Optimal. Leaf size=32 \[ 2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}+\frac {2}{5} (-1+x)^{5/2} \]
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Rubi [A]
time = 0.00, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} \frac {2}{5} (x-1)^{5/2}+\frac {4}{3} (x-1)^{3/2}+2 \sqrt {x-1} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {-1+x}} \, dx &=\int \left (\frac {1}{\sqrt {-1+x}}+2 \sqrt {-1+x}+(-1+x)^{3/2}\right ) \, dx\\ &=2 \sqrt {-1+x}+\frac {4}{3} (-1+x)^{3/2}+\frac {2}{5} (-1+x)^{5/2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.66 \begin {gather*} \frac {2}{15} \sqrt {-1+x} \left (8+4 x+3 x^2\right ) \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 = 2.33, size = 56, normalized size = 1.75 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {2 \left (8+4 x+3 x^2\right ) \sqrt {-1+x}}{15},\text {Abs}\left [x\right ]>1\right \}\right \},\frac {I 8 x \sqrt {1-x}}{15}+\frac {I 2 x^2 \sqrt {1-x}}{5}+\frac {I 16 \sqrt {1-x}}{15}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.05, size = 23, normalized size = 0.72
method | result | size |
trager | \(\left (\frac {2}{5} x^{2}+\frac {8}{15} x +\frac {16}{15}\right ) \sqrt {-1+x}\) | \(17\) |
gosper | \(\frac {2 \sqrt {-1+x}\, \left (3 x^{2}+4 x +8\right )}{15}\) | \(18\) |
risch | \(\frac {2 \sqrt {-1+x}\, \left (3 x^{2}+4 x +8\right )}{15}\) | \(18\) |
derivativedivides | \(\frac {4 \left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {2 \left (-1+x \right )^{\frac {5}{2}}}{5}+2 \sqrt {-1+x}\) | \(23\) |
default | \(\frac {4 \left (-1+x \right )^{\frac {3}{2}}}{3}+\frac {2 \left (-1+x \right )^{\frac {5}{2}}}{5}+2 \sqrt {-1+x}\) | \(23\) |
meijerg | \(-\frac {\sqrt {-\mathrm {signum}\left (-1+x \right )}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (6 x^{2}+8 x +16\right ) \sqrt {1-x}}{15}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (-1+x \right )}}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 22, normalized size = 0.69 \begin {gather*} \frac {2}{5} \, {\left (x - 1\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x - 1\right )}^{\frac {3}{2}} + 2 \, \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 17, normalized size = 0.53 \begin {gather*} \frac {2}{15} \, {\left (3 \, x^{2} + 4 \, x + 8\right )} \sqrt {x - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.61, size = 76, normalized size = 2.38 \begin {gather*} \begin {cases} \frac {2 x^{2} \sqrt {x - 1}}{5} + \frac {8 x \sqrt {x - 1}}{15} + \frac {16 \sqrt {x - 1}}{15} & \text {for}\: \left |{x}\right | > 1 \\\frac {2 i x^{2} \sqrt {1 - x}}{5} + \frac {8 i x \sqrt {1 - x}}{15} + \frac {16 i \sqrt {1 - x}}{15} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 39, normalized size = 1.22 \begin {gather*} 2 \left (\frac {1}{5} \sqrt {x-1} \left (x-1\right )^{2}+\frac {2}{3} \sqrt {x-1} \left (x-1\right )+\sqrt {x-1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 19, normalized size = 0.59 \begin {gather*} \frac {2\,\sqrt {x-1}\,\left (10\,x+3\,{\left (x-1\right )}^2+5\right )}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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