Optimal. Leaf size=83 \[ \frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{21 \sqrt {1-x} \sqrt {1+x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {47, 40, 39}
\begin {gather*} \frac {8 x}{21 \sqrt {1-x} \sqrt {x+1}}+\frac {4 x}{21 (1-x)^{3/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (x+1)^{3/2}}+\frac {1}{7 (1-x)^{7/2} (x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rule 47
Rubi steps
\begin {align*} \int \frac {1}{(1-x)^{9/2} (1+x)^{5/2}} \, dx &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {5}{7} \int \frac {1}{(1-x)^{7/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4}{7} \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8}{21} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{7 (1-x)^{7/2} (1+x)^{3/2}}+\frac {1}{7 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{21 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{21 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 45, normalized size = 0.54 \begin {gather*} \frac {6+9 x-24 x^2+4 x^3+16 x^4-8 x^5}{21 (1-x)^{7/2} (1+x)^{3/2}} \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 = 56.75, size = 388, normalized size = 4.67 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (6+9 x-24 x^2+4 x^3+16 x^4-8 x^5\right ) \sqrt {\frac {1-x}{1+x}}}{21 \left (1-3 x+2 x^2+2 x^3-3 x^4+x^5\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-140 I \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{336+336 x-672 \left (1+x\right )^2-168 \left (1+x\right )^4+21 \left (1+x\right )^5+504 \left (1+x\right )^3}-\frac {35 I \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{336+336 x-672 \left (1+x\right )^2-168 \left (1+x\right )^4+21 \left (1+x\right )^5+504 \left (1+x\right )^3}-\frac {8 I \left (1+x\right )^5 \sqrt {1-\frac {2}{1+x}}}{336+336 x-672 \left (1+x\right )^2-168 \left (1+x\right )^4+21 \left (1+x\right )^5+504 \left (1+x\right )^3}-\frac {7 I \sqrt {1-\frac {2}{1+x}}}{336+336 x-672 \left (1+x\right )^2-168 \left (1+x\right )^4+21 \left (1+x\right )^5+504 \left (1+x\right )^3}+\frac {I 56 \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}}{336+336 x-672 \left (1+x\right )^2-168 \left (1+x\right )^4+21 \left (1+x\right )^5+504 \left (1+x\right )^3}+\frac {I 140 \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{336+336 x-672 \left (1+x\right )^2-168 \left (1+x\right )^4+21 \left (1+x\right )^5+504 \left (1+x\right )^3}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 86, normalized size = 1.04
method | result | size |
gosper | \(-\frac {8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6}{21 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {7}{2}}}\) | \(40\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{5}-16 x^{4}-4 x^{3}+24 x^{2}-9 x -6\right )}{21 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{3} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(66\) |
default | \(\frac {1}{7 \left (1-x \right )^{\frac {7}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {1}{7 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{21 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{7 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{21 \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{21 \sqrt {1+x}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 91, normalized size = 1.10 \begin {gather*} \frac {8 \, x}{21 \, \sqrt {-x^{2} + 1}} + \frac {4 \, x}{21 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} + \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x^{2} - 2 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} - \frac {1}{7 \, {\left ({\left (-x^{2} + 1\right )}^{\frac {3}{2}} x - {\left (-x^{2} + 1\right )}^{\frac {3}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 101, normalized size = 1.22 \begin {gather*} \frac {6 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 24 \, x^{3} - 6 \, x^{2} - {\left (8 \, x^{5} - 16 \, x^{4} - 4 \, x^{3} + 24 \, x^{2} - 9 \, x - 6\right )} \sqrt {x + 1} \sqrt {-x + 1} - 12 \, x + 6}{21 \, {\left (x^{6} - 2 \, x^{5} - x^{4} + 4 \, x^{3} - x^{2} - 2 \, 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 = 71.34, size = 593, normalized size = 7.14 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{5}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {56 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {140 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {35 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {7 \sqrt {-1 + \frac {2}{x + 1}}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} & \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}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {56 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} + \frac {140 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {35 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} - \frac {7 i \sqrt {1 - \frac {2}{x + 1}}}{336 x + 21 \left (x + 1\right )^{5} - 168 \left (x + 1\right )^{4} + 504 \left (x + 1\right )^{3} - 672 \left (x + 1\right )^{2} + 336} & \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. 190 vs.
\(2 (59) = 118\).
time = 0.02, size = 341, normalized size = 4.11 \begin {gather*} -2 \left (\frac {-\frac {1}{7}\cdot 302231454903657293676544 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}-906694364710971881029632 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}-\frac {1}{3}\cdot 32943228584498645010743296 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}+\frac {77824599637691753121710080 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{2475880078570760549798248448}+\frac {10815 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{6}+763 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}+63 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}+3}{172032 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{7}}+\frac {2 \left (-\frac {7}{192} \sqrt {-x+1} \sqrt {-x+1}+\frac {5}{64}\right ) \sqrt {-x+1} \sqrt {x+1}}{\left (x+1\right )^{2}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.41, size = 86, normalized size = 1.04 \begin {gather*} \frac {9\,x\,\sqrt {1-x}+6\,\sqrt {1-x}-24\,x^2\,\sqrt {1-x}+4\,x^3\,\sqrt {1-x}+16\,x^4\,\sqrt {1-x}-8\,x^5\,\sqrt {1-x}}{\left (21\,x+21\right )\,{\left (x-1\right )}^4\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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