Optimal. Leaf size=52 \[ e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]
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Rubi [A] time = 0.234901, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {6707, 2176, 2180, 2204} \[ e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\frac{1}{2} \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6707
Rule 2176
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int e^{a+b x+c x^2} (b+2 c x) \sqrt{a+b x+c x^2} \, dx &=\operatorname{Subst}\left (\int e^x \sqrt{x} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,a+b x+c x^2\right )\\ &=e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )\\ &=e^{a+b x+c x^2} \sqrt{a+b x+c x^2}-\frac{1}{2} \sqrt{\pi } \text{erfi}\left (\sqrt{a+b x+c x^2}\right )\\ \end{align*}
Mathematica [A] time = 0.0475016, size = 46, normalized size = 0.88 \[ \frac{\sqrt{a+x (b+c x)} \text{Gamma}\left (\frac{3}{2},-a-x (b+c x)\right )}{\sqrt{-a-x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 44, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }}{2}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) }+{{\rm e}^{c{x}^{2}+bx+a}}\sqrt{c{x}^{2}+bx+a} \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 \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}\,{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 (\sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.8545, size = 78, normalized size = 1.5 \begin{align*} \frac{\left (\sqrt{- a - b x - c x^{2}} e^{a + b x + c x^{2}} + \frac{\sqrt{\pi } \operatorname{erfc}{\left (\sqrt{- a - b x - c x^{2}} \right )}}{2}\right ) \sqrt{a + b x + c x^{2}}}{\sqrt{- a - b x - c x^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25469, size = 63, normalized size = 1.21 \begin{align*} -\frac{1}{2} \, \sqrt{\pi } i \operatorname{erf}\left (-\sqrt{c x^{2} + b x + a} i\right ) + \sqrt{c x^{2} + b x + a} e^{\left (c x^{2} + b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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