Optimal. Leaf size=49 \[ 2^{m+1} 3^{-m-1} \sqrt {3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \]
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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 = {67, 12, 65} \[ 2^{m+1} 3^{-m-1} \sqrt {3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 67
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 \[ 2^{m+1} 3^{-m-1} \sqrt {3 x-2} (-x)^m x^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (-x\right )^{m}}{\sqrt {3 \, x - 2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x\right )^{m}}{\sqrt {3 \, x - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 44, normalized size = 0.90 \[ \frac {\sqrt {2}\, \sqrt {-\mathrm {signum}\left (x -\frac {2}{3}\right )}\, x \left (-x \right )^{m} \hypergeom \left (\left [\frac {1}{2}, m +1\right ], \left [m +2\right ], \frac {3 x}{2}\right )}{2 \left (m +1\right ) \sqrt {\mathrm {signum}\left (x -\frac {2}{3}\right )}} \]
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 \[ \int \frac {\left (-x\right )^{m}}{\sqrt {3 \, x - 2}}\,{d x} \]
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 \[ \int \frac {{\left (-x\right )}^m}{\sqrt {3\,x-2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.28, size = 42, normalized size = 0.86 \[ - \frac {\sqrt {2} i x x^{m} e^{i \pi m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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