Optimal. Leaf size=32 \[ 2^{p-1} (2 x+1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{2} (2 x+1)^2\right ) \]
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Rubi [A] time = 0.0137135, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {619, 245} \[ 2^{p-1} (2 x+1) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{2} (2 x+1)^2\right ) \]
Antiderivative was successfully verified.
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Rule 619
Rule 245
Rubi steps
\begin{align*} \int \left (3+4 x+4 x^2\right )^p \, dx &=2^{-3+p} \operatorname{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*}
Mathematica [A] time = 0.0072556, size = 32, normalized size = 1. \[ 2^{p-3} (8 x+4) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{32} (8 x+4)^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 3.586, size = 0, normalized size = 0. \begin{align*} \int \left ( 4\,{x}^{2}+4\,x+3 \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (4 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (4 \, x^{2} + 4 \, x + 3\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (4 x^{2} + 4 x + 3\right )^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (4 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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