Optimal. Leaf size=68 \[ \frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]
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Rubi [A] time = 0.0254584, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2235, 2234, 2204} \[ \frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b c+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 2235
Rule 2234
Rule 2204
Rubi steps
\begin{align*} \int e^{(a+b x) (c+d x)} \, dx &=\int e^{a c+(b c+a d) x+b d x^2} \, dx\\ &=e^{-\frac{(b c-a d)^2}{4 b d}} \int e^{\frac{(b c+a d+2 b d x)^2}{4 b d}} \, dx\\ &=\frac{e^{-\frac{(b c-a d)^2}{4 b d}} \sqrt{\pi } \text{erfi}\left (\frac{b c+a d+2 b d x}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0164936, size = 68, normalized size = 1. \[ \frac{\sqrt{\pi } e^{-\frac{(b c-a d)^2}{4 b d}} \text{Erfi}\left (\frac{a d+b (c+2 d x)}{2 \sqrt{b} \sqrt{d}}\right )}{2 \sqrt{b} \sqrt{d}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 60, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{\pi }}{2}{{\rm e}^{ac-{\frac{ \left ( ad+bc \right ) ^{2}}{4\,bd}}}}{\it Erf} \left ( -\sqrt{-bd}x+{\frac{ad+bc}{2}{\frac{1}{\sqrt{-bd}}}} \right ){\frac{1}{\sqrt{-bd}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00926, size = 78, normalized size = 1.15 \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (\sqrt{-b d} x - \frac{b c + a d}{2 \, \sqrt{-b d}}\right ) e^{\left (a c - \frac{{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{2 \, \sqrt{-b d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53094, size = 171, normalized size = 2.51 \begin{align*} -\frac{\sqrt{\pi } \sqrt{-b d} \operatorname{erf}\left (\frac{{\left (2 \, b d x + b c + a d\right )} \sqrt{-b d}}{2 \, b d}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, b d} \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*} e^{a c} \int e^{a d x} e^{b c x} e^{b d x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21417, size = 92, normalized size = 1.35 \begin{align*} -\frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-b d}{\left (2 \, x + \frac{b c + a d}{b d}\right )}\right ) e^{\left (-\frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, \sqrt{-b d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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