Optimal. Leaf size=155 \[ \frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{2 d^2 \sqrt {b^2-d}}-\frac {b e^c \text {erf}\left (x \sqrt {b^2-d}\right )}{4 d \left (b^2-d\right )^{3/2}}+\frac {b x e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {\text {erf}(b x) e^{c+d x^2}}{2 d^2}+\frac {x^2 \text {erf}(b x) e^{c+d x^2}}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.16, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6385, 6382, 2205, 2212} \[ \frac {b e^c \text {Erf}\left (x \sqrt {b^2-d}\right )}{2 d^2 \sqrt {b^2-d}}-\frac {b e^c \text {Erf}\left (x \sqrt {b^2-d}\right )}{4 d \left (b^2-d\right )^{3/2}}+\frac {b x e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}-\frac {\text {Erf}(b x) e^{c+d x^2}}{2 d^2}+\frac {x^2 \text {Erf}(b x) e^{c+d x^2}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2205
Rule 2212
Rule 6382
Rule 6385
Rubi steps
\begin {align*} \int e^{c+d x^2} x^3 \text {erf}(b x) \, dx &=\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{2 d}-\frac {\int e^{c+d x^2} x \text {erf}(b x) \, dx}{d}-\frac {b \int e^{c-\left (b^2-d\right ) x^2} x^2 \, dx}{d \sqrt {\pi }}\\ &=\frac {b e^{c-\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}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{2 d}+\frac {b \int e^{c-\left (b^2-d\right ) x^2} \, dx}{d^2 \sqrt {\pi }}-\frac {b \int e^{c+\left (-b^2+d\right ) x^2} \, dx}{2 \left (b^2-d\right ) d \sqrt {\pi }}\\ &=\frac {b e^{c-\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}(b x)}{2 d^2}+\frac {e^{c+d x^2} x^2 \text {erf}(b x)}{2 d}+\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{2 \sqrt {b^2-d} d^2}-\frac {b e^c \text {erf}\left (\sqrt {b^2-d} x\right )}{4 \left (b^2-d\right )^{3/2} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 99, normalized size = 0.64 \[ \frac {e^c \left (\frac {b \left (3 d-2 b^2\right ) \text {erfi}\left (x \sqrt {d-b^2}\right )}{\left (d-b^2\right )^{3/2}}+\frac {2 b d x e^{x^2 \left (d-b^2\right )}}{\sqrt {\pi } \left (b^2-d\right )}+2 e^{d x^2} \left (d x^2-1\right ) \text {erf}(b x)\right )}{4 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.44, size = 149, normalized size = 0.96 \[ \frac {\pi {\left (2 \, b^{3} - 3 \, b d\right )} \sqrt {b^{2} - d} \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c} + 2 \, \sqrt {\pi } {\left (b^{3} d - b d^{2}\right )} x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )} + 2 \, {\left (\pi {\left (b^{4} d - 2 \, b^{2} d^{2} + d^{3}\right )} x^{2} - \pi {\left (b^{4} - 2 \, b^{2} d + d^{2}\right )}\right )} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{4 \, \pi {\left (b^{4} d^{2} - 2 \, b^{2} d^{3} + d^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 124, normalized size = 0.80 \[ \frac {{\left (d x^{2} - 1\right )} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{2 \, d^{2}} + \frac {b d {\left (\frac {2 \, x e^{\left (-b^{2} x^{2} + d x^{2} + c\right )}}{b^{2} - d} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{{\left (b^{2} - d\right )}^{\frac {3}{2}}}\right )} - \frac {2 \, \sqrt {\pi } b \operatorname {erf}\left (-\sqrt {b^{2} - d} x\right ) e^{c}}{\sqrt {b^{2} - d}}}{4 \, \sqrt {\pi } d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.20, size = 168, normalized size = 1.08 \[ \frac {\frac {\erf \left (b x \right ) {\mathrm e}^{c} \left (\frac {b^{4} x^{2} {\mathrm e}^{d \,x^{2}}}{2 d}-\frac {b^{4} {\mathrm e}^{d \,x^{2}}}{2 d^{2}}\right )}{b^{3}}-\frac {{\mathrm e}^{c} \left (\frac {b^{2} \left (\frac {b x \,{\mathrm e}^{\left (-1+\frac {d}{b^{2}}\right ) b^{2} x^{2}}}{-2+\frac {2 d}{b^{2}}}-\frac {\sqrt {\pi }\, \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{4 \left (-1+\frac {d}{b^{2}}\right ) \sqrt {1-\frac {d}{b^{2}}}}\right )}{d}-\frac {b^{4} \sqrt {\pi }\, \erf \left (\sqrt {1-\frac {d}{b^{2}}}\, b x \right )}{2 d^{2} \sqrt {1-\frac {d}{b^{2}}}}\right )}{\sqrt {\pi }\, b^{3}}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (d x^{2} e^{c} - e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2}\right )}}{2 \, d^{2}} - \frac {-\frac {b d x^{3} e^{c} \Gamma \left (\frac {3}{2}, {\left (b^{2} - d\right )} x^{2}\right )}{2 \, \left ({\left (b^{2} - d\right )} x^{2}\right )^{\frac {3}{2}}} - \frac {\sqrt {\pi } b \operatorname {erf}\left (\sqrt {b^{2} - d} x\right ) e^{c}}{2 \, \sqrt {b^{2} - d}}}{\sqrt {\pi } d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.53, size = 131, normalized size = 0.85 \[ \frac {b\,x\,{\mathrm {e}}^{-b^2\,x^2+d\,x^2+c}}{2\,\sqrt {\pi }\,\left (b^2\,d-d^2\right )}-\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{d\,x^2+c}}{2\,d^2}-\frac {x^2\,{\mathrm {e}}^{d\,x^2+c}}{2\,d}\right )+\frac {b\,{\mathrm {e}}^c\,\mathrm {erf}\left (x\,\sqrt {b^2-d}\right )}{2\,d^2\,\sqrt {b^2-d}}+\frac {b\,{\mathrm {e}}^c\,\mathrm {erfi}\left (x\,\sqrt {d-b^2}\right )}{4\,d\,{\left (d-b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x^{3} e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________