Optimal. Leaf size=184 \[ \frac {(a+b x) (b c-a d) \text {erfi}(a+b x)^2}{b^2}-\frac {2 e^{(a+b x)^2} (b c-a d) \text {erfi}(a+b x)}{\sqrt {\pi } b^2}+\frac {\sqrt {\frac {2}{\pi }} (b c-a d) \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {d \text {erfi}(a+b x)^2}{4 b^2}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{\sqrt {\pi } b^2}+\frac {d e^{2 (a+b x)^2}}{2 \pi b^2} \]
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Rubi [A] time = 0.18, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6369, 6354, 6384, 2204, 6366, 6387, 6375, 30, 2209} \[ \frac {(a+b x) (b c-a d) \text {Erfi}(a+b x)^2}{b^2}-\frac {2 e^{(a+b x)^2} (b c-a d) \text {Erfi}(a+b x)}{\sqrt {\pi } b^2}+\frac {\sqrt {\frac {2}{\pi }} (b c-a d) \text {Erfi}\left (\sqrt {2} (a+b x)\right )}{b^2}+\frac {d (a+b x)^2 \text {Erfi}(a+b x)^2}{2 b^2}+\frac {d \text {Erfi}(a+b x)^2}{4 b^2}-\frac {d e^{(a+b x)^2} (a+b x) \text {Erfi}(a+b x)}{\sqrt {\pi } b^2}+\frac {d e^{2 (a+b x)^2}}{2 \pi b^2} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2204
Rule 2209
Rule 6354
Rule 6366
Rule 6369
Rule 6375
Rule 6384
Rule 6387
Rubi steps
\begin {align*} \int (c+d x) \text {erfi}(a+b x)^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (b c \left (1-\frac {a d}{b c}\right ) \text {erfi}(x)^2+d x \text {erfi}(x)^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac {d \operatorname {Subst}\left (\int x \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac {(b c-a d) \operatorname {Subst}\left (\int \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^2}\\ &=\frac {(b c-a d) (a+b x) \text {erfi}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}-\frac {(2 d) \operatorname {Subst}\left (\int e^{x^2} x^2 \text {erfi}(x) \, dx,x,a+b x\right )}{b^2 \sqrt {\pi }}-\frac {(4 (b c-a d)) \operatorname {Subst}\left (\int e^{x^2} x \text {erfi}(x) \, dx,x,a+b x\right )}{b^2 \sqrt {\pi }}\\ &=-\frac {2 (b c-a d) e^{(a+b x)^2} \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}+\frac {(b c-a d) (a+b x) \text {erfi}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {(2 d) \operatorname {Subst}\left (\int e^{2 x^2} x \, dx,x,a+b x\right )}{b^2 \pi }+\frac {(4 (b c-a d)) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b^2 \pi }+\frac {d \operatorname {Subst}\left (\int e^{x^2} \text {erfi}(x) \, dx,x,a+b x\right )}{b^2 \sqrt {\pi }}\\ &=\frac {d e^{2 (a+b x)^2}}{2 b^2 \pi }-\frac {2 (b c-a d) e^{(a+b x)^2} \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}+\frac {(b c-a d) (a+b x) \text {erfi}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {(b c-a d) \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^2}+\frac {d \operatorname {Subst}(\int x \, dx,x,\text {erfi}(a+b x))}{2 b^2}\\ &=\frac {d e^{2 (a+b x)^2}}{2 b^2 \pi }-\frac {2 (b c-a d) e^{(a+b x)^2} \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}+\frac {d \text {erfi}(a+b x)^2}{4 b^2}+\frac {(b c-a d) (a+b x) \text {erfi}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {(b c-a d) \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 128, normalized size = 0.70 \[ \frac {\pi \text {erfi}(a+b x)^2 \left (-2 a^2 d+4 a b c+4 b^2 c x+2 b^2 d x^2+d\right )-4 \sqrt {\pi } e^{(a+b x)^2} \text {erfi}(a+b x) (-a d+2 b c+b d x)+4 \sqrt {2 \pi } (b c-a d) \text {erfi}\left (\sqrt {2} (a+b x)\right )+2 d e^{2 (a+b x)^2}}{4 \pi b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 167, normalized size = 0.91 \[ \frac {4 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} {\left (b c - a d\right )} \operatorname {erfi}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 4 \, \sqrt {\pi } {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} + {\left (2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x + \pi {\left (4 \, a b^{2} c - {\left (2 \, a^{2} - 1\right )} b d\right )}\right )} \operatorname {erfi}\left (b x + a\right )^{2} + 2 \, b d e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )}}{4 \, \pi b^{3}} \]
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 \[ \int {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) \erfi \left (b x +a \right )^{2}\, dx \]
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 \[ \int {\left (d x + c\right )} \operatorname {erfi}\left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {erfi}\left (a+b\,x\right )}^2\,\left (c+d\,x\right ) \,d x \]
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 \[ \int \left (c + d x\right ) \operatorname {erfi}^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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