Optimal. Leaf size=69 \[ \frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \sin ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {51, 38, 41, 222}
\begin {gather*} \frac {3 \text {ArcSin}(x)}{8}-\frac {1}{5} (1-x)^{5/2} (x+1)^{5/2}+\frac {1}{4} (1-x)^{3/2} x (x+1)^{3/2}+\frac {3}{8} \sqrt {1-x} x \sqrt {x+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 41
Rule 51
Rule 222
Rubi steps
\begin {align*} \int (1-x)^{3/2} (1+x)^{5/2} \, dx &=-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\int (1-x)^{3/2} (1+x)^{3/2} \, dx\\ &=\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{4} \int \sqrt {1-x} \sqrt {1+x} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\frac {3}{8} \sqrt {1-x} x \sqrt {1+x}+\frac {1}{4} (1-x)^{3/2} x (1+x)^{3/2}-\frac {1}{5} (1-x)^{5/2} (1+x)^{5/2}+\frac {3}{8} \sin ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 68, normalized size = 0.99 \begin {gather*} \frac {\sqrt {1-x} \left (-8+17 x+41 x^2+6 x^3-18 x^4-8 x^5\right )}{40 \sqrt {1+x}}-\frac {3}{4} \tan ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {1+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 99, normalized size = 1.43
method | result | size |
risch | \(\frac {\left (8 x^{4}+10 x^{3}-16 x^{2}-25 x +8\right ) \sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{40 \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(87\) |
default | \(\frac {\left (1-x \right )^{\frac {3}{2}} \left (1+x \right )^{\frac {7}{2}}}{5}+\frac {3 \sqrt {1-x}\, \left (1+x \right )^{\frac {7}{2}}}{20}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {5}{2}}}{20}-\frac {\sqrt {1-x}\, \left (1+x \right )^{\frac {3}{2}}}{8}-\frac {3 \sqrt {1-x}\, \sqrt {1+x}}{8}+\frac {3 \sqrt {\left (1+x \right ) \left (1-x \right )}\, \arcsin \left (x \right )}{8 \sqrt {1+x}\, \sqrt {1-x}}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 40, normalized size = 0.58 \begin {gather*} -\frac {1}{5} \, {\left (-x^{2} + 1\right )}^{\frac {5}{2}} + \frac {1}{4} \, {\left (-x^{2} + 1\right )}^{\frac {3}{2}} x + \frac {3}{8} \, \sqrt {-x^{2} + 1} x + \frac {3}{8} \, \arcsin \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 57, normalized size = 0.83 \begin {gather*} -\frac {1}{40} \, {\left (8 \, x^{4} + 10 \, x^{3} - 16 \, x^{2} - 25 \, x + 8\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {3}{4} \, \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\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 = 60.64, size = 245, normalized size = 3.55 \begin {gather*} \begin {cases} - \frac {3 i \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} - \frac {i \left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {x - 1}} + \frac {19 i \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {x - 1}} - \frac {23 i \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {x - 1}} - \frac {i \left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {x - 1}} + \frac {3 i \sqrt {x + 1}}{4 \sqrt {x - 1}} & \text {for}\: \left |{x + 1}\right | > 2 \\\frac {3 \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {x + 1}}{2} \right )}}{4} + \frac {\left (x + 1\right )^{\frac {11}{2}}}{5 \sqrt {1 - x}} - \frac {19 \left (x + 1\right )^{\frac {9}{2}}}{20 \sqrt {1 - x}} + \frac {23 \left (x + 1\right )^{\frac {7}{2}}}{20 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {5}{2}}}{40 \sqrt {1 - x}} + \frac {\left (x + 1\right )^{\frac {3}{2}}}{8 \sqrt {1 - x}} - \frac {3 \sqrt {x + 1}}{4 \sqrt {1 - x}} & \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. 114 vs.
\(2 (49) = 98\).
time = 1.24, size = 114, normalized size = 1.65 \begin {gather*} -\frac {1}{120} \, {\left ({\left (2 \, {\left (3 \, {\left (4 \, x - 17\right )} {\left (x + 1\right )} + 133\right )} {\left (x + 1\right )} - 295\right )} {\left (x + 1\right )} + 195\right )} \sqrt {x + 1} \sqrt {-x + 1} - \frac {1}{12} \, {\left ({\left (2 \, {\left (3 \, x - 10\right )} {\left (x + 1\right )} + 43\right )} {\left (x + 1\right )} - 39\right )} \sqrt {x + 1} \sqrt {-x + 1} + \sqrt {x + 1} {\left (x - 2\right )} \sqrt {-x + 1} + \sqrt {x + 1} \sqrt {-x + 1} + \frac {3}{4} \, \arcsin \left (\frac {1}{2} \, \sqrt {2} \sqrt {x + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (1-x\right )}^{3/2}\,{\left (x+1\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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