Optimal. Leaf size=34 \[ -\frac {(-x)^{m+1} \, _2F_1\left (\frac {1}{2},m+1;m+2;-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {64} \[ -\frac {(-x)^{m+1} \, _2F_1\left (\frac {1}{2},m+1;m+2;-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \]
Antiderivative was successfully verified.
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Rule 64
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 = 0.01, size = 32, normalized size = 0.94 \[ \frac {x (-x)^m \, _2F_1\left (\frac {1}{2},m+1;m+2;-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, 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 [A] time = 0.06, size = 30, normalized size = 0.88 \[ \frac {\sqrt {2}\, x \left (-x \right )^{m} \hypergeom \left (\left [\frac {1}{2}, m +1\right ], \left [m +2\right ], -\frac {3 x}{2}\right )}{2 m +2} \]
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 = -0.03 \[ \int \frac {{\left (-x\right )}^m}{\sqrt {3\,x+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.20, size = 44, normalized size = 1.29 \[ \frac {2 \cdot 2^{m} \sqrt {3} \cdot 3^{- m} \sqrt {x + \frac {2}{3}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - m \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 \left (x + \frac {2}{3}\right ) e^{2 i \pi }}{2}} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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