Optimal. Leaf size=37 \[ 3^{-p-1} 5^p (3 x+2) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {1}{5} (3 x+2)^2\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 37, 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 = {619, 245} \[ 3^{-p-1} 5^p (3 x+2) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {1}{5} (3 x+2)^2\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rule 619
Rubi steps
\begin {align*} \int \left (3+4 x+3 x^2\right )^p \, dx &=\frac {1}{2} \left (3^{-1-p} 5^p\right ) \operatorname {Subst}\left (\int \left (1+\frac {x^2}{20}\right )^p \, dx,x,4+6 x\right )\\ &=3^{-1-p} 5^p (2+3 x) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {1}{5} (2+3 x)^2\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 1.00 \[ 3^{-p-1} 5^p (3 x+2) \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {1}{5} (3 x+2)^2\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (3 \, x^{2} + 4 \, x + 3\right )}^{p}, x\right ) \]
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 \[ \int {\left (3 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.55, size = 0, normalized size = 0.00 \[ \int \left (3 x^{2}+4 x +3\right )^{p}\, dx \]
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 \[ \int {\left (3 \, x^{2} + 4 \, x + 3\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (3\,x^2+4\,x+3\right )}^p \,d x \]
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 \[ \int \left (3 x^{2} + 4 x + 3\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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