Optimal. Leaf size=342 \[ \frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d^2 \sqrt {b^2-d}}-\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 d \left (b^2-d\right )^{3/2}}-\frac {a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}+\frac {b x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {a^2 b^3 e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \left (b^2-d\right )^{5/2}}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d^2}+\frac {x^2 e^{c+d x^2} \text {erf}(a+b x)}{2 d} \]
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Rubi [A] time = 0.51, antiderivative size = 342, 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 = {6385, 6382, 2234, 2205, 2241, 2240} \[ \frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {Erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d^2 \sqrt {b^2-d}}-\frac {a^2 b^3 e^{\frac {a^2 d}{b^2-d}+c} \text {Erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 d \left (b^2-d\right )^{5/2}}-\frac {b e^{\frac {a^2 d}{b^2-d}+c} \text {Erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 d \left (b^2-d\right )^{3/2}}-\frac {a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )^2}+\frac {b x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {e^{c+d x^2} \text {Erf}(a+b x)}{2 d^2}+\frac {x^2 e^{c+d x^2} \text {Erf}(a+b x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2205
Rule 2234
Rule 2240
Rule 2241
Rule 6382
Rule 6385
Rubi steps
\begin {align*} \int e^{c+d x^2} x^3 \text {erf}(a+b x) \, dx &=\frac {e^{c+d x^2} x^2 \text {erf}(a+b x)}{2 d}-\frac {\int e^{c+d x^2} x \text {erf}(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 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(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 \left (b^2-d\right ) d \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}{\left (b^2-d\right ) d \sqrt {\pi }}\\ &=-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(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}{\left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{d^2 \sqrt {\pi }}-\frac {\left (b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{2 \left (b^2-d\right ) d \sqrt {\pi }}\\ &=-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(a+b x)}{2 d}+\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d^2}-\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d}-\frac {\left (a^2 b^3 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{\left (b^2-d\right )^2 d \sqrt {\pi }}\\ &=-\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {e^{c+d x^2} \text {erf}(a+b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(a+b x)}{2 d}+\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \sqrt {b^2-d} d^2}-\frac {a^2 b^3 e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{5/2} d}-\frac {b e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d}\\ \end {align*}
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Mathematica [A] time = 5.45, size = 240, normalized size = 0.70 \[ \frac {e^c \left (-\frac {b d e^{-a^2-2 a b x+x^2 \left (d-b^2\right )} \left (\sqrt {\pi } \sqrt {b^2-d} \left (\left (2 a^2+1\right ) b^2-d\right ) e^{\frac {\left (a b+x \left (b^2-d\right )\right )^2}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )+2 \left (b^2-d\right ) \left (a b+x \left (d-b^2\right )\right )\right )}{\sqrt {\pi } \left (b^2-d\right )^3}+\frac {2 b e^{\frac {a^2 d}{b^2-d}} \text {erfi}\left (\frac {x \left (d-b^2\right )-a b}{\sqrt {d-b^2}}\right )}{\sqrt {d-b^2}}+2 e^{d x^2} \left (d x^2-1\right ) \text {erf}(a+b x)\right )}{4 d^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 267, normalized size = 0.78 \[ \frac {\pi {\left (2 \, b^{5} - {\left (2 \, a^{2} + 5\right )} b^{3} d + 3 \, b d^{2}\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\frac {a b + {\left (b^{2} - d\right )} x}{\sqrt {b^{2} - d}}\right ) e^{\left (\frac {b^{2} c + {\left (a^{2} - c\right )} d}{b^{2} - d}\right )} + 2 \, {\left (\pi {\left (b^{6} d - 3 \, b^{4} d^{2} + 3 \, b^{2} d^{3} - d^{4}\right )} x^{2} - \pi {\left (b^{6} - 3 \, b^{4} d + 3 \, b^{2} d^{2} - d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (a b^{4} d - a b^{2} d^{2} - {\left (b^{5} d - 2 \, b^{3} d^{2} + b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )}}{4 \, \pi {\left (b^{6} d^{2} - 3 \, b^{4} d^{3} + 3 \, b^{2} d^{4} - d^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 271, normalized size = 0.79 \[ \frac {{\left (d x^{2} - 1\right )} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{2 \, d^{2}} - \frac {\frac {2 \, \sqrt {\pi } b \operatorname {erf}\left (-\sqrt {b^{2} - d} {\left (\frac {a b}{b^{2} - d} + x\right )}\right ) e^{\left (\frac {b^{2} c + a^{2} d - c d}{b^{2} - d}\right )}}{\sqrt {b^{2} - d}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, a^{2} b^{2} + b^{2} - d\right )} \operatorname {erf}\left (-\sqrt {b^{2} - d} {\left (\frac {a b}{b^{2} - d} + x\right )}\right ) e^{\left (\frac {b^{2} c + a^{2} d - c d}{b^{2} - d}\right )}}{\sqrt {b^{2} - d}} + 2 \, {\left ({\left (\frac {a b}{b^{2} - d} + x\right )} b^{2} - 2 \, a b - {\left (\frac {a b}{b^{2} - d} + x\right )} d\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x + d x^{2} - a^{2} + c\right )}\right )} b d}{b^{4} - 2 \, b^{2} d + d^{2}}}{4 \, \sqrt {\pi } d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{3} \erf \left (b x +a \right )\, dx \]
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 \[ \frac {{\left (d x^{2} e^{c} - e^{c}\right )} \operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2}\right )}}{2 \, d^{2}} - \frac {-\frac {{\left (\frac {\sqrt {\pi } {\left (a b + {\left (b^{2} - d\right )} x\right )} a^{2} b^{2} {\left (\operatorname {erf}\left (\sqrt {\frac {{\left (a b + {\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}}\right ) - 1\right )}}{{\left (-b^{2} + d\right )}^{\frac {5}{2}} \sqrt {\frac {{\left (a b + {\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}}} - \frac {2 \, a b e^{\left (-\frac {{\left (a b + {\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}\right )}}{{\left (-b^{2} + d\right )}^{\frac {3}{2}}} - \frac {{\left (a b + {\left (b^{2} - d\right )} x\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (a b + {\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}\right )}{{\left (-b^{2} + d\right )}^{\frac {5}{2}} \left (\frac {{\left (a b + {\left (b^{2} - d\right )} x\right )}^{2}}{b^{2} - d}\right )^{\frac {3}{2}}}\right )} b d e^{\left (\frac {a^{2} b^{2}}{b^{2} - d} - a^{2} + c\right )}}{2 \, \sqrt {-b^{2} + d}} - \frac {\sqrt {\pi } b \operatorname {erf}\left (\frac {a b}{\sqrt {b^{2} - d}} + \sqrt {b^{2} - d} x\right ) e^{\left (\frac {a^{2} b^{2}}{b^{2} - d} - a^{2} + c\right )}}{2 \, \sqrt {b^{2} - d}}}{\sqrt {\pi } d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 386, normalized size = 1.13 \[ \frac {\mathrm {erfi}\left (\frac {a\,b-x\,\left (d-b^2\right )}{\sqrt {d-b^2}}\right )\,\left (b^3\,{\mathrm {e}}^{\frac {c\,d}{d-b^2}-\frac {a^2\,d}{d-b^2}-\frac {b^2\,c}{d-b^2}}+2\,a^2\,b^3\,{\mathrm {e}}^{\frac {c\,d}{d-b^2}-\frac {a^2\,d}{d-b^2}-\frac {b^2\,c}{d-b^2}}-b\,d\,{\mathrm {e}}^{\frac {c\,d}{d-b^2}-\frac {a^2\,d}{d-b^2}-\frac {b^2\,c}{d-b^2}}\right )}{4\,d\,{\left (d-b^2\right )}^{5/2}}-\frac {\frac {a\,b^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2+d\,x^2+c}}{2\,{\left (d-b^2\right )}^2}+\frac {b\,x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2+d\,x^2+c}}{2\,\left (d-b^2\right )}}{d\,\sqrt {\pi }}-\mathrm {erf}\left (a+b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )+\frac {b\,\mathrm {erf}\left (\frac {a\,b\,1{}\mathrm {i}-x\,\left (d-b^2\right )\,1{}\mathrm {i}}{\sqrt {d-b^2}}\right )\,{\mathrm {e}}^{c-a^2-\frac {a^2\,b^2}{d-b^2}}\,1{}\mathrm {i}}{2\,d^2\,\sqrt {d-b^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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