Optimal. Leaf size=102 \[ \frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{63 \sqrt {1-x} \sqrt {1+x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {47, 39}
\begin {gather*} \frac {8 x}{63 \sqrt {1-x} \sqrt {x+1}}+\frac {4}{63 (1-x)^{3/2} \sqrt {x+1}}+\frac {4}{63 (1-x)^{5/2} \sqrt {x+1}}+\frac {5}{63 (1-x)^{7/2} \sqrt {x+1}}+\frac {1}{9 (1-x)^{9/2} \sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{11/2} (1+x)^{3/2}} \, dx &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{9} \int \frac {1}{(1-x)^{9/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {20}{63} \int \frac {1}{(1-x)^{7/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{21} \int \frac {1}{(1-x)^{5/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8}{63} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{9 (1-x)^{9/2} \sqrt {1+x}}+\frac {5}{63 (1-x)^{7/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{5/2} \sqrt {1+x}}+\frac {4}{63 (1-x)^{3/2} \sqrt {1+x}}+\frac {8 x}{63 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 45, normalized size = 0.44 \begin {gather*} \frac {20-17 x-16 x^2+44 x^3-32 x^4+8 x^5}{63 (-1+x)^4 \sqrt {1-x^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 = 88.37, size = 388, normalized size = 3.80 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-20+17 x+16 x^2-44 x^3+32 x^4-8 x^5\right ) \sqrt {\frac {1-x}{1+x}}}{63 \left (-1+5 x-10 x^2+10 x^3-5 x^4+x^5\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-315 I \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{3024+5040 x-5040 \left (1+x\right )^2-630 \left (1+x\right )^4+63 \left (1+x\right )^5+2520 \left (1+x\right )^3}-\frac {252 I \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{3024+5040 x-5040 \left (1+x\right )^2-630 \left (1+x\right )^4+63 \left (1+x\right )^5+2520 \left (1+x\right )^3}-\frac {8 I \left (1+x\right )^5 \sqrt {1-\frac {2}{1+x}}}{3024+5040 x-5040 \left (1+x\right )^2-630 \left (1+x\right )^4+63 \left (1+x\right )^5+2520 \left (1+x\right )^3}+\frac {I 63 \sqrt {1-\frac {2}{1+x}}}{3024+5040 x-5040 \left (1+x\right )^2-630 \left (1+x\right )^4+63 \left (1+x\right )^5+2520 \left (1+x\right )^3}+\frac {I 72 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}}{3024+5040 x-5040 \left (1+x\right )^2-630 \left (1+x\right )^4+63 \left (1+x\right )^5+2520 \left (1+x\right )^3}+\frac {I 420 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{3024+5040 x-5040 \left (1+x\right )^2-630 \left (1+x\right )^4+63 \left (1+x\right )^5+2520 \left (1+x\right )^3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 86, normalized size = 0.84
method | result | size |
gosper | \(\frac {8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20}{63 \sqrt {1+x}\, \left (1-x \right )^{\frac {9}{2}}}\) | \(40\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-32 x^{4}+44 x^{3}-16 x^{2}-17 x +20\right )}{63 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(66\) |
default | \(\frac {1}{9 \left (1-x \right )^{\frac {9}{2}} \sqrt {1+x}}+\frac {5}{63 \left (1-x \right )^{\frac {7}{2}} \sqrt {1+x}}+\frac {4}{63 \left (1-x \right )^{\frac {5}{2}} \sqrt {1+x}}+\frac {4}{63 \left (1-x \right )^{\frac {3}{2}} \sqrt {1+x}}+\frac {8}{63 \sqrt {1-x}\, \sqrt {1+x}}-\frac {8 \sqrt {1-x}}{63 \sqrt {1+x}}\) | \(86\) |
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. 201 vs.
\(2 (72) = 144\).
time = 0.27, size = 201, normalized size = 1.97 \begin {gather*} \frac {8 \, x}{63 \, \sqrt {-x^{2} + 1}} + \frac {1}{9 \, {\left (\sqrt {-x^{2} + 1} x^{4} - 4 \, \sqrt {-x^{2} + 1} x^{3} + 6 \, \sqrt {-x^{2} + 1} x^{2} - 4 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {5}{63 \, {\left (\sqrt {-x^{2} + 1} x^{3} - 3 \, \sqrt {-x^{2} + 1} x^{2} + 3 \, \sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} + \frac {4}{63 \, {\left (\sqrt {-x^{2} + 1} x^{2} - 2 \, \sqrt {-x^{2} + 1} x + \sqrt {-x^{2} + 1}\right )}} - \frac {4}{63 \, {\left (\sqrt {-x^{2} + 1} x - \sqrt {-x^{2} + 1}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 91, normalized size = 0.89 \begin {gather*} \frac {20 \, x^{6} - 80 \, x^{5} + 100 \, x^{4} - 100 \, x^{2} - {\left (8 \, x^{5} - 32 \, x^{4} + 44 \, x^{3} - 16 \, x^{2} - 17 \, x + 20\right )} \sqrt {x + 1} \sqrt {-x + 1} + 80 \, x - 20}{63 \, {\left (x^{6} - 4 \, x^{5} + 5 \, x^{4} - 5 \, x^{2} + 4 \, 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 = 117.56, size = 593, normalized size = 5.81 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {72 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {252 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {315 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {63 \sqrt {-1 + \frac {2}{x + 1}}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{5}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {72 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {252 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {420 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} - \frac {315 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} + \frac {63 i \sqrt {1 - \frac {2}{x + 1}}}{5040 x + 63 \left (x + 1\right )^{5} - 630 \left (x + 1\right )^{4} + 2520 \left (x + 1\right )^{3} - 5040 \left (x + 1\right )^{2} + 3024} & \text {otherwise} \end {cases} \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. 227 vs.
\(2 (72) = 144\).
time = 0.02, size = 387, normalized size = 3.79 \begin {gather*} 2 \left (\frac {\frac {1}{9}\cdot 1329227995784915872903807060280344576 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{9}+\frac {1}{7}\cdot 17279963945203906347749491783644479488 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}+21267647932558653966460912964485513216 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}+138239711561631250781995934269155835904 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}-\frac {580872634158008236458963685342510579712 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{43556142965880123323311949751266331066368}+\frac {-55062 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{8}-6552 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{6}-1008 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}-117 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-7}{2064384 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{9}}-\frac {\sqrt {-x+1} \sqrt {x+1}}{64 \left (x+1\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 80, normalized size = 0.78 \begin {gather*} \frac {17\,x\,\sqrt {1-x}-20\,\sqrt {1-x}+16\,x^2\,\sqrt {1-x}-44\,x^3\,\sqrt {1-x}+32\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{63\,{\left (x-1\right )}^5\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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