Optimal. Leaf size=115 \[ -\frac {(b c-a d)^2 \text {erfi}(a+b x)}{2 b^2 d}-\frac {e^{(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}+\frac {d \text {erfi}(a+b x)}{4 b^2}-\frac {d e^{(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d} \]
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Rubi [A] time = 0.11, antiderivative size = 115, 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 = {6363, 2226, 2204, 2209, 2212} \[ -\frac {(b c-a d)^2 \text {Erfi}(a+b x)}{2 b^2 d}-\frac {e^{(a+b x)^2} (b c-a d)}{\sqrt {\pi } b^2}+\frac {d \text {Erfi}(a+b x)}{4 b^2}-\frac {d e^{(a+b x)^2} (a+b x)}{2 \sqrt {\pi } b^2}+\frac {(c+d x)^2 \text {Erfi}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2209
Rule 2212
Rule 2226
Rule 6363
Rubi steps
\begin {align*} \int (c+d x) \text {erfi}(a+b x) \, dx &=\frac {(c+d x)^2 \text {erfi}(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 {erfi}(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 {erfi}(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 {erfi}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfi}(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 {erfi}(a+b x)}{4 b^2}-\frac {(b c-a d)^2 \text {erfi}(a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \text {erfi}(a+b x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 78, normalized size = 0.68 \[ \frac {\sqrt {\pi } \text {erfi}(a+b x) \left (-2 a^2 d+4 a b c+4 b^2 c x+2 b^2 d x^2+d\right )-2 e^{(a+b x)^2} (-a d+2 b c+b d x)}{4 \sqrt {\pi } b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 89, normalized size = 0.77 \[ -\frac {2 \, \sqrt {\pi } {\left (b d x + 2 \, b c - a d\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (2 \, \pi b^{2} d x^{2} + 4 \, \pi b^{2} c x + \pi {\left (4 \, a b c - {\left (2 \, a^{2} - 1\right )} d\right )}\right )} \operatorname {erfi}\left (b x + a\right )}{4 \, \pi b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 117, normalized size = 1.02 \[ \frac {\frac {\erfi \left (b x +a \right ) \left (b x +a \right )^{2} d}{2 b}-\frac {\erfi \left (b x +a \right ) a d \left (b x +a \right )}{b}+\erfi \left (b x +a \right ) c \left (b x +a \right )-\frac {d \left (\frac {\left (b x +a \right ) {\mathrm e}^{\left (b x +a \right )^{2}}}{2}-\frac {\sqrt {\pi }\, \erfi \left (b x +a \right )}{4}\right )-a d \,{\mathrm e}^{\left (b x +a \right )^{2}}+{\mathrm e}^{\left (b x +a \right )^{2}} b c}{\sqrt {\pi }\, b}}{b} \]
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 \[ \int {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 106, normalized size = 0.92 \[ \frac {\frac {{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {a\,d}{2}-b\,c\right )}{b^2}-\frac {d\,x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2}}{2\,b}}{\sqrt {\pi }}+\mathrm {erfi}\left (a+b\,x\right )\,\left (\frac {d\,x^2}{2}+c\,x\right )+\frac {\mathrm {erfi}\left (a+b\,x\right )\,\left (-2\,d\,a^2\,b+4\,c\,a\,b^2+d\,b\right )}{4\,b^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.07, size = 178, normalized size = 1.55 \[ \begin {cases} - \frac {a^{2} d \operatorname {erfi}{\left (a + b x \right )}}{2 b^{2}} + \frac {a c \operatorname {erfi}{\left (a + b x \right )}}{b} + \frac {a d e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt {\pi } b^{2}} + c x \operatorname {erfi}{\left (a + b x \right )} + \frac {d x^{2} \operatorname {erfi}{\left (a + b x \right )}}{2} - \frac {c e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{\sqrt {\pi } b} - \frac {d x e^{a^{2}} e^{b^{2} x^{2}} e^{2 a b x}}{2 \sqrt {\pi } b} + \frac {d \operatorname {erfi}{\left (a + b x \right )}}{4 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \operatorname {erfi}{\relax (a )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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