Optimal. Leaf size=31 \[ \frac {x^{1+m} \, _2F_1\left (\frac {1}{2},1+m;2+m;\frac {3 x}{2}\right )}{\sqrt {2} (1+m)} \]
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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {66}
\begin {gather*} \frac {x^{m+1} \, _2F_1\left (\frac {1}{2},m+1;m+2;\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rubi steps
\begin {align*} \int \frac {x^m}{\sqrt {2-3 x}} \, dx &=\frac {x^{1+m} \, _2F_1\left (\frac {1}{2},1+m;2+m;\frac {3 x}{2}\right )}{\sqrt {2} (1+m)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 31, normalized size = 1.00 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (\frac {1}{2},1+m;2+m;\frac {3 x}{2}\right )}{\sqrt {2} (1+m)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 5 in
optimal.
time = 2.28, size = 33, normalized size = 1.06 \begin {gather*} -I \left (\frac {2}{3}\right )^{1+m} \sqrt {-2+3 x} \text {hyper}\left [\left \{\frac {1}{2},-m\right \},\left \{\frac {3}{2}\right \},\frac {\left (-2+3 x\right ) \text {exp\_polar}\left [I \text {Pi}\right ]}{2}\right ] \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 29, normalized size = 0.94
method | result | size |
meijerg | \(\frac {x^{1+m} \hypergeom \left (\left [\frac {1}{2}, 1+m \right ], \left [2+m \right ], \frac {3 x}{2}\right ) \sqrt {2}}{2+2 m}\) | \(29\) |
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 [F]
time = 0.33, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = 0.61, size = 46, normalized size = 1.48 \begin {gather*} - \frac {2 \cdot 2^{m} \sqrt {3} \cdot 3^{- m} i \sqrt {x - \frac {2}{3}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - m \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 \left (x - \frac {2}{3}\right ) e^{i \pi }}{2}} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \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 = -0.03 \begin {gather*} \int \frac {x^m}{\sqrt {2-3\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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