Optimal. Leaf size=118 \[ -\frac {(b c-a d)^2 \text {erf}(a+b x)}{2 b^2 d}+\frac {e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}-\frac {d \text {erf}(a+b x)}{4 b^2}+\frac {d e^{-(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {erf}(a+b x)}{2 d} \]
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Rubi [A] time = 0.12, antiderivative size = 118, 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 = {6361, 2226, 2205, 2209, 2212} \[ -\frac {(b c-a d)^2 \text {Erf}(a+b x)}{2 b^2 d}+\frac {e^{-(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}-\frac {d \text {Erf}(a+b x)}{4 b^2}+\frac {d e^{-(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {Erf}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2209
Rule 2212
Rule 2226
Rule 6361
Rubi steps
\begin {align*} \int (c+d x) \text {erf}(a+b x) \, dx &=\frac {(c+d x)^2 \text {erf}(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 {erf}(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 {erf}(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 {erf}(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 {erf}(a+b x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 88, normalized size = 0.75 \[ \frac {e^{-(a+b x)^2} \left (-\sqrt {\pi } e^{(a+b x)^2} \text {erf}(a+b x) \left (2 a^2 d-4 a b c-4 b^2 c x-2 b^2 d x^2+d\right )-2 a d+4 b c+2 b d x\right )}{4 \sqrt {\pi } b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 91, normalized size = 0.77 \[ \frac {2 \, \sqrt {\pi } {\left (b d x + 2 \, b c - a d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} + {\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi {\left (4 \, a b c - {\left (2 \, a^{2} + 1\right )} d\right )}\right )} \operatorname {erf}\left (b x + a\right )}{4 \, \pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 149, normalized size = 1.26 \[ \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} \operatorname {erf}\left (b x + a\right ) - \frac {4 \, \sqrt {\pi } {\left (\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} c - \frac {\sqrt {\pi } {\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}\right )} d}{b}}{4 \, \pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 111, normalized size = 0.94 \[ \frac {\frac {\erf \left (b x +a \right ) \left (\frac {\left (b x +a \right )^{2} d}{2}-a d \left (b x +a \right )+b c \left (b x +a \right )\right )}{b}-\frac {d \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \erf \left (b x +a \right )}{4}\right )+a d \,{\mathrm e}^{-\left (b x +a \right )^{2}}-{\mathrm e}^{-\left (b x +a \right )^{2}} b c}{\sqrt {\pi }\, b}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} \operatorname {erf}\left (b x + a\right ) - \frac {-\frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a b^{2} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {3}{2}}} + \frac {b^{2} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {3}{2}}}\right )} b c}{\sqrt {-b^{2}}} - \frac {{\left (\frac {\sqrt {\pi } {\left (b^{2} x + a b\right )} a^{2} b^{3} {\left (\operatorname {erf}\left (\frac {\sqrt {{\left (b^{2} x + a b\right )}^{2}}}{b}\right ) - 1\right )}}{\sqrt {{\left (b^{2} x + a b\right )}^{2}} \left (-b^{2}\right )^{\frac {5}{2}}} - \frac {{\left (b^{2} x + a b\right )}^{3} b^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}{{\left ({\left (b^{2} x + a b\right )}^{2}\right )}^{\frac {3}{2}} \left (-b^{2}\right )^{\frac {5}{2}}} + \frac {2 \, a b^{3} e^{\left (-\frac {{\left (b^{2} x + a b\right )}^{2}}{b^{2}}\right )}}{\left (-b^{2}\right )^{\frac {5}{2}}}\right )} b d}{2 \, \sqrt {-b^{2}}}}{\sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 126, normalized size = 1.07 \[ \mathrm {erf}\left (a+b\,x\right )\,\left (\frac {d\,x^2}{2}+c\,x\right )-\frac {\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (\frac {a\,d}{2}-b\,c\right )}{b^2}-\frac {d\,x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{2\,b}}{\sqrt {\pi }}+\frac {\sqrt {\pi }\,\mathrm {erfi}\left (a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )\,\left (\frac {2\,d\,a^2+d}{2\,\sqrt {\pi }}-\frac {2\,a\,b\,c}{\sqrt {\pi }}\right )\,1{}\mathrm {i}}{2\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.56, size = 178, normalized size = 1.51 \[ \begin {cases} - \frac {a^{2} d \operatorname {erf}{\left (a + b x \right )}}{2 b^{2}} + \frac {a c \operatorname {erf}{\left (a + b x \right )}}{b} - \frac {a d e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b^{2}} + c x \operatorname {erf}{\left (a + b x \right )} + \frac {d x^{2} \operatorname {erf}{\left (a + b x \right )}}{2} + \frac {c e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{\sqrt {\pi } b} + \frac {d x e^{- a^{2}} e^{- b^{2} x^{2}} e^{- 2 a b x}}{2 \sqrt {\pi } b} - \frac {d \operatorname {erf}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \operatorname {erf}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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