Optimal. Leaf size=130 \[ \frac {1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac {11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac {11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac {55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac {55}{128} x \sqrt {x+1} \sqrt {1-x}+\frac {55}{128} \sin ^{-1}(x) \]
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Rubi [A] time = 0.03, 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 = {49, 38, 41, 216} \[ \frac {1}{9} (x+1)^{7/2} (1-x)^{11/2}+\frac {11}{72} (x+1)^{7/2} (1-x)^{9/2}+\frac {11}{56} (x+1)^{7/2} (1-x)^{7/2}+\frac {11}{48} x (x+1)^{5/2} (1-x)^{5/2}+\frac {55}{192} x (x+1)^{3/2} (1-x)^{3/2}+\frac {55}{128} x \sqrt {x+1} \sqrt {1-x}+\frac {55}{128} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 49
Rule 216
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 = 0.07, size = 75, normalized size = 0.58 \[ \frac {\sqrt {1-x^2} \left (-896 x^8+3024 x^7-1024 x^6-7224 x^5+8448 x^4+3066 x^3-10240 x^2+4599 x+3712\right )-6930 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{8064} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 77, normalized size = 0.59 \[ -\frac {1}{8064} \, {\left (896 \, x^{8} - 3024 \, x^{7} + 1024 \, x^{6} + 7224 \, x^{5} - 8448 \, x^{4} - 3066 \, x^{3} + 10240 \, x^{2} - 4599 \, x - 3712\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {55}{64} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.40, size = 323, normalized size = 2.48 \[ -\frac {1}{40320} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (2 \, {\left (7 \, {\left (8 \, x - 65\right )} {\left (x + 1\right )} + 2073\right )} {\left (x + 1\right )} - 9833\right )} {\left (x + 1\right )} + 75293\right )} {\left (x + 1\right )} - 310203\right )} {\left (x + 1\right )} + 216993\right )} {\left (x + 1\right )} - 205275\right )} {\left (x + 1\right )} + 69615\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{6720} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, {\left (7 \, x - 50\right )} {\left (x + 1\right )} + 1219\right )} {\left (x + 1\right )} - 12463\right )} {\left (x + 1\right )} + 64233\right )} {\left (x + 1\right )} - 53963\right )} {\left (x + 1\right )} + 59465\right )} {\left (x + 1\right )} - 23205\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{840} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, {\left (6 \, x - 37\right )} {\left (x + 1\right )} + 661\right )} {\left (x + 1\right )} - 4551\right )} {\left (x + 1\right )} + 4781\right )} {\left (x + 1\right )} - 6335\right )} {\left (x + 1\right )} + 2835\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{40} \, {\left ({\left (2 \, {\left ({\left (4 \, {\left (5 \, x - 26\right )} {\left (x + 1\right )} + 321\right )} {\left (x + 1\right )} - 451\right )} {\left (x + 1\right )} + 745\right )} {\left (x + 1\right )} - 405\right )} \sqrt {x + 1} \sqrt {-x + 1} + \frac {1}{4} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{3} \, {\left ({\left (2 \, x - 5\right )} {\left (x + 1\right )} + 9\right )} \sqrt {x + 1} \sqrt {-x + 1} - \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {55}{64} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 155, normalized size = 1.19 \[ \frac {55 \sqrt {\left (x +1\right ) \left (-x +1\right )}\, \arcsin \relax (x )}{128 \sqrt {x +1}\, \sqrt {-x +1}}+\frac {\left (-x +1\right )^{\frac {11}{2}} \left (x +1\right )^{\frac {7}{2}}}{9}+\frac {11 \left (-x +1\right )^{\frac {9}{2}} \left (x +1\right )^{\frac {7}{2}}}{72}+\frac {11 \left (-x +1\right )^{\frac {7}{2}} \left (x +1\right )^{\frac {7}{2}}}{56}+\frac {11 \left (-x +1\right )^{\frac {5}{2}} \left (x +1\right )^{\frac {7}{2}}}{48}+\frac {11 \left (-x +1\right )^{\frac {3}{2}} \left (x +1\right )^{\frac {7}{2}}}{48}+\frac {11 \sqrt {-x +1}\, \left (x +1\right )^{\frac {7}{2}}}{64}-\frac {11 \sqrt {-x +1}\, \left (x +1\right )^{\frac {5}{2}}}{192}-\frac {55 \sqrt {-x +1}\, \left (x +1\right )^{\frac {3}{2}}}{384}-\frac {55 \sqrt {-x +1}\, \sqrt {x +1}}{128} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.01, size = 78, normalized size = 0.60 \[ \frac {1}{9} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x^{2} - \frac {3}{8} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} x + \frac {29}{63} \, {\left (-x^{2} + 1\right )}^{\frac {7}{2}} + \frac {11}{48} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} x + \frac {55}{192} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {55}{128} \, \sqrt {-x^{2} + 1} x + \frac {55}{128} \, \arcsin \relax (x) \]
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 \[ \int {\left (1-x\right )}^{11/2}\,{\left (x+1\right )}^{5/2} \,d x \]
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 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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