Optimal. Leaf size=86 \[ \frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6383, 2234, 2205} \[ \frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {Erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \sqrt {b^2-d}}+\frac {e^{c+d x^2} \text {Erfc}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2205
Rule 2234
Rule 6383
Rubi steps
\begin {align*} \int e^{c+d x^2} x \text {erfc}(a+b x) \, dx &=\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}+\frac {b \int e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} \, dx}{d \sqrt {\pi }}\\ &=\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}+\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d \sqrt {\pi }}\\ &=\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d}+\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 81, normalized size = 0.94 \[ \frac {e^c \left (\frac {b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {x \left (d-b^2\right )-a b}{\sqrt {d-b^2}}\right )}{\sqrt {d-b^2}}+e^{d x^2} \text {erfc}(a+b x)\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.51, size = 108, normalized size = 1.26 \[ \frac {\sqrt {b^{2} - d} b \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} + {\left (b^{2} - {\left (b^{2} - d\right )} \operatorname {erf}\left (b x + a\right ) - d\right )} e^{\left (d x^{2} + c\right )}}{2 \, {\left (b^{2} d - d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.25, size = 175, normalized size = 2.03 \[ \frac {\frac {b \,{\mathrm e}^{\frac {\left (b x +a \right )^{2} d}{b^{2}}-\frac {2 a d \left (b x +a \right )}{b^{2}}+\frac {a^{2} d}{b^{2}}+c}}{2 d}-\frac {\erf \left (b x +a \right ) b \,{\mathrm e}^{\frac {\left (b x +a \right )^{2} d}{b^{2}}-\frac {2 a d \left (b x +a \right )}{b^{2}}+\frac {a^{2} d}{b^{2}}+c}}{2 d}+\frac {b \,{\mathrm e}^{\frac {a^{2} d}{b^{2}}+c -\frac {a^{2} d^{2}}{b^{4} \left (-1+\frac {d}{b^{2}}\right )}} \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, \left (b x +a \right )+\frac {a d}{b^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{2 d \sqrt {1-\frac {d}{b^{2}}}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________