Optimal. Leaf size=119 \[ \frac {(b c-a d)^2 \text {erf}(a+b x)}{2 b^2 d}-\frac {e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}+\frac {d \text {erf}(a+b x)}{4 b^2}-\frac {d e^{-(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {erfc}(a+b x)}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6362, 2226, 2205, 2209, 2212} \[ \frac {(b c-a d)^2 \text {Erf}(a+b x)}{2 b^2 d}-\frac {e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}+\frac {d \text {Erf}(a+b x)}{4 b^2}-\frac {d e^{-(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {Erfc}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2205
Rule 2209
Rule 2212
Rule 2226
Rule 6362
Rubi steps
\begin {align*} \int (c+d x) \text {erfc}(a+b x) \, dx &=\frac {(c+d x)^2 \text {erfc}(a+b x)}{2 d}+\frac {b \int e^{-(a+b x)^2} (c+d x)^2 \, dx}{d \sqrt {\pi }}\\ &=\frac {(c+d x)^2 \text {erfc}(a+b x)}{2 d}+\frac {b \int \left (\frac {(b c-a d)^2 e^{-(a+b x)^2}}{b^2}+\frac {2 d (b c-a d) e^{-(a+b x)^2} (a+b x)}{b^2}+\frac {d^2 e^{-(a+b x)^2} (a+b x)^2}{b^2}\right ) \, dx}{d \sqrt {\pi }}\\ &=\frac {(c+d x)^2 \text {erfc}(a+b x)}{2 d}+\frac {d \int e^{-(a+b x)^2} (a+b x)^2 \, dx}{b \sqrt {\pi }}+\frac {(2 (b c-a d)) \int e^{-(a+b x)^2} (a+b x) \, dx}{b \sqrt {\pi }}+\frac {(b c-a d)^2 \int e^{-(a+b x)^2} \, dx}{b d \sqrt {\pi }}\\ &=-\frac {(b c-a d) e^{-(a+b x)^2}}{b^2 \sqrt {\pi }}-\frac {d e^{-(a+b x)^2} (a+b x)}{2 b^2 \sqrt {\pi }}+\frac {(b c-a d)^2 \text {erf}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfc}(a+b x)}{2 d}+\frac {d \int e^{-(a+b x)^2} \, dx}{2 b \sqrt {\pi }}\\ &=-\frac {(b c-a d) e^{-(a+b x)^2}}{b^2 \sqrt {\pi }}-\frac {d e^{-(a+b x)^2} (a+b x)}{2 b^2 \sqrt {\pi }}+\frac {d \text {erf}(a+b x)}{4 b^2}+\frac {(b c-a d)^2 \text {erf}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfc}(a+b x)}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.15, size = 104, normalized size = 0.87 \[ \frac {e^{-(a+b x)^2} \left (\sqrt {\pi } e^{(a+b x)^2} \left (2 a^2 d-4 a b c+d\right ) \text {erf}(a+b x)+2 \sqrt {\pi } b^2 x e^{(a+b x)^2} (2 c+d x) \text {erfc}(a+b x)+2 a d-4 b c-2 b d x\right )}{4 \sqrt {\pi } b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.53, size = 110, normalized size = 0.92 \[ \frac {2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x - 2 \, \sqrt {\pi } {\left (b d x + 2 \, b c - a d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - {\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi {\left (4 \, a b c - {\left (2 \, a^{2} + 1\right )} d\right )}\right )} \operatorname {erf}\left (b x + a\right )}{4 \, \pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.30, size = 158, normalized size = 1.33 \[ \frac {1}{2} \, d x^{2} - {\left (x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }}\right )} c - \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b}\right )} d + c x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 122, normalized size = 1.03 \[ \frac {\frac {\mathrm {erfc}\left (b x +a \right ) \left (b x +a \right )^{2} d}{2 b}-\frac {\mathrm {erfc}\left (b x +a \right ) a d \left (b x +a \right )}{b}+\mathrm {erfc}\left (b x +a \right ) c \left (b x +a \right )+\frac {d \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )+a d \,{\mathrm e}^{-\left (b x +a \right )^{2}}-{\mathrm e}^{-\left (b x +a \right )^{2}} b c}{\sqrt {\pi }\, b}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \operatorname {erfc}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.22, size = 119, normalized size = 1.00 \[ c\,x\,\mathrm {erfc}\left (a+b\,x\right )-{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {c}{b\,\sqrt {\pi }}-\frac {a\,d}{2\,b^2\,\sqrt {\pi }}\right )-\frac {\mathrm {erfc}\left (a+b\,x\right )\,\left (\frac {d\,a^2}{2}-b\,c\,a+\frac {d}{4}\right )}{b^2}+\frac {d\,x^2\,\mathrm {erfc}\left (a+b\,x\right )}{2}-\frac {d\,x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{2\,b\,\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 1.45, size = 178, normalized size = 1.50 \[ \begin {cases} - \frac {a^{2} d \operatorname {erfc}{\left (a + b x \right )}}{2 b^{2}} + \frac {a c \operatorname {erfc}{\left (a + b x \right )}}{b} + \frac {a d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b^{2}} + c x \operatorname {erfc}{\left (a + b x \right )} + \frac {d x^{2} \operatorname {erfc}{\left (a + b x \right )}}{2} - \frac {c e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} - \frac {d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b} - \frac {d \operatorname {erfc}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \operatorname {erfc}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________