Integrand size = 15, antiderivative size = 37 \[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=-\left (\frac {3}{2}\right )^{-1-m} \sqrt {-2-3 x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+\frac {3 x}{2}\right ) \]
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Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {67} \[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=-\left (\frac {3}{2}\right )^{-m-1} \sqrt {-3 x-2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},\frac {3 x}{2}+1\right ) \]
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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 = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=-\frac {2}{3} \left (1+\frac {1}{2} (-2-3 x)\right )^{-m} \sqrt {-2-3 x} x^{-m} \left (-x^2\right )^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m,\frac {3}{2},1+\frac {3 x}{2}\right ) \]
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Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.84
method | result | size |
meijerg | \(-\frac {i \sqrt {2}\, \left (-x \right )^{m} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},1+m ;2+m ;-\frac {3 x}{2}\right )}{2 \left (1+m \right )}\) | \(31\) |
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\[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=\int { \frac {\left (-x\right )^{m}}{\sqrt {-3 \, x - 2}} \,d x } \]
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Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.30 \[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=- \frac {2 \cdot 2^{m} \sqrt {3} \cdot 3^{- m} i \sqrt {x + \frac {2}{3}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - m \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 \left (x + \frac {2}{3}\right ) e^{2 i \pi }}{2}} \right )}}{3} \]
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\[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=\int { \frac {\left (-x\right )^{m}}{\sqrt {-3 \, x - 2}} \,d x } \]
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\[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=\int { \frac {\left (-x\right )^{m}}{\sqrt {-3 \, x - 2}} \,d x } \]
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Timed out. \[ \int \frac {(-x)^m}{\sqrt {-2-3 x}} \, dx=\int \frac {{\left (-x\right )}^m}{\sqrt {-3\,x-2}} \,d x \]
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