Optimal. Leaf size=21 \[ \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]
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Rubi [A] time = 0.263932, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6707, 2180, 2204} \[ \sqrt{\pi } \text{Erfi}\left (\sqrt{a+b x+c x^2}\right ) \]
Antiderivative was successfully verified.
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Rule 6707
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{e^{a+b x+c x^2} (b+2 c x)}{\sqrt{a+b x+c x^2}} \, dx &=\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,a+b x+c x^2\right )\\ &=2 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )\\ &=\sqrt{\pi } \text{erfi}\left (\sqrt{a+b x+c x^2}\right )\\ \end{align*}
Mathematica [B] time = 0.0589681, size = 46, normalized size = 2.19 \[ \frac{\sqrt{-a-x (b+c x)} \text{Gamma}\left (\frac{1}{2},-a-x (b+c x)\right )}{\sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 18, normalized size = 0.9 \begin{align*}{\it erfi} \left ( \sqrt{c{x}^{2}+bx+a} \right ) \sqrt{\pi } \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 \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{\sqrt{c x^{2} + b x + a}}\,{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 (\frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{\sqrt{c x^{2} + b x + a}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 5.72087, size = 49, normalized size = 2.33 \begin{align*} \frac{\sqrt{\pi } \sqrt{- a - b x - c x^{2}} \operatorname{erfc}{\left (\sqrt{- a - b x - c x^{2}} \right )}}{\sqrt{a + b x + c x^{2}}} \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 \frac{{\left (2 \, c x + b\right )} e^{\left (c x^{2} + b x + a\right )}}{\sqrt{c x^{2} + b x + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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