Integrand size = 13, antiderivative size = 36 \[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\left (\frac {3}{2}\right )^{-1-m} \sqrt {-2+3 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\frac {3 x}{2}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {67} \[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\left (\frac {3}{2}\right )^{-m-1} \sqrt {3 x-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\frac {3 x}{2}\right ) \]
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Rule 67
Rubi steps \begin{align*} \text {integral}& = \left (\frac {3}{2}\right )^{-1-m} \sqrt {-2+3 x} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1-\frac {3 x}{2}\right ) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\left (\frac {3}{2}\right )^{-1-m} \sqrt {-2+3 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1-\frac {3 x}{2}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 5.
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.19
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \sqrt {-\operatorname {signum}\left (-\frac {2}{3}+x \right )}\, x^{1+m} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},1+m ;2+m ;\frac {3 x}{2}\right )}{2 \sqrt {\operatorname {signum}\left (-\frac {2}{3}+x \right )}\, \left (1+m \right )}\) | \(43\) |
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\[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\int { \frac {x^{m}}{\sqrt {3 \, x - 2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 1.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=- \frac {\sqrt {2} i x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {3 x}{2}} \right )}}{2 \Gamma \left (m + 2\right )} \]
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\[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\int { \frac {x^{m}}{\sqrt {3 \, x - 2}} \,d x } \]
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\[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\int { \frac {x^{m}}{\sqrt {3 \, x - 2}} \,d x } \]
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Timed out. \[ \int \frac {x^m}{\sqrt {-2+3 x}} \, dx=\int \frac {x^m}{\sqrt {3\,x-2}} \,d x \]
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