Optimal. Leaf size=54 \[ -\frac {2^{1+2 p} (-2-2 x)^{-1-p} \left (3+4 x+x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {3+x}{2}\right )}{1+p} \]
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Rubi [A]
time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {638}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 638
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*}
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Mathematica [A]
time = 0.06, size = 48, normalized size = 0.89 \begin {gather*} \frac {2^p (1+x) (3+x)^{-p} \left (3+4 x+x^2\right )^p \, _2F_1\left (-p,1+p;2+p;\frac {1}{2} (-1-x)\right )}{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.13, size = 0, normalized size = 0.00 \[\int \left (x^{2}+4 x +3\right )^{p}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^{2} + 4 x + 3\right )^{p}\, dx \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 = -0.02 \begin {gather*} \int {\left (x^2+4\,x+3\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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