Integrand size = 12, antiderivative size = 32 \[ \int \left (3+4 x+4 x^2\right )^p \, dx=2^{-1+p} (1+2 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {1}{2} (1+2 x)^2\right ) \]
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Time = 0.01 (sec) , antiderivative size = 32, 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} \[ \int \left (3+4 x+4 x^2\right )^p \, dx=2^{p-1} (2 x+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {1}{2} (2 x+1)^2\right ) \]
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Rule 251
Rule 633
Rubi steps \begin{align*} \text {integral}& = 2^{-3+p} \text {Subst}\left (\int \left (1+\frac {x^2}{32}\right )^p \, dx,x,4+8 x\right ) \\ & = 2^{-1+p} (1+2 x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {1}{2} (1+2 x)^2\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \left (3+4 x+4 x^2\right )^p \, dx=2^{-3+p} (4+8 x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {1}{32} (4+8 x)^2\right ) \]
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\[\int \left (4 x^{2}+4 x +3\right )^{p}d x\]
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\[ \int \left (3+4 x+4 x^2\right )^p \, dx=\int { {\left (4 \, x^{2} + 4 \, x + 3\right )}^{p} \,d x } \]
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\[ \int \left (3+4 x+4 x^2\right )^p \, dx=\int \left (4 x^{2} + 4 x + 3\right )^{p}\, dx \]
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\[ \int \left (3+4 x+4 x^2\right )^p \, dx=\int { {\left (4 \, x^{2} + 4 \, x + 3\right )}^{p} \,d x } \]
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\[ \int \left (3+4 x+4 x^2\right )^p \, dx=\int { {\left (4 \, x^{2} + 4 \, x + 3\right )}^{p} \,d x } \]
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Timed out. \[ \int \left (3+4 x+4 x^2\right )^p \, dx=\int {\left (4\,x^2+4\,x+3\right )}^p \,d x \]
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