Optimal. Leaf size=25 \[ \frac {1}{40} \left (1+x^3\right )^{2/3} \left (-3+2 x^3+5 x^6\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.08, 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 = {272, 45}
\begin {gather*} \frac {1}{8} \left (x^3+1\right )^{8/3}-\frac {1}{5} \left (x^3+1\right )^{5/3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int x^5 \left (1+x^3\right )^{2/3} \, dx &=\frac {1}{3} \text {Subst}\left (\int x (1+x)^{2/3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (-(1+x)^{2/3}+(1+x)^{5/3}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{5} \left (1+x^3\right )^{5/3}+\frac {1}{8} \left (1+x^3\right )^{8/3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{40} \left (1+x^3\right )^{5/3} \left (-3+5 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
2.
time = 0.26, size = 17, normalized size = 0.68
method | result | size |
meijerg | \(\frac {x^{6} \hypergeom \left (\left [-\frac {2}{3}, 2\right ], \left [3\right ], -x^{3}\right )}{6}\) | \(17\) |
trager | \(\left (\frac {1}{8} x^{6}+\frac {1}{20} x^{3}-\frac {3}{40}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}\) | \(21\) |
risch | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (5 x^{6}+2 x^{3}-3\right )}{40}\) | \(22\) |
gosper | \(\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (5 x^{3}-3\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{40}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.25, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.33, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{40} \, {\left (5 \, x^{6} + 2 \, x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.22, size = 37, normalized size = 1.48 \begin {gather*} \frac {x^{6} \left (x^{3} + 1\right )^{\frac {2}{3}}}{8} + \frac {x^{3} \left (x^{3} + 1\right )^{\frac {2}{3}}}{20} - \frac {3 \left (x^{3} + 1\right )^{\frac {2}{3}}}{40} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.39, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} {\left (x^3+1\right )}^{2/3}\,\left (\frac {x^6}{8}+\frac {x^3}{20}-\frac {3}{40}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________