Optimal. Leaf size=25 \[ \frac {3}{80} \left (x^2-1\right )^{2/3} \left (5 x^4-2 x^2-3\right ) \]
________________________________________________________________________________________
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 = {266, 43} \begin {gather*} \frac {3}{16} \left (x^2-1\right )^{8/3}+\frac {3}{10} \left (x^2-1\right )^{5/3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^3 \left (-1+x^2\right )^{2/3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (-1+x)^{2/3} x \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left ((-1+x)^{2/3}+(-1+x)^{5/3}\right ) \, dx,x,x^2\right )\\ &=\frac {3}{10} \left (-1+x^2\right )^{5/3}+\frac {3}{16} \left (-1+x^2\right )^{8/3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {3}{80} \left (x^2-1\right )^{5/3} \left (5 x^2+3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {3}{80} \left (-1+x^2\right )^{5/3} \left (3+5 x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.45, size = 21, normalized size = 0.84 \begin {gather*} \frac {3}{80} \, {\left (5 \, x^{4} - 2 \, x^{2} - 3\right )} {\left (x^{2} - 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.29, size = 19, normalized size = 0.76 \begin {gather*} \frac {3}{16} \, {\left (x^{2} - 1\right )}^{\frac {8}{3}} + \frac {3}{10} \, {\left (x^{2} - 1\right )}^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.08, size = 21, normalized size = 0.84
method | result | size |
trager | \(\left (\frac {3}{16} x^{4}-\frac {3}{40} x^{2}-\frac {9}{80}\right ) \left (x^{2}-1\right )^{\frac {2}{3}}\) | \(21\) |
risch | \(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}} \left (5 x^{4}-2 x^{2}-3\right )}{80}\) | \(22\) |
gosper | \(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (5 x^{2}+3\right ) \left (x^{2}-1\right )^{\frac {2}{3}}}{80}\) | \(23\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{2}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, 2\right ], \relax [3], x^{2}\right ) x^{4}}{4 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {2}{3}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 19, normalized size = 0.76 \begin {gather*} \frac {3}{16} \, {\left (x^{2} - 1\right )}^{\frac {8}{3}} + \frac {3}{10} \, {\left (x^{2} - 1\right )}^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.21, size = 21, normalized size = 0.84 \begin {gather*} -{\left (x^2-1\right )}^{2/3}\,\left (-\frac {3\,x^4}{16}+\frac {3\,x^2}{40}+\frac {9}{80}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.37, size = 41, normalized size = 1.64 \begin {gather*} \frac {3 x^{4} \left (x^{2} - 1\right )^{\frac {2}{3}}}{16} - \frac {3 x^{2} \left (x^{2} - 1\right )^{\frac {2}{3}}}{40} - \frac {9 \left (x^{2} - 1\right )^{\frac {2}{3}}}{80} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________