Optimal. Leaf size=107 \[ \frac {e^{x (a d+b c)+a c+b d x^2}}{2 b d}-\frac {\sqrt {\pi } (a d+b c) e^{-\frac {(b c-a d)^2}{4 b d}} \text {erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{4 b^{3/2} d^{3/2}} \]
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Rubi [A] time = 0.10, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2244, 2240, 2234, 2204} \[ \frac {e^{x (a d+b c)+a c+b d x^2}}{2 b d}-\frac {\sqrt {\pi } (a d+b c) e^{-\frac {(b c-a d)^2}{4 b d}} \text {Erfi}\left (\frac {a d+b c+2 b d x}{2 \sqrt {b} \sqrt {d}}\right )}{4 b^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2234
Rule 2240
Rule 2244
Rubi steps
\begin {align*} \int e^{(a+b x) (c+d x)} x \, dx &=\int e^{a c+(b c+a d) x+b d x^2} x \, dx\\ &=\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b d}-\frac {(b c+a d) \int e^{a c+(b c+a d) x+b d x^2} \, dx}{2 b d}\\ &=\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b d}-\frac {\left ((b c+a d) e^{-\frac {(b c-a d)^2}{4 b d}}\right ) \int e^{\frac {(b c+a d+2 b d x)^2}{4 b d}} \, dx}{2 b d}\\ &=\frac {e^{a c+(b c+a d) x+b d x^2}}{2 b d}-\frac {(b c+a d) 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 )}{4 b^{3/2} d^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 116, normalized size = 1.08 \[ \frac {e^{-\frac {(b c-a d)^2}{4 b d}} \left (2 \sqrt {b} \sqrt {d} e^{\frac {(a d+b (c+2 d x))^2}{4 b d}}-\sqrt {\pi } (a d+b c) \text {erfi}\left (\frac {a d+b (c+2 d x)}{2 \sqrt {b} \sqrt {d}}\right )\right )}{4 b^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 107, normalized size = 1.00 \[ \frac {\sqrt {\pi } {\left (b c + a d\right )} \sqrt {-b d} \operatorname {erf}\left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d}}{2 \, b d}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )} + 2 \, b d e^{\left (b d x^{2} + a c + {\left (b c + a d\right )} x\right )}}{4 \, b^{2} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 104, normalized size = 0.97 \[ \frac {\frac {\sqrt {\pi } {\left (b c + a d\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-b d} {\left (2 \, x + \frac {b c + a d}{b d}\right )}\right ) e^{\left (-\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{4 \, b d}\right )}}{\sqrt {-b d}} + 2 \, e^{\left (b d x^{2} + b c x + a d x + a c\right )}}{4 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 102, normalized size = 0.95 \[ \frac {\left (a d +b c \right ) \sqrt {\pi }\, \erf \left (-\sqrt {-b d}\, x +\frac {a d +b c}{2 \sqrt {-b d}}\right ) {\mathrm e}^{a c -\frac {\left (a d +b c \right )^{2}}{4 b d}}}{4 \sqrt {-b d}\, b d}+\frac {{\mathrm e}^{b d \,x^{2}+a c +\left (a d +b c \right ) x}}{2 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.38, size = 143, normalized size = 1.34 \[ -\frac {{\left (\frac {\sqrt {\pi } {\left (2 \, b d x + b c + a d\right )} {\left (b c + a d\right )} {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}\right ) - 1\right )}}{\left (b d\right )^{\frac {3}{2}} \sqrt {-\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{b d}}} - \frac {2 \, b d e^{\left (\frac {{\left (2 \, b d x + b c + a d\right )}^{2}}{4 \, b d}\right )}}{\left (b d\right )^{\frac {3}{2}}}\right )} e^{\left (a c - \frac {{\left (b c + a d\right )}^{2}}{4 \, b d}\right )}}{4 \, \sqrt {b d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.67, size = 95, normalized size = 0.89 \[ \frac {{\mathrm {e}}^{a\,c+a\,d\,x+b\,c\,x+b\,d\,x^2}}{2\,b\,d}-\frac {\sqrt {\pi }\,{\mathrm {e}}^{\frac {a\,c}{2}-\frac {a^2\,d}{4\,b}-\frac {b\,c^2}{4\,d}}\,\mathrm {erfi}\left (\frac {\frac {a\,d}{2}+\frac {b\,c}{2}+b\,d\,x}{\sqrt {b\,d}}\right )\,\left (a\,d+b\,c\right )}{4\,b\,d\,\sqrt {b\,d}} \]
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 \[ e^{a c} \int x e^{a d x} e^{b c x} e^{b d x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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