Optimal. Leaf size=68 \[ \frac {e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {b c+a d+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \]
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Rubi [A]
time = 0.02, 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 = {2267, 2266,
2235} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2267
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 \begin {gather*} \frac {e^{-\frac {(b c-a d)^2}{4 b d}} \sqrt {\pi } \text {erfi}\left (\frac {a d+b (c+2 d x)}{2 \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 60, normalized size = 0.88
method | result | size |
risch | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {\left (a d -c b \right )^{2}}{4 b d}} \erf \left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right )}{2 \sqrt {-b d}}\) | \(57\) |
default | \(-\frac {\sqrt {\pi }\, {\mathrm e}^{c a -\frac {\left (a d +c b \right )^{2}}{4 b d}} \erf \left (-\sqrt {-b d}\, x +\frac {a d +c b}{2 \sqrt {-b d}}\right )}{2 \sqrt {-b d}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 58, normalized size = 0.85 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 74, normalized size = 1.09 \begin {gather*} -\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 {gather*}
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 \begin {gather*} e^{a c} \int e^{a d x} e^{b c x} e^{b d x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.15, size = 68, normalized size = 1.00 \begin {gather*} -\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 {gather*}
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 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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