^4 \left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {2}{3} \int \frac {x^2}{\left (2+x^6\right )^{3/2}} \, dx\\ &=-\frac {1}{18 x^9 \sqrt {2+x^6}}+\frac {1}{9 x^3 \sqrt {2+x^6}}+\frac {x^3}{9 \sqrt {2+x^6}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 28, normalized size = 0.57 \[ \frac {2 x^{12}+2 x^6-1}{18 x^9 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.89, size = 44, normalized size = 0.90 \[ \frac {2 \, x^{15} + 4 \, x^{9} + {\left (2 \, x^{12} + 2 \, x^{6} - 1\right )} \sqrt {x^{6} + 2}}{18 \, {\left (x^{15} + 2 \, x^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 25, normalized size = 0.51 \[ \frac {2 x^{12}+2 x^{6}-1}{18 \sqrt {x^{6}+2}\, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.11, size = 37, normalized size = 0.76 \[ \frac {x^{3}}{24 \, \sqrt {x^{6} + 2}} + \frac {\sqrt {x^{6} + 2}}{12 \, x^{3}} - \frac {{\left (x^{6} + 2\right )}^{\frac {3}{2}}}{72 \, x^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.24, size = 28, normalized size = 0.57 \[ -\frac {24\,x^6-8\,{\left (x^6+2\right )}^2+36}{72\,x^9\,\sqrt {x^6+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.53, size = 70, normalized size = 1.43 \[ \frac {2 x^{12} \sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} + \frac {2 x^{6} \sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} - \frac {\sqrt {1 + \frac {2}{x^{6}}}}{18 x^{12} + 36 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________