Optimal. Leaf size=54 \[ -\frac{2^{2 p+1} (-2 x-2)^{-p-1} \left (x^2+4 x+3\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{x+3}{2}\right )}{p+1} \]
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Rubi [A] time = 0.0102118, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {624} \[ -\frac{2^{2 p+1} (-2 x-2)^{-p-1} \left (x^2+4 x+3\right )^{p+1} \, _2F_1\left (-p,p+1;p+2;\frac{x+3}{2}\right )}{p+1} \]
Antiderivative was successfully verified.
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Rule 624
Rubi steps
\begin{align*} \int \left (3+4 x+x^2\right )^p \, dx &=-\frac{2^{1+2 p} (-2-2 x)^{-1-p} \left (3+4 x+x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac{3+x}{2}\right )}{1+p}\\ \end{align*}
Mathematica [A] time = 0.0203232, size = 48, normalized size = 0.89 \[ \frac{2^p (x+1) (x+3)^{-p} \left (x^2+4 x+3\right )^p \, _2F_1\left (-p,p+1;p+2;\frac{1}{2} (-x-1)\right )}{p+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.841, size = 0, normalized size = 0. \begin{align*} \int \left ({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 (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 (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 (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 (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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