Optimal. Leaf size=31 \[ \frac {x^{m+1} \, _2F_1\left (\frac {1}{2},m+1;m+2;-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 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 = {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 = 31, normalized size = 1.00 \[ \frac {x^{m+1} \, _2F_1\left (\frac {1}{2},m+1;m+2;-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\sqrt {3 \, x + 2}}, x\right ) \]
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 \[ \int \frac {x^{m}}{\sqrt {3 \, x + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 29, normalized size = 0.94 \[ \frac {\sqrt {2}\, x^{m +1} \hypergeom \left (\left [\frac {1}{2}, m +1\right ], \left [m +2\right ], -\frac {3 x}{2}\right )}{2 m +2} \]
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 \[ \int \frac {x^{m}}{\sqrt {3 \, x + 2}}\,{d x} \]
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 \[ \int \frac {x^m}{\sqrt {3\,x+2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.20, size = 37, normalized size = 1.19 \[ \frac {\sqrt {2} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {3 x e^{i \pi }}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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