Optimal. Leaf size=28 \[ \frac {\left (x^3+1\right )^{2/3} \left (-x^6-6 x^3-5\right )}{20 x^8} \]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {453, 264} \begin {gather*} -\frac {\left (x^3+1\right )^{5/3}}{4 x^8}-\frac {\left (x^3+1\right )^{5/3}}{20 x^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 264
Rule 453
Rubi steps
\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{x^9} \, dx &=-\frac {\left (1+x^3\right )^{5/3}}{4 x^8}+\frac {1}{4} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (1+x^3\right )^{5/3}}{4 x^8}-\frac {\left (1+x^3\right )^{5/3}}{20 x^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 21, normalized size = 0.75 \begin {gather*} -\frac {\left (x^3+1\right )^{5/3} \left (x^3+5\right )}{20 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.09, size = 28, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (-5-6 x^3-x^6\right )}{20 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 22, normalized size = 0.79 \begin {gather*} -\frac {{\left (x^{6} + 6 \, x^{3} + 5\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{20 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 23, normalized size = 0.82
method | result | size |
trager | \(-\frac {\left (x^{6}+6 x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(23\) |
risch | \(-\frac {x^{9}+7 x^{6}+11 x^{3}+5}{20 x^{8} \left (x^{3}+1\right )^{\frac {1}{3}}}\) | \(28\) |
gosper | \(-\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+5\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{20 x^{8}}\) | \(29\) |
meijerg | \(-\frac {\left (x^{3}+1\right )^{\frac {5}{3}}}{5 x^{5}}-\frac {\left (-\frac {3}{5} x^{6}+\frac {2}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{4 x^{8}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.65, size = 25, normalized size = 0.89 \begin {gather*} \frac {{\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{4 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.29, size = 22, normalized size = 0.79 \begin {gather*} -\frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6+6\,x^3+5\right )}{20\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 2.38, size = 139, normalized size = 4.96 \begin {gather*} \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 x^{2} \Gamma \left (- \frac {2}{3}\right )} + \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {5}{3}\right )}{3 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {4 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{5} \Gamma \left (- \frac {2}{3}\right )} - \frac {10 \left (x^{3} + 1\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{8} \Gamma \left (- \frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________