Optimal. Leaf size=37 \[ 5^{-p-1} 11^p (5 x+2) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{11} (5 x+2)^2\right ) \]
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Rubi [A] time = 0.0155656, antiderivative size = 37, 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} \[ 5^{-p-1} 11^p (5 x+2) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{11} (5 x+2)^2\right ) \]
Antiderivative was successfully verified.
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Rule 619
Rule 245
Rubi steps
\begin{align*} \int \left (3+4 x+5 x^2\right )^p \, dx &=\frac{1}{2} \left (5^{-1-p} 11^p\right ) \operatorname{Subst}\left (\int \left (1+\frac{x^2}{44}\right )^p \, dx,x,4+10 x\right )\\ &=5^{-1-p} 11^p (2+5 x) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{11} (2+5 x)^2\right )\\ \end{align*}
Mathematica [A] time = 0.0104419, size = 37, normalized size = 1. \[ 5^{-p-1} 11^p (5 x+2) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{1}{11} (5 x+2)^2\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 3.641, size = 0, normalized size = 0. \begin{align*} \int \left ( 5\,{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 (5 \, 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 (5 \, 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 (5 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 (5 \, 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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