Optimal. Leaf size=49 \[ 2^{1+m} 3^{-1-m} (-x)^m x^{-m} \sqrt {-2+3 x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {69, 12, 67}
\begin {gather*} 2^{m+1} 3^{-m-1} \sqrt {3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 67
Rule 69
Rubi steps
\begin {align*} \int \frac {(-x)^m}{\sqrt {-2+3 x}} \, dx &=\left (\left (\frac {2}{3}\right )^m (-x)^m x^{-m}\right ) \int \frac {\left (\frac {3}{2}\right )^m x^m}{\sqrt {-2+3 x}} \, dx\\ &=\left ((-x)^m x^{-m}\right ) \int \frac {x^m}{\sqrt {-2+3 x}} \, dx\\ &=2^{1+m} 3^{-1-m} (-x)^m x^{-m} \sqrt {-2+3 x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 49, normalized size = 1.00 \begin {gather*} 2^{1+m} 3^{-1-m} (-x)^m x^{-m} \sqrt {-2+3 x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {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 5 in
optimal.
time = 2.31, size = 36, normalized size = 0.73 \begin {gather*} \frac {-I \sqrt {2} E^{I \text {Pi} m} x^{1+m} \text {hyper}\left [\left \{\frac {1}{2},1+m\right \},\left \{2+m\right \},\frac {3 x}{2}\right ]}{2+2 m} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
5.
time = 0.13, size = 44, normalized size = 0.90
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \left (-x \right )^{m} \sqrt {-\mathrm {signum}\left (x -\frac {2}{3}\right )}\, x \hypergeom \left (\left [\frac {1}{2}, 1+m \right ], \left [2+m \right ], \frac {3 x}{2}\right )}{2 \sqrt {\mathrm {signum}\left (x -\frac {2}{3}\right )}\, \left (1+m \right )}\) | \(44\) |
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 = 0.31, 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 = 0.62, size = 42, normalized size = 0.86 \begin {gather*} - \frac {\sqrt {2} i x x^{m} e^{i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \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 = -0.02 \begin {gather*} \int \frac {{\left (-x\right )}^m}{\sqrt {3\,x-2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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