Optimal. Leaf size=63 \[ \frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{15 \sqrt {1-x} \sqrt {1+x}} \]
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Rubi [A]
time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 3, 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}{15 \sqrt {1-x} \sqrt {x+1}}+\frac {4 x}{15 (1-x)^{3/2} (x+1)^{3/2}}+\frac {1}{5 (1-x)^{5/2} (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)^{7/2} (1+x)^{5/2}} \, dx &=\frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4}{5} \int \frac {1}{(1-x)^{5/2} (1+x)^{5/2}} \, dx\\ &=\frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8}{15} \int \frac {1}{(1-x)^{3/2} (1+x)^{3/2}} \, dx\\ &=\frac {1}{5 (1-x)^{5/2} (1+x)^{3/2}}+\frac {4 x}{15 (1-x)^{3/2} (1+x)^{3/2}}+\frac {8 x}{15 \sqrt {1-x} \sqrt {1+x}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 40, normalized size = 0.63 \begin {gather*} \frac {3+12 x-12 x^2-8 x^3+8 x^4}{15 (1-x)^{5/2} (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 = 22.17, size = 285, normalized size = 4.52 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {\left (-3-12 x+12 x^2+8 x^3-8 x^4\right ) \sqrt {\frac {1-x}{1+x}}}{15 \left (-1+2 x-2 x^3+x^4\right )},\frac {1}{\text {Abs}\left [1+x\right ]}>\frac {1}{2}\right \}\right \},\frac {-60 I \left (1+x\right )^2 \sqrt {1-\frac {2}{1+x}}}{-120-120 x-90 \left (1+x\right )^3+15 \left (1+x\right )^4+180 \left (1+x\right )^2}-\frac {8 I \left (1+x\right )^4 \sqrt {1-\frac {2}{1+x}}}{-120-120 x-90 \left (1+x\right )^3+15 \left (1+x\right )^4+180 \left (1+x\right )^2}+\frac {I 5 \sqrt {1-\frac {2}{1+x}}}{-120-120 x-90 \left (1+x\right )^3+15 \left (1+x\right )^4+180 \left (1+x\right )^2}+\frac {I 20 \left (1+x\right ) \sqrt {1-\frac {2}{1+x}}}{-120-120 x-90 \left (1+x\right )^3+15 \left (1+x\right )^4+180 \left (1+x\right )^2}+\frac {I 40 \left (1+x\right )^3 \sqrt {1-\frac {2}{1+x}}}{-120-120 x-90 \left (1+x\right )^3+15 \left (1+x\right )^4+180 \left (1+x\right )^2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.16, size = 72, normalized size = 1.14
method | result | size |
gosper | \(\frac {8 x^{4}-8 x^{3}-12 x^{2}+12 x +3}{15 \left (1+x \right )^{\frac {3}{2}} \left (1-x \right )^{\frac {5}{2}}}\) | \(35\) |
risch | \(\frac {\sqrt {\left (1+x \right ) \left (1-x \right )}\, \left (8 x^{4}-8 x^{3}-12 x^{2}+12 x +3\right )}{15 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}} \left (-1+x \right )^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}}\) | \(61\) |
default | \(\frac {1}{5 \left (1-x \right )^{\frac {5}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{15 \left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {3}{2}}}+\frac {4}{5 \sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{15 \left (1+x \right )^{\frac {3}{2}}}-\frac {8 \sqrt {1-x}}{15 \sqrt {1+x}}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 52, normalized size = 0.83 \begin {gather*} \frac {8 \, x}{15 \, \sqrt {-x^{2} + 1}} + \frac {4 \, x}{15 \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}}} - \frac {1}{5 \, {\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 = 0.29, size = 84, normalized size = 1.33 \begin {gather*} \frac {3 \, x^{5} - 3 \, x^{4} - 6 \, x^{3} + 6 \, x^{2} - {\left (8 \, x^{4} - 8 \, x^{3} - 12 \, x^{2} + 12 \, x + 3\right )} \sqrt {x + 1} \sqrt {-x + 1} + 3 \, x - 3}{15 \, {\left (x^{5} - x^{4} - 2 \, x^{3} + 2 \, x^{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 = 23.86, size = 425, normalized size = 6.75 \begin {gather*} \begin {cases} - \frac {8 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {40 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} - \frac {60 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {20 \sqrt {-1 + \frac {2}{x + 1}} \left (x + 1\right )}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {5 \sqrt {-1 + \frac {2}{x + 1}}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} & \text {for}\: \frac {1}{\left |{x + 1}\right |} > \frac {1}{2} \\- \frac {8 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{4}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {40 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{3}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} - \frac {60 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )^{2}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {20 i \sqrt {1 - \frac {2}{x + 1}} \left (x + 1\right )}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} + \frac {5 i \sqrt {1 - \frac {2}{x + 1}}}{- 120 x + 15 \left (x + 1\right )^{4} - 90 \left (x + 1\right )^{3} + 180 \left (x + 1\right )^{2} - 120} & \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. 148 vs.
\(2 (45) = 90\).
time = 0.01, size = 269, normalized size = 4.27 \begin {gather*} 2 \left (\frac {\frac {1}{5}\cdot 1099511627776 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}+\frac {1}{3}\cdot 14293651161088 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{3}-\frac {45079976738816 \left (-2 \sqrt {x+1}+2 \sqrt {2}\right )}{\sqrt {-x+1}}}{1125899906842624}+\frac {-1230 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{4}-65 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{2}-3}{15360 \left (-\frac {-2 \sqrt {x+1}+2 \sqrt {2}}{2 \sqrt {-x+1}}\right )^{5}}+\frac {2 \left (\frac {11}{192} \sqrt {-x+1} \sqrt {-x+1}-\frac 1{8}\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 = 0.38, size = 75, normalized size = 1.19 \begin {gather*} -\frac {12\,x\,\sqrt {1-x}+3\,\sqrt {1-x}-12\,x^2\,\sqrt {1-x}-8\,x^3\,\sqrt {1-x}+8\,x^4\,\sqrt {1-x}}{\left (15\,x+15\right )\,{\left (x-1\right )}^3\,\sqrt {x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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