Optimal. Leaf size=21 \[ (1+x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-2 (1+x)^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 21, normalized size of antiderivative = 1.00, 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 = {633, 251}
\begin {gather*} (x+1) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-2 (x+1)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rule 633
Rubi steps
\begin {align*} \int \left (3+4 x+2 x^2\right )^p \, dx &=\frac {1}{4} \text {Subst}\left (\int \left (1+\frac {x^2}{8}\right )^p \, dx,x,4+4 x\right )\\ &=(1+x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-2 (1+x)^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} (1+x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-2 (1+x)^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (2 x^{2}+4 x +3\right )^{p}\, dx\]
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 (2 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 = -0.05 \begin {gather*} \int {\left (2\,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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