Optimal. Leaf size=83 \[ -\frac {4^{1+p} (1-x)^{\frac {1}{2}+p} (c x)^{-2 (1+p)} \left (\frac {x}{1+x}\right )^{2 (1+p)} (1+x)^{\frac {3}{2}+p} \, _2F_1\left (\frac {1}{2}+p,2 (1+p);\frac {3}{2}+p;\frac {1-x}{1+x}\right )}{1+2 p} \]
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Rubi [A]
time = 0.01, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {134}
\begin {gather*} -\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)} \, _2F_1\left (p+\frac {1}{2},2 (p+1);p+\frac {3}{2};\frac {1-x}{x+1}\right )}{2 p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 134
Rubi steps
\begin {align*} \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} \, _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*}
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Mathematica [A]
time = 0.08, size = 82, normalized size = 0.99 \begin {gather*} -\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} \, _2F_1\left (\frac {1}{2}+p,2+2 p;\frac {3}{2}+p;\frac {1-x}{1+x}\right )}{c^2 (1+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (1-x \right )^{-\frac {1}{2}+p} \left (1+x \right )^{\frac {1}{2}+p} \left (c x \right )^{-2 p -2}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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