Integrand size = 12, antiderivative size = 31 \[ \int \left (3+4 x-2 x^2\right )^p \, dx=-5^p (1-x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {2}{5} (1-x)^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {633, 251} \[ \int \left (3+4 x-2 x^2\right )^p \, dx=-5^p (1-x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {2}{5} (1-x)^2\right ) \]
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Rule 251
Rule 633
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{4} 5^p \text {Subst}\left (\int \left (1-\frac {x^2}{40}\right )^p \, dx,x,4-4 x\right )\right ) \\ & = -5^p (1-x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};\frac {2}{5} (1-x)^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \left (3+4 x-2 x^2\right )^p \, dx=5^p (-1+x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},\frac {2}{5} (-1+x)^2\right ) \]
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\[\int \left (-2 x^{2}+4 x +3\right )^{p}d x\]
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\[ \int \left (3+4 x-2 x^2\right )^p \, dx=\int { {\left (-2 \, x^{2} + 4 \, x + 3\right )}^{p} \,d x } \]
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\[ \int \left (3+4 x-2 x^2\right )^p \, dx=\int \left (- 2 x^{2} + 4 x + 3\right )^{p}\, dx \]
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\[ \int \left (3+4 x-2 x^2\right )^p \, dx=\int { {\left (-2 \, x^{2} + 4 \, x + 3\right )}^{p} \,d x } \]
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\[ \int \left (3+4 x-2 x^2\right )^p \, dx=\int { {\left (-2 \, x^{2} + 4 \, x + 3\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (3+4 x-2 x^2\right )^p \, dx=\int {\left (-2\,x^2+4\,x+3\right )}^p \,d x \]
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