Optimal. Leaf size=121 \[ -\frac {8}{9} \left (1-x^3\right )^{3/2}+\frac {32}{15} \left (1-x^3\right )^{5/2}-\frac {22}{7} \left (1-x^3\right )^{7/2}+\frac {86}{27} \left (1-x^3\right )^{9/2}-\frac {74}{33} \left (1-x^3\right )^{11/2}+\frac {14}{13} \left (1-x^3\right )^{13/2}-\frac {14}{45} \left (1-x^3\right )^{15/2}+\frac {2}{51} \left (1-x^3\right )^{17/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.08, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1835, 1634}
\begin {gather*} \frac {2}{51} \left (1-x^3\right )^{17/2}-\frac {14}{45} \left (1-x^3\right )^{15/2}+\frac {14}{13} \left (1-x^3\right )^{13/2}-\frac {74}{33} \left (1-x^3\right )^{11/2}+\frac {86}{27} \left (1-x^3\right )^{9/2}-\frac {22}{7} \left (1-x^3\right )^{7/2}+\frac {32}{15} \left (1-x^3\right )^{5/2}-\frac {8}{9} \left (1-x^3\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1634
Rule 1835
Rubi steps
\begin {align*} \int x^5 \sqrt {1-x^3} \left (1+x^9\right )^2 \, dx &=\frac {1}{3} \text {Subst}\left (\int \sqrt {1-x} x \left (1+x^3\right )^2 \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (4 \sqrt {1-x}-16 (1-x)^{3/2}+33 (1-x)^{5/2}-43 (1-x)^{7/2}+37 (1-x)^{9/2}-21 (1-x)^{11/2}+7 (1-x)^{13/2}-(1-x)^{15/2}\right ) \, dx,x,x^3\right )\\ &=-\frac {8}{9} \left (1-x^3\right )^{3/2}+\frac {32}{15} \left (1-x^3\right )^{5/2}-\frac {22}{7} \left (1-x^3\right )^{7/2}+\frac {86}{27} \left (1-x^3\right )^{9/2}-\frac {74}{33} \left (1-x^3\right )^{11/2}+\frac {14}{13} \left (1-x^3\right )^{13/2}-\frac {14}{45} \left (1-x^3\right )^{15/2}+\frac {2}{51} \left (1-x^3\right )^{17/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.03, size = 57, normalized size = 0.47 \begin {gather*} \frac {2 \sqrt {1-x^3} \left (-173014-86507 x^3+126561 x^6-22160 x^9-19390 x^{12}+135702 x^{15}-3234 x^{18}-3003 x^{21}+45045 x^{24}\right )}{2297295} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.50, size = 125, normalized size = 1.03
method | result | size |
trager | \(\left (\frac {2}{51} x^{24}-\frac {2}{765} x^{21}-\frac {28}{9945} x^{18}+\frac {1436}{12155} x^{15}-\frac {1108}{65637} x^{12}-\frac {8864}{459459} x^{9}+\frac {84374}{765765} x^{6}-\frac {173014}{2297295} x^{3}-\frac {346028}{2297295}\right ) \sqrt {-x^{3}+1}\) | \(53\) |
gosper | \(\frac {2 \sqrt {-x^{3}+1}\, \left (45045 x^{21}+42042 x^{18}+38808 x^{15}+174510 x^{12}+155120 x^{9}+132960 x^{6}+259521 x^{3}+173014\right ) \left (-1+x \right ) \left (x^{2}+x +1\right )}{2297295}\) | \(58\) |
risch | \(-\frac {2 \left (45045 x^{24}-3003 x^{21}-3234 x^{18}+135702 x^{15}-19390 x^{12}-22160 x^{9}+126561 x^{6}-86507 x^{3}-173014\right ) \left (x^{3}-1\right )}{2297295 \sqrt {-x^{3}+1}}\) | \(59\) |
default | \(\frac {2 x^{24} \sqrt {-x^{3}+1}}{51}-\frac {2 x^{21} \sqrt {-x^{3}+1}}{765}-\frac {28 x^{18} \sqrt {-x^{3}+1}}{9945}+\frac {1436 x^{15} \sqrt {-x^{3}+1}}{12155}-\frac {1108 x^{12} \sqrt {-x^{3}+1}}{65637}-\frac {8864 x^{9} \sqrt {-x^{3}+1}}{459459}+\frac {84374 x^{6} \sqrt {-x^{3}+1}}{765765}-\frac {173014 x^{3} \sqrt {-x^{3}+1}}{2297295}-\frac {346028 \sqrt {-x^{3}+1}}{2297295}\) | \(125\) |
elliptic | \(\frac {2 x^{24} \sqrt {-x^{3}+1}}{51}-\frac {2 x^{21} \sqrt {-x^{3}+1}}{765}-\frac {28 x^{18} \sqrt {-x^{3}+1}}{9945}+\frac {1436 x^{15} \sqrt {-x^{3}+1}}{12155}-\frac {1108 x^{12} \sqrt {-x^{3}+1}}{65637}-\frac {8864 x^{9} \sqrt {-x^{3}+1}}{459459}+\frac {84374 x^{6} \sqrt {-x^{3}+1}}{765765}-\frac {173014 x^{3} \sqrt {-x^{3}+1}}{2297295}-\frac {346028 \sqrt {-x^{3}+1}}{2297295}\) | \(125\) |
meijerg | \(-\frac {-\frac {8192 \sqrt {\pi }}{109395}+\frac {4 \sqrt {\pi }\, \left (-x^{3}+1\right )^{\frac {3}{2}} \left (6435 x^{21}+6006 x^{18}+5544 x^{15}+5040 x^{12}+4480 x^{9}+3840 x^{6}+3072 x^{3}+2048\right )}{109395}}{6 \sqrt {\pi }}+\frac {\frac {512 \sqrt {\pi }}{3465}-\frac {4 \sqrt {\pi }\, \left (-x^{3}+1\right )^{\frac {3}{2}} \left (315 x^{12}+280 x^{9}+240 x^{6}+192 x^{3}+128\right )}{3465}}{3 \sqrt {\pi }}-\frac {-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (-x^{3}+1\right )^{\frac {3}{2}} \left (3 x^{3}+2\right )}{15}}{6 \sqrt {\pi }}\) | \(143\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.50, size = 89, normalized size = 0.74 \begin {gather*} \frac {2}{51} \, {\left (-x^{3} + 1\right )}^{\frac {17}{2}} - \frac {14}{45} \, {\left (-x^{3} + 1\right )}^{\frac {15}{2}} + \frac {14}{13} \, {\left (-x^{3} + 1\right )}^{\frac {13}{2}} - \frac {74}{33} \, {\left (-x^{3} + 1\right )}^{\frac {11}{2}} + \frac {86}{27} \, {\left (-x^{3} + 1\right )}^{\frac {9}{2}} - \frac {22}{7} \, {\left (-x^{3} + 1\right )}^{\frac {7}{2}} + \frac {32}{15} \, {\left (-x^{3} + 1\right )}^{\frac {5}{2}} - \frac {8}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 53, normalized size = 0.44 \begin {gather*} \frac {2}{2297295} \, {\left (45045 \, x^{24} - 3003 \, x^{21} - 3234 \, x^{18} + 135702 \, x^{15} - 19390 \, x^{12} - 22160 \, x^{9} + 126561 \, x^{6} - 86507 \, x^{3} - 173014\right )} \sqrt {-x^{3} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 1.18, size = 133, normalized size = 1.10 \begin {gather*} \frac {2 x^{24} \sqrt {1 - x^{3}}}{51} - \frac {2 x^{21} \sqrt {1 - x^{3}}}{765} - \frac {28 x^{18} \sqrt {1 - x^{3}}}{9945} + \frac {1436 x^{15} \sqrt {1 - x^{3}}}{12155} - \frac {1108 x^{12} \sqrt {1 - x^{3}}}{65637} - \frac {8864 x^{9} \sqrt {1 - x^{3}}}{459459} + \frac {84374 x^{6} \sqrt {1 - x^{3}}}{765765} - \frac {173014 x^{3} \sqrt {1 - x^{3}}}{2297295} - \frac {346028 \sqrt {1 - x^{3}}}{2297295} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.30, size = 138, normalized size = 1.14 \begin {gather*} \frac {2}{51} \, {\left (x^{3} - 1\right )}^{8} \sqrt {-x^{3} + 1} + \frac {14}{45} \, {\left (x^{3} - 1\right )}^{7} \sqrt {-x^{3} + 1} + \frac {14}{13} \, {\left (x^{3} - 1\right )}^{6} \sqrt {-x^{3} + 1} + \frac {74}{33} \, {\left (x^{3} - 1\right )}^{5} \sqrt {-x^{3} + 1} + \frac {86}{27} \, {\left (x^{3} - 1\right )}^{4} \sqrt {-x^{3} + 1} + \frac {22}{7} \, {\left (x^{3} - 1\right )}^{3} \sqrt {-x^{3} + 1} + \frac {32}{15} \, {\left (x^{3} - 1\right )}^{2} \sqrt {-x^{3} + 1} - \frac {8}{9} \, {\left (-x^{3} + 1\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 3.27, size = 124, normalized size = 1.02 \begin {gather*} \frac {84374\,x^6\,\sqrt {1-x^3}}{765765}-\frac {173014\,x^3\,\sqrt {1-x^3}}{2297295}-\frac {8864\,x^9\,\sqrt {1-x^3}}{459459}-\frac {1108\,x^{12}\,\sqrt {1-x^3}}{65637}+\frac {1436\,x^{15}\,\sqrt {1-x^3}}{12155}-\frac {28\,x^{18}\,\sqrt {1-x^3}}{9945}-\frac {2\,x^{21}\,\sqrt {1-x^3}}{765}+\frac {2\,x^{24}\,\sqrt {1-x^3}}{51}-\frac {346028\,\sqrt {1-x^3}}{2297295} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________