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 = 0.03, antiderivative size = 68, normalized size of antiderivative = 1.00, 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 2204
Rule 2234
Rule 2235
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*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^{-\frac {(b c-a d)^2}{4 b d}} \text {erfi}\left (\frac {a d+b (c+2 d x)}{2 \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 74, normalized size = 1.09 \[ -\frac {\sqrt {\pi } \sqrt {-b d} \operatorname {erf}\left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d}}{2 \, b d}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 68, normalized size = 1.00 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-b d} {\left (2 \, x + \frac {b c + a d}{b d}\right )}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{2 \, \sqrt {-b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 60, normalized size = 0.88 \[ -\frac {\sqrt {\pi }\, \erf \left (-\sqrt {-b d}\, x +\frac {a d +b c}{2 \sqrt {-b d}}\right ) {\mathrm e}^{a c -\frac {\left (a d +b c \right )^{2}}{4 b d}}}{2 \sqrt {-b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.85, size = 58, normalized size = 0.85 \[ \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b d} x - \frac {b c + a d}{2 \, \sqrt {-b d}}\right ) e^{\left (a c - \frac {{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{2 \, \sqrt {-b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 60, normalized size = 0.88 \[ -\frac {\sqrt {\pi }\,{\mathrm {e}}^{\frac {a\,c}{2}-\frac {a^2\,d}{4\,b}-\frac {b\,c^2}{4\,d}}\,\mathrm {erf}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{2\,\sqrt {b\,d}}\right )\,1{}\mathrm {i}}{2\,\sqrt {b\,d}} \]
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 \[ e^{a c} \int e^{a d x} e^{b c x} e^{b d x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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