Optimal. Leaf size=81 \[ \frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 \sqrt {x+1}}{35 \sqrt {1-x}}+\frac {2 \sqrt {x+1}}{35 (1-x)^{3/2}}+\frac {3 \sqrt {x+1}}{35 (1-x)^{5/2}}+\frac {\sqrt {x+1}}{7 (1-x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{9/2} \sqrt {1+x}} \, dx &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3}{7} \int \frac {1}{(1-x)^{7/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {6}{35} \int \frac {1}{(1-x)^{5/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2}{35} \int \frac {1}{(1-x)^{3/2} \sqrt {1+x}} \, dx\\ &=\frac {\sqrt {1+x}}{7 (1-x)^{7/2}}+\frac {3 \sqrt {1+x}}{35 (1-x)^{5/2}}+\frac {2 \sqrt {1+x}}{35 (1-x)^{3/2}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 35, normalized size = 0.43 \begin {gather*} \frac {\sqrt {1+x} \left (12-13 x+8 x^2-2 x^3\right )}{35 (1-x)^{7/2}} \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 = 20.02, size = 350, normalized size = 4.32 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {-12+13 x-8 x^2+2 x^3}{35 \left (-1+3 x-3 x^2+x^3\right ) \sqrt {\frac {1-x}{1+x}}},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-35 I \left (1+x\right )}{-280 \sqrt {1-\frac {2}{1+x}}-210 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+35 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+420 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}-\frac {2 I \left (1+x\right )^3}{-280 \sqrt {1-\frac {2}{1+x}}-210 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+35 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+420 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}+\frac {I 14 \left (1+x\right )^2}{-280 \sqrt {1-\frac {2}{1+x}}-210 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+35 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+420 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}+\frac {I 35}{-280 \sqrt {1-\frac {2}{1+x}}-210 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}+35 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}+420 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 58, normalized size = 0.72
method | result | size |
gosper | \(-\frac {\sqrt {1+x}\, \left (2 x^{3}-8 x^{2}+13 x -12\right )}{35 \left (1-x \right )^{\frac {7}{2}}}\) | \(30\) |
default | \(\frac {\sqrt {1+x}}{7 \left (1-x \right )^{\frac {7}{2}}}+\frac {3 \sqrt {1+x}}{35 \left (1-x \right )^{\frac {5}{2}}}+\frac {2 \sqrt {1+x}}{35 \left (1-x \right )^{\frac {3}{2}}}+\frac {2 \sqrt {1+x}}{35 \sqrt {1-x}}\) | \(58\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (2 x^{4}-6 x^{3}+5 x^{2}+x -12\right )}{35 \sqrt {1-x}\, \sqrt {1+x}\, \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(59\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 95, normalized size = 1.17 \begin {gather*} \frac {\sqrt {-x^{2} + 1}}{7 \, {\left (x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1\right )}} - \frac {3 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}} + \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x^{2} - 2 \, x + 1\right )}} - \frac {2 \, \sqrt {-x^{2} + 1}}{35 \, {\left (x - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.29, size = 71, normalized size = 0.88 \begin {gather*} \frac {12 \, x^{4} - 48 \, x^{3} + 72 \, x^{2} - {\left (2 \, x^{3} - 8 \, x^{2} + 13 \, x - 12\right )} \sqrt {x + 1} \sqrt {-x + 1} - 48 \, x + 12}{35 \, {\left (x^{4} - 4 \, x^{3} + 6 \, 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 = 19.20, size = 542, normalized size = 6.69 \begin {gather*} \begin {cases} \frac {2 \left (x + 1\right )^{3}}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {14 \left (x + 1\right )^{2}}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} + \frac {35 \left (x + 1\right )}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} - \frac {35}{35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {-1 + \frac {2}{x + 1}}} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {2 i \left (x + 1\right )^{3}}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} + \frac {14 i \left (x + 1\right )^{2}}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} - \frac {35 i \left (x + 1\right )}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} + \frac {35 i}{35 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3} - 210 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2} + 420 \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right ) - 280 \sqrt {1 - \frac {2}{x + 1}}} & \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. 171 vs.
\(2 (57) = 114\).
time = 0.01, size = 293, normalized size = 3.62 \begin {gather*} -2 \left (\frac {-\frac {1}{7}\cdot 73786976294838206464 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}-\frac {1}{5}\cdot 516508834063867445248 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}-516508834063867445248 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}+\frac {1291272085159668613120 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{151115727451828646838272}+\frac {1225 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{6}+245 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}+49 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+5}{71680 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.34, size = 67, normalized size = 0.83 \begin {gather*} -\frac {x\,\sqrt {1-x}-12\,\sqrt {1-x}+5\,x^2\,\sqrt {1-x}-6\,x^3\,\sqrt {1-x}+2\,x^4\,\sqrt {1-x}}{35\,{\left (x-1\right )}^4\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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