Optimal. Leaf size=61 \[ -\frac {2}{3} \sqrt {1-x^3}+\frac {2}{3} \left (1-x^3\right )^{3/2}-\frac {2}{5} \left (1-x^3\right )^{5/2}+\frac {2}{21} \left (1-x^3\right )^{7/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} \frac {2}{21} \left (1-x^3\right )^{7/2}-\frac {2}{5} \left (1-x^3\right )^{5/2}+\frac {2}{3} \left (1-x^3\right )^{3/2}-\frac {2 \sqrt {1-x^3}}{3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\sqrt {1-x^3}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{\sqrt {1-x}}-3 \sqrt {1-x}+3 (1-x)^{3/2}-(1-x)^{5/2}\right ) \, dx,x,x^3\right )\\ &=-\frac {2}{3} \sqrt {1-x^3}+\frac {2}{3} \left (1-x^3\right )^{3/2}-\frac {2}{5} \left (1-x^3\right )^{5/2}+\frac {2}{21} \left (1-x^3\right )^{7/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 32, normalized size = 0.52 \begin {gather*} -\frac {2}{105} \sqrt {1-x^3} \left (16+8 x^3+6 x^6+5 x^9\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.14, size = 55, normalized size = 0.90
method | result | size |
trager | \(\left (-\frac {2}{21} x^{9}-\frac {4}{35} x^{6}-\frac {16}{105} x^{3}-\frac {32}{105}\right ) \sqrt {-x^{3}+1}\) | \(28\) |
risch | \(\frac {2 \left (5 x^{9}+6 x^{6}+8 x^{3}+16\right ) \left (x^{3}-1\right )}{105 \sqrt {-x^{3}+1}}\) | \(34\) |
gosper | \(\frac {2 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (5 x^{9}+6 x^{6}+8 x^{3}+16\right )}{105 \sqrt {-x^{3}+1}}\) | \(38\) |
meijerg | \(\frac {\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (40 x^{9}+48 x^{6}+64 x^{3}+128\right ) \sqrt {-x^{3}+1}}{140}}{3 \sqrt {\pi }}\) | \(43\) |
default | \(-\frac {2 x^{9} \sqrt {-x^{3}+1}}{21}-\frac {4 x^{6} \sqrt {-x^{3}+1}}{35}-\frac {16 x^{3} \sqrt {-x^{3}+1}}{105}-\frac {32 \sqrt {-x^{3}+1}}{105}\) | \(55\) |
elliptic | \(-\frac {2 x^{9} \sqrt {-x^{3}+1}}{21}-\frac {4 x^{6} \sqrt {-x^{3}+1}}{35}-\frac {16 x^{3} \sqrt {-x^{3}+1}}{105}-\frac {32 \sqrt {-x^{3}+1}}{105}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.32, size = 45, normalized size = 0.74 \begin {gather*} \frac {2}{21} \, {\left (-x^{3} + 1\right )}^{\frac {7}{2}} - \frac {2}{5} \, {\left (-x^{3} + 1\right )}^{\frac {5}{2}} + \frac {2}{3} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 28, normalized size = 0.46 \begin {gather*} -\frac {2}{105} \, {\left (5 \, x^{9} + 6 \, x^{6} + 8 \, x^{3} + 16\right )} \sqrt {-x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.23, size = 58, normalized size = 0.95 \begin {gather*} - \frac {2 x^{9} \sqrt {1 - x^{3}}}{21} - \frac {4 x^{6} \sqrt {1 - x^{3}}}{35} - \frac {16 x^{3} \sqrt {1 - x^{3}}}{105} - \frac {32 \sqrt {1 - x^{3}}}{105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.92, size = 59, normalized size = 0.97 \begin {gather*} -\frac {2}{21} \, {\left (x^{3} - 1\right )}^{3} \sqrt {-x^{3} + 1} - \frac {2}{5} \, {\left (x^{3} - 1\right )}^{2} \sqrt {-x^{3} + 1} + \frac {2}{3} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} - \frac {2}{3} \, \sqrt {-x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.04, size = 54, normalized size = 0.89 \begin {gather*} -\frac {16\,x^3\,\sqrt {1-x^3}}{105}-\frac {4\,x^6\,\sqrt {1-x^3}}{35}-\frac {2\,x^9\,\sqrt {1-x^3}}{21}-\frac {32\,\sqrt {1-x^3}}{105} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________