Optimal. Leaf size=304 \[ \frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {Erfi}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {Erfi}(a+b x)}{2 d}-\frac {a^2 b^3 e^{c+\frac {a^2 d}{b^2+d}} \text {Erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d \left (b^2+d\right )^{5/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {Erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {Erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d^2 \sqrt {b^2+d}} \]
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Rubi [A]
time = 0.32, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6522, 6519,
2266, 2235, 2273, 2272} \begin {gather*} \frac {b e^{\frac {a^2 d}{b^2+d}+c} \text {Erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )}{2 d^2 \sqrt {b^2+d}}+\frac {b e^{\frac {a^2 d}{b^2+d}+c} \text {Erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {a b^2 e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt {\pi } d \left (b^2+d\right )^2}-\frac {b x e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 \sqrt {\pi } d \left (b^2+d\right )}-\frac {a^2 b^3 e^{\frac {a^2 d}{b^2+d}+c} \text {Erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )}{2 d \left (b^2+d\right )^{5/2}}-\frac {e^{c+d x^2} \text {Erfi}(a+b x)}{2 d^2}+\frac {x^2 e^{c+d x^2} \text {Erfi}(a+b x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2235
Rule 2266
Rule 2272
Rule 2273
Rule 6519
Rule 6522
Rubi steps
\begin {align*} \int e^{c+d x^2} x^3 \text {erfi}(a+b x) \, dx &=\frac {e^{c+d x^2} x^2 \text {erfi}(a+b x)}{2 d}-\frac {\int e^{c+d x^2} x \text {erfi}(a+b x) \, dx}{d}-\frac {b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2 \, dx}{d \sqrt {\pi }}\\ &=-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(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^2 \sqrt {\pi }}+\frac {b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{2 d \left (b^2+d\right ) \sqrt {\pi }}+\frac {\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right ) \sqrt {\pi }}\\ &=\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(a+b x)}{2 d}-\frac {\left (a^2 b^3\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {\left (b e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d^2 \sqrt {\pi }}+\frac {\left (b e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{2 d \left (b^2+d\right ) \sqrt {\pi }}\\ &=\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(a+b x)}{2 d}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d^2 \sqrt {b^2+d}}-\frac {\left (a^2 b^3 e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d \left (b^2+d\right )^2 \sqrt {\pi }}\\ &=\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erfi}(a+b x)}{2 d}-\frac {a^2 b^3 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d \left (b^2+d\right )^{5/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{3/2}}+\frac {b e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d^2 \sqrt {b^2+d}}\\ \end {align*}
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Mathematica [A]
time = 1.48, size = 206, normalized size = 0.68 \begin {gather*} \frac {e^c \left (2 e^{d x^2} \left (-1+d x^2\right ) \text {Erfi}(a+b x)+\frac {2 b e^{\frac {a^2 d}{b^2+d}} \text {Erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{\sqrt {b^2+d}}-\frac {b d e^{\frac {a^2 d}{b^2+d}} \left (2 \left (b^2+d\right ) e^{\frac {\left (a b+\left (b^2+d\right ) x\right )^2}{b^2+d}} \left (-a b+\left (b^2+d\right ) x\right )+\left (\left (-1+2 a^2\right ) b^2-d\right ) \sqrt {b^2+d} \sqrt {\pi } \text {Erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )\right )}{\left (b^2+d\right )^3 \sqrt {\pi }}\right )}{4 d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d \,x^{2}+c} x^{3} \erfi \left (b x +a \right )\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 262, normalized size = 0.86 \begin {gather*} -\frac {\pi {\left (2 \, b^{5} - {\left (2 \, a^{2} - 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt {-b^{2} - d} \operatorname {erf}\left (\frac {{\left (a b + {\left (b^{2} + d\right )} x\right )} \sqrt {-b^{2} - d}}{b^{2} + d}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} + c\right )} d}{b^{2} + d}\right )} - 2 \, {\left (\pi {\left (b^{6} d + 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} + d^{4}\right )} x^{2} - \pi {\left (b^{6} + 3 \, b^{4} d + 3 \, b^{2} d^{2} + d^{3}\right )}\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (a b^{4} d + a b^{2} d^{2} - {\left (b^{5} d + 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (b^{2} x^{2} + 2 \, a b x + d x^{2} + a^{2} + c\right )}}{4 \, \pi {\left (b^{6} d^{2} + 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} + d^{5}\right )}} \end {gather*}
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 \begin {gather*} e^{c} \int x^{3} e^{d x^{2}} \operatorname {erfi}{\left (a + b x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.13, size = 336, normalized size = 1.11 \begin {gather*} \frac {\mathrm {erfi}\left (\frac {a\,b+x\,\left (b^2+d\right )}{\sqrt {b^2+d}}\right )\,\left (b^3\,{\mathrm {e}}^{\frac {c\,d}{b^2+d}+\frac {a^2\,d}{b^2+d}+\frac {b^2\,c}{b^2+d}}-2\,a^2\,b^3\,{\mathrm {e}}^{\frac {c\,d}{b^2+d}+\frac {a^2\,d}{b^2+d}+\frac {b^2\,c}{b^2+d}}+b\,d\,{\mathrm {e}}^{\frac {c\,d}{b^2+d}+\frac {a^2\,d}{b^2+d}+\frac {b^2\,c}{b^2+d}}\right )}{4\,d\,{\left (b^2+d\right )}^{5/2}}-\frac {\frac {b\,x\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2+d\,x^2+c}}{2\,\left (b^2+d\right )}-\frac {a\,b^2\,{\mathrm {e}}^{a^2+2\,a\,b\,x+b^2\,x^2+d\,x^2+c}}{2\,{\left (b^2+d\right )}^2}}{d\,\sqrt {\pi }}-\mathrm {erfi}\left (a+b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )-\frac {b\,{\mathrm {e}}^{c+a^2-\frac {a^2\,b^2}{b^2+d}}\,\mathrm {erf}\left (\frac {a\,b\,1{}\mathrm {i}+x\,\left (b^2+d\right )\,1{}\mathrm {i}}{\sqrt {b^2+d}}\right )\,1{}\mathrm {i}}{2\,d^2\,\sqrt {b^2+d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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