Optimal. Leaf size=61 \[ \frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {2 (x+1)^{7/2}}{693 (1-x)^{7/2}}+\frac {2 (x+1)^{7/2}}{99 (1-x)^{9/2}}+\frac {(x+1)^{7/2}}{11 (1-x)^{11/2}} \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)^{13/2}} \, dx &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2}{11} \int \frac {(1+x)^{5/2}}{(1-x)^{11/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2}{99} \int \frac {(1+x)^{5/2}}{(1-x)^{9/2}} \, dx\\ &=\frac {(1+x)^{7/2}}{11 (1-x)^{11/2}}+\frac {2 (1+x)^{7/2}}{99 (1-x)^{9/2}}+\frac {2 (1+x)^{7/2}}{693 (1-x)^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 30, normalized size = 0.49 \begin {gather*} \frac {(1+x)^{7/2} \left (79-18 x+2 x^2\right )}{693 (1-x)^{11/2}} \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 = 103.28, size = 475, normalized size = 7.79 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \left (-55+143 x-26 \left (1+x\right )^2+2 \left (1+x\right )^3\right ) \left (1+x\right )^{\frac {7}{2}}}{693 \sqrt {-1+x} \left (1-6 x+15 x^2-20 x^3+15 x^4-6 x^5+x^6\right )},\text {Abs}\left [1+x\right ]>2\right \}\right \},\frac {-143 \left (1+x\right )^{\frac {9}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}-\frac {2 \left (1+x\right )^{\frac {13}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}+\frac {26 \left (1+x\right )^{\frac {11}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \left (1+x\right )^2 \sqrt {1-x}}+\frac {198 \left (1+x\right )^{\frac {7}{2}}}{-133056 \left (1+x\right ) \sqrt {1-x}-110880 \left (1+x\right )^3 \sqrt {1-x}-8316 \left (1+x\right )^5 \sqrt {1-x}+693 \left (1+x\right )^6 \sqrt {1-x}+41580 \left (1+x\right )^4 \sqrt {1-x}+44352 \sqrt {1-x}+166320 \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. \(113\) vs.
\(2(43)=86\).
time = 0.14, size = 114, normalized size = 1.87
method | result | size |
gosper | \(\frac {\left (1+x \right )^{\frac {7}{2}} \left (2 x^{2}-18 x +79\right )}{693 \left (1-x \right )^{\frac {11}{2}}}\) | \(25\) |
risch | \(-\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{6}-10 x^{5}+19 x^{4}+216 x^{3}+404 x^{2}+298 x +79\right )}{693 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{5} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(71\) |
default | \(\frac {\left (1+x \right )^{\frac {5}{2}}}{3 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \left (1+x \right )^{\frac {3}{2}}}{12 \left (1-x \right )^{\frac {11}{2}}}+\frac {5 \sqrt {1+x}}{22 \left (1-x \right )^{\frac {11}{2}}}-\frac {5 \sqrt {1+x}}{396 \left (1-x \right )^{\frac {9}{2}}}-\frac {5 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {7}{2}}}-\frac {\sqrt {1+x}}{231 \left (1-x \right )^{\frac {5}{2}}}-\frac {2 \sqrt {1+x}}{693 \left (1-x \right )^{\frac {3}{2}}}-\frac {2 \sqrt {1+x}}{693 \sqrt {1-x}}\) | \(114\) |
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. 269 vs.
\(2 (43) = 86\).
time = 0.26, size = 269, normalized size = 4.41 \begin {gather*} \frac {{\left (-x^{2} + 1\right )}^{\frac {5}{2}}}{3 \, {\left (x^{8} - 8 \, x^{7} + 28 \, x^{6} - 56 \, x^{5} + 70 \, x^{4} - 56 \, x^{3} + 28 \, x^{2} - 8 \, x + 1\right )}} + \frac {5 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}}{12 \, {\left (x^{7} - 7 \, x^{6} + 21 \, x^{5} - 35 \, x^{4} + 35 \, x^{3} - 21 \, x^{2} + 7 \, x - 1\right )}} + \frac {5 \, \sqrt {-x^{2} + 1}}{22 \, {\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}}{396 \, {\left (x^{5} - 5 \, x^{4} + 10 \, x^{3} - 10 \, x^{2} + 5 \, x - 1\right )}} - \frac {5 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} + \frac {\sqrt {-x^{2} + 1}}{231 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\left (x^{2} - 2 \, x + 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{693 \, {\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. 100 vs.
\(2 (43) = 86\).
time = 0.30, size = 100, normalized size = 1.64 \begin {gather*} \frac {79 \, x^{6} - 474 \, x^{5} + 1185 \, x^{4} - 1580 \, x^{3} + 1185 \, x^{2} + {\left (2 \, x^{5} - 12 \, x^{4} + 31 \, x^{3} + 185 \, x^{2} + 219 \, x + 79\right )} \sqrt {x + 1} \sqrt {-x + 1} - 474 \, x + 79}{693 \, {\left (x^{6} - 6 \, x^{5} + 15 \, x^{4} - 20 \, x^{3} + 15 \, x^{2} - 6 \, x + 1\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 [A]
time = 0.02, size = 105, normalized size = 1.72 \begin {gather*} \frac {2 \left (\left (\frac {1}{693} \sqrt {x+1} \sqrt {x+1}-\frac 1{63}\right ) \sqrt {x+1} \sqrt {x+1}+\frac 1{14}\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 )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 94, normalized size = 1.54 \begin {gather*} \frac {\sqrt {1-x}\,\left (\frac {73\,x\,\sqrt {x+1}}{231}+\frac {79\,\sqrt {x+1}}{693}+\frac {185\,x^2\,\sqrt {x+1}}{693}+\frac {31\,x^3\,\sqrt {x+1}}{693}-\frac {4\,x^4\,\sqrt {x+1}}{231}+\frac {2\,x^5\,\sqrt {x+1}}{693}\right )}{x^6-6\,x^5+15\,x^4-20\,x^3+15\,x^2-6\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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