Optimal. Leaf size=45 \[ -\frac {1}{3} (-1+x)^{3/2}-\frac {1}{5} (-1+x)^{5/2}-\frac {1}{3} (1+x)^{3/2}+\frac {1}{5} (1+x)^{5/2} \]
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Rubi [A]
time = 0.02, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2129, 45}
\begin {gather*} -\frac {1}{5} (x-1)^{5/2}-\frac {1}{3} (x-1)^{3/2}+\frac {1}{5} (x+1)^{5/2}-\frac {1}{3} (x+1)^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2129
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=-\left (\frac {1}{2} \int \sqrt {-1+x} x \, dx\right )+\frac {1}{2} \int x \sqrt {1+x} \, dx\\ &=-\left (\frac {1}{2} \int \left (\sqrt {-1+x}+(-1+x)^{3/2}\right ) \, dx\right )+\frac {1}{2} \int \left (-\sqrt {1+x}+(1+x)^{3/2}\right ) \, dx\\ &=-\frac {1}{3} (-1+x)^{3/2}-\frac {1}{5} (-1+x)^{5/2}-\frac {1}{3} (1+x)^{3/2}+\frac {1}{5} (1+x)^{5/2}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 34, normalized size = 0.76 \begin {gather*} \frac {1}{15} \left ((1+x)^{3/2} (-2+3 x)+\sqrt {-1+x} \left (2+x-3 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 30, normalized size = 0.67
method | result | size |
default | \(-\frac {\left (x -1\right )^{\frac {3}{2}}}{3}-\frac {\left (x -1\right )^{\frac {5}{2}}}{5}-\frac {\left (x +1\right )^{\frac {3}{2}}}{3}+\frac {\left (x +1\right )^{\frac {5}{2}}}{5}\) | \(30\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.57, size = 33, normalized size = 0.73 \begin {gather*} \frac {1}{15} \, {\left (3 \, x^{2} + x - 2\right )} \sqrt {x + 1} - \frac {1}{15} \, {\left (3 \, x^{2} - x - 2\right )} \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x - 1} + \sqrt {x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 33, normalized size = 0.73 \begin {gather*} \frac {1}{5} \, {\left (x + 1\right )}^{\frac {5}{2}} - \frac {1}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {1}{15} \, {\left ({\left (3 \, x - 4\right )} {\left (x + 1\right )} + 2\right )} \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 51, normalized size = 1.13 \begin {gather*} \frac {x\,\sqrt {x-1}}{15}+\frac {x\,\sqrt {x+1}}{15}+\frac {2\,\sqrt {x-1}}{15}-\frac {2\,\sqrt {x+1}}{15}-\frac {x^2\,\sqrt {x-1}}{5}+\frac {x^2\,\sqrt {x+1}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 15, normalized size = 0.33 \begin {gather*} \frac {4 \left (x -1\right )^{\frac {3}{2}}}{3}-2 \sqrt {x -1} \end {gather*}
Warning: Unable to verify antiderivative.
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