Optimal. Leaf size=90 \[ \frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac {5}{16} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 6, 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}{7} (1-x)^{7/2} (x+1)^{7/2}+\frac {1}{6} (1-x)^{5/2} x (x+1)^{5/2}+\frac {5}{24} (1-x)^{3/2} x (x+1)^{3/2}+\frac {5}{16} \sqrt {1-x} x \sqrt {x+1}+\frac {5}{16} \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)^{7/2} (1+x)^{5/2} \, dx &=\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\int (1-x)^{5/2} (1+x)^{5/2} \, dx\\ &=\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac {5}{6} \int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac {5}{8} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac {5}{16} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac {5}{16} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {5}{16} \sqrt {1-x} x \sqrt {1+x}+\frac {5}{24} (1-x)^{3/2} x (1+x)^{3/2}+\frac {1}{6} (1-x)^{5/2} x (1+x)^{5/2}+\frac {1}{7} (1-x)^{7/2} (1+x)^{7/2}+\frac {5}{16} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 78, normalized size = 0.87 \begin {gather*} \frac {\sqrt {1-x} \left (48+279 x+87 x^2-326 x^3-38 x^4+200 x^5+8 x^6-48 x^7\right )}{336 \sqrt {1+x}}-\frac {5}{8} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 127, normalized size = 1.41
method | result | size |
risch | \(\frac {\left (48 x^{6}-56 x^{5}-144 x^{4}+182 x^{3}+144 x^{2}-231 x -48\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{336 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(97\) |
default | \(\frac {\left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {7}{2}}}{7}+\frac {\left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {7}{2}}}{6}+\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{6}+\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{8}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{24}-\frac {5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{48}-\frac {5 \sqrt {1-x}\, \sqrt {1+x}}{16}+\frac {5 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{16 \sqrt {1+x}\, \sqrt {1-x}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 52, normalized size = 0.58 \begin {gather*} \frac {1}{7} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {1}{6} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {5}{24} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {5}{16} \, \sqrt {-x^{2} + 1} x + \frac {5}{16} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 67, normalized size = 0.74 \begin {gather*} -\frac {1}{336} \, {\left (48 \, x^{6} - 56 \, x^{5} - 144 \, x^{4} + 182 \, x^{3} + 144 \, x^{2} - 231 \, x - 48\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {5}{8} \, \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(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 239 vs.
\(2 (64) = 128\).
time = 0.04, size = 756, normalized size = 8.40 \begin {gather*} -2 \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 )+4 \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 )+2 \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 )+2 \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 )+4 \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 )}^{7/2}\,{\left (x+1\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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