Integrand size = 30, antiderivative size = 83 \[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=-\frac {4^{1+p} (1-x)^{\frac {1}{2}+p} (c x)^{-2 (1+p)} \left (\frac {x}{1+x}\right )^{2 (1+p)} (1+x)^{\frac {3}{2}+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+p,2 (1+p),\frac {3}{2}+p,\frac {1-x}{1+x}\right )}{1+2 p} \]
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Time = 0.02 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {134} \[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=-\frac {4^{p+1} (1-x)^{p+\frac {1}{2}} \left (\frac {x}{x+1}\right )^{2 (p+1)} (x+1)^{p+\frac {3}{2}} (c x)^{-2 (p+1)} \operatorname {Hypergeometric2F1}\left (p+\frac {1}{2},2 (p+1),p+\frac {3}{2},\frac {1-x}{x+1}\right )}{2 p+1} \]
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Rule 134
Rubi steps \begin{align*} \text {integral}& = -\frac {4^{1+p} (1-x)^{\frac {1}{2}+p} (c x)^{-2 (1+p)} \left (\frac {x}{1+x}\right )^{2 (1+p)} (1+x)^{\frac {3}{2}+p} \, _2F_1\left (\frac {1}{2}+p,2 (1+p);\frac {3}{2}+p;\frac {1-x}{1+x}\right )}{1+2 p} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.99 \[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=-\frac {4^{1+p} (1-x)^{\frac {1}{2}+p} (c x)^{-2 p} \left (\frac {x}{1+x}\right )^{2 p} (1+x)^{-\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}+p,2+2 p,\frac {3}{2}+p,\frac {1-x}{1+x}\right )}{c^2 (1+2 p)} \]
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\[\int \left (1-x \right )^{-\frac {1}{2}+p} \left (1+x \right )^{\frac {1}{2}+p} \left (c x \right )^{-2-2 p}d x\]
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\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int { \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}} \,d x } \]
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\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int \left (c x\right )^{- 2 p - 2} \left (1 - x\right )^{p - \frac {1}{2}} \left (x + 1\right )^{p + \frac {1}{2}}\, dx \]
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\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int { \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}} \,d x } \]
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\[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int { \frac {{\left (x + 1\right )}^{p + \frac {1}{2}} {\left (-x + 1\right )}^{p - \frac {1}{2}}}{\left (c x\right )^{2 \, p + 2}} \,d x } \]
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Timed out. \[ \int (1-x)^{-\frac {1}{2}+p} (c x)^{-2 (1+p)} (1+x)^{\frac {1}{2}+p} \, dx=\int \frac {{\left (1-x\right )}^{p-\frac {1}{2}}\,{\left (x+1\right )}^{p+\frac {1}{2}}}{{\left (c\,x\right )}^{2\,p+2}} \,d x \]
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