Optimal. Leaf size=130 \[ \frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.02, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222}
\begin {gather*} \frac {1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac {11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac {11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac {55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac {55}{128} x \sqrt {x+1} \sqrt {1-x}+\frac {55}{128} \sin ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps
\begin {align*} \int (1-x)^{11/2} (1+x)^{5/2} \, dx &=\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {11}{9} \int (1-x)^{9/2} (1+x)^{5/2} \, dx\\ &=\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {11}{8} \int (1-x)^{7/2} (1+x)^{5/2} \, dx\\ &=\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {11}{8} \int (1-x)^{5/2} (1+x)^{5/2} \, dx\\ &=\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{48} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{64} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {55}{128} \sqrt {1-x} x \sqrt {1+x}+\frac {55}{192} (1-x)^{3/2} x (1+x)^{3/2}+\frac {11}{48} (1-x)^{5/2} x (1+x)^{5/2}+\frac {11}{56} (1-x)^{7/2} (1+x)^{7/2}+\frac {11}{72} (1-x)^{9/2} (1+x)^{7/2}+\frac {1}{9} (1-x)^{11/2} (1+x)^{7/2}+\frac {55}{128} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 88, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1-x} \left (3712+8311 x-5641 x^2-7174 x^3+11514 x^4+1224 x^5-8248 x^6+2000 x^7+2128 x^8-896 x^9\right )}{8064 \sqrt {1+x}}-\frac {55}{64} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 155, normalized size = 1.19
method | result | size |
risch | \(\frac {\left (896 x^{8}-3024 x^{7}+1024 x^{6}+7224 x^{5}-8448 x^{4}-3066 x^{3}+10240 x^{2}-4599 x -3712\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{8064 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {55 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\) | \(107\) |
default | \(\frac {\left (1-x \right )^{\frac {11}{2}} \left (1+x \right )^{\frac {7}{2}}}{9}+\frac {11 \left (1-x \right )^{\frac {9}{2}} \left (1+x \right )^{\frac {7}{2}}}{72}+\frac {11 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{56}+\frac {11 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{48}+\frac {11 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{48}+\frac {11 \sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{64}-\frac {11 \sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{192}-\frac {55 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{384}-\frac {55 \sqrt {1-x}\, \sqrt {1+x}}{128}+\frac {55 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{128 \sqrt {1+x}\, \sqrt {1-x}}\) | \(155\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 78, normalized size = 0.60 \begin {gather*} \frac {1}{9} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x^{2} - \frac {3}{8} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x + \frac {29}{63} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {11}{48} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {55}{192} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {55}{128} \, \sqrt {-x^{2} + 1} x + \frac {55}{128} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 77, normalized size = 0.59 \begin {gather*} -\frac {1}{8064} \, {\left (896 \, x^{8} - 3024 \, x^{7} + 1024 \, x^{6} + 7224 \, x^{5} - 8448 \, x^{4} - 3066 \, x^{3} + 10240 \, x^{2} - 4599 \, x - 3712\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {55}{64} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 365 vs.
\(2 (92) = 184\).
time = 0.06, size = 1174, normalized size = 9.03 \begin {gather*} -2 \left (2 \left (\left (\left (\left (\left (\left (\left (\left (-\frac {73}{288}+\frac {1}{36} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {691}{672}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {9833}{4032}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {75293}{20160}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {34467}{8960}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {10333}{3840}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {1955}{1536}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {221}{512}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {35}{128} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )+8 \left (2 \left (\left (\left (\left (\left (\left (\left (\frac {57}{224}-\frac {1}{32} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {1219}{1344}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {12463}{6720}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {21411}{8960}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {7709}{3840}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {1699}{1536}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {221}{512}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {35}{128} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-8 \left (2 \left (\left (\left (\left (\left (\left (\frac {1}{28} \sqrt {-x+1} \sqrt {-x+1}-\frac {43}{168}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {661}{840}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {1517}{1120}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {683}{480}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {181}{192}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {27}{64}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {5}{16} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-8 \left (2 \left (\left (\left (\left (\left (\frac {31}{120}-\frac {1}{24} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {107}{160}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {451}{480}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {149}{192}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {27}{64}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {5}{16} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )+20 \left (2 \left (\left (\left (\left (\frac {1}{20} \sqrt {-x+1} \sqrt {-x+1}-\frac {21}{80}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {133}{240}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {59}{96}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {13}{32}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {3}{8} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-8 \left (2 \left (\left (\left (\frac {13}{48}-\frac {1}{16} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {-x+1}-\frac {43}{96}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {13}{32}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {3}{8} \arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right )-8 \left (2 \left (\left (\frac {1}{12} \sqrt {-x+1} \sqrt {-x+1}-\frac {7}{24}\right ) \sqrt {-x+1} \sqrt {-x+1}+\frac {3}{8}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )+8 \left (2 \left (\frac {3}{8}-\frac {1}{8} \sqrt {-x+1} \sqrt {-x+1}\right ) \sqrt {-x+1} \sqrt {x+1}+\frac {\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )}{2}\right )-2 \left (\frac {1}{2} \sqrt {-x+1} \sqrt {x+1}+\arcsin \left (\frac {\sqrt {-x+1}}{\sqrt {2}}\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{11/2}\,{\left (x+1\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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