Integrand size = 15, antiderivative size = 34 \[ \int \frac {(-x)^m}{\sqrt {2+3 x}} \, dx=-\frac {(-x)^{1+m} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-\frac {3 x}{2}\right )}{\sqrt {2} (1+m)} \]
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Time = 0.01 (sec) , antiderivative size = 34, 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 = {66} \[ \int \frac {(-x)^m}{\sqrt {2+3 x}} \, dx=-\frac {(-x)^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},m+1,m+2,-\frac {3 x}{2}\right )}{\sqrt {2} (m+1)} \]
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Rule 66
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.94 \[ \int \frac {(-x)^m}{\sqrt {2+3 x}} \, dx=\frac {(-x)^m x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,2+m,-\frac {3 x}{2}\right )}{\sqrt {2} (1+m)} \]
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Time = 0.10 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88
method | result | size |
meijerg | \(\frac {\sqrt {2}\, \left (-x \right )^{m} x {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {1}{2},1+m ;2+m ;-\frac {3 x}{2}\right )}{2+2 m}\) | \(30\) |
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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 = 1.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.29 \[ \int \frac {(-x)^m}{\sqrt {2+3 x}} \, dx=\frac {2 \cdot 2^{m} \sqrt {3} \cdot 3^{- m} \sqrt {x + \frac {2}{3}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - m \\ \frac {3}{2} \end {matrix}\middle | {\frac {3 \left (x + \frac {2}{3}\right ) e^{2 i \pi }}{2}} \right )}}{3} \]
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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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