Optimal. Leaf size=41 \[ \frac {(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{7/2}}{63 (1-x)^{7/2}} \]
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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 37}
\begin {gather*} \frac {(x+1)^{7/2}}{63 (1-x)^{7/2}}+\frac {(x+1)^{7/2}}{9 (1-x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx &=\frac {(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac {1}{9} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{9 (1-x)^{9/2}}+\frac {(1+x)^{7/2}}{63 (1-x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 23, normalized size = 0.56 \begin {gather*} -\frac {(-8+x) (1+x)^{7/2}}{63 (1-x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in
optimal.
time = 37.91, size = 191, normalized size = 4.66 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-8+x\right ) \left (1+x\right )^{\frac {7}{2}}}{63 \sqrt {-1+x} \left (1-4 x+6 x^2-4 x^3+x^4\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},-\frac {\left (1+x\right )^{\frac {9}{2}}}{-2016 \left (1+x\right ) \sqrt {1-x}-504 \left (1+x\right )^3 \sqrt {1-x}+63 \left (1+x\right )^4 \sqrt {1-x}+1008 \sqrt {1-x}+1512 \left (1+x\right )^2 \sqrt {1-x}}+\frac {9 \left (1+x\right )^{\frac {7}{2}}}{-2016 \left (1+x\right ) \sqrt {1-x}-504 \left (1+x\right )^3 \sqrt {1-x}+63 \left (1+x\right )^4 \sqrt {1-x}+1008 \sqrt {1-x}+1512 \left (1+x\right )^2 \sqrt {1-x}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs.
\(2(29)=58\).
time = 0.14, size = 100, normalized size = 2.44
method | result | size |
gosper | \(-\frac {\left (1+x \right )^{\frac {7}{2}} \left (x -8\right )}{63 \left (1-x \right )^{\frac {9}{2}}}\) | \(18\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (x^{5}-4 x^{4}-26 x^{3}-44 x^{2}-31 x -8\right )}{63 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(64\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{2 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{6 \left (1-x \right )^{\frac {9}{2}}}+\frac {5 \sqrt {1+x}}{9 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{126 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{42 \left (1-x \right )^{\frac {5}{2}}}-\frac {\sqrt {1+x}}{63 \left (1-x \right )^{\frac {3}{2}}}-\frac {\sqrt {1+x}}{63 \sqrt {1-x}}\) | \(100\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 218 vs.
\(2 (29) = 58\).
time = 0.28, size = 218, normalized size = 5.32 \begin {gather*} -\frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{2 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} - \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{6 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{9 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{126 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{42 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{63 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 83 vs.
\(2 (29) = 58\).
time = 0.29, size = 83, normalized size = 2.02 \begin {gather*} \frac {8 \, x^{5} - 40 \, x^{4} + 80 \, x^{3} - 80 \, x^{2} + {\left (x^{4} - 5 \, x^{3} - 21 \, x^{2} - 23 \, x - 8\right )} \sqrt {x + 1} \sqrt {-x + 1} + 40 \, x - 8}{63 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 67.28, size = 280, normalized size = 6.83 \begin {gather*} \begin {cases} \frac {i \left (x + 1\right )^{\frac {9}{2}}}{63 \sqrt {x - 1} \left (x + 1\right )^{4} - 504 \sqrt {x - 1} \left (x + 1\right )^{3} + 1512 \sqrt {x - 1} \left (x + 1\right )^{2} - 2016 \sqrt {x - 1} \left (x + 1\right ) + 1008 \sqrt {x - 1}} - \frac {9 i \left (x + 1\right )^{\frac {7}{2}}}{63 \sqrt {x - 1} \left (x + 1\right )^{4} - 504 \sqrt {x - 1} \left (x + 1\right )^{3} + 1512 \sqrt {x - 1} \left (x + 1\right )^{2} - 2016 \sqrt {x - 1} \left (x + 1\right ) + 1008 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\- \frac {\left (x + 1\right )^{\frac {9}{2}}}{63 \sqrt {1 - x} \left (x + 1\right )^{4} - 504 \sqrt {1 - x} \left (x + 1\right )^{3} + 1512 \sqrt {1 - x} \left (x + 1\right )^{2} - 2016 \sqrt {1 - x} \left (x + 1\right ) + 1008 \sqrt {1 - x}} + \frac {9 \left (x + 1\right )^{\frac {7}{2}}}{63 \sqrt {1 - x} \left (x + 1\right )^{4} - 504 \sqrt {1 - x} \left (x + 1\right )^{3} + 1512 \sqrt {1 - x} \left (x + 1\right )^{2} - 2016 \sqrt {1 - x} \left (x + 1\right ) + 1008 \sqrt {1 - x}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.02, size = 87, normalized size = 2.12 \begin {gather*} \frac {2 \left (\frac 1{14}-\frac {1}{126} \sqrt {x+1} \sqrt {x+1}\right ) \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {x+1} \sqrt {-x+1}}{\left (-x+1\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 80, normalized size = 1.95 \begin {gather*} -\frac {\sqrt {1-x}\,\left (\frac {23\,x\,\sqrt {x+1}}{63}+\frac {8\,\sqrt {x+1}}{63}+\frac {x^2\,\sqrt {x+1}}{3}+\frac {5\,x^3\,\sqrt {x+1}}{63}-\frac {x^4\,\sqrt {x+1}}{63}\right )}{x^5-5\,x^4+10\,x^3-10\,x^2+5\,x-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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