Integrand size = 19, antiderivative size = 118 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {2 e^c x}{b^5 \sqrt {\pi }}+\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}-\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2} \]
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Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6520, 6517, 8, 12, 30} \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {2 e^c x}{\sqrt {\pi } b^5}+\frac {2 e^c x^3}{3 \sqrt {\pi } b^3}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}+\frac {e^{b^2 x^2+c} \text {erf}(b x)}{b^6}-\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{b^4}-\frac {e^c x^5}{5 \sqrt {\pi } b} \]
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Rule 8
Rule 12
Rule 30
Rule 6517
Rule 6520
Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}-\frac {2 \int e^{c+b^2 x^2} x^3 \text {erf}(b x) \, dx}{b^2}-\frac {\int e^c x^4 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}+\frac {2 \int e^{c+b^2 x^2} x \text {erf}(b x) \, dx}{b^4}+\frac {2 \int e^c x^2 \, dx}{b^3 \sqrt {\pi }}-\frac {e^c \int x^4 \, dx}{b \sqrt {\pi }} \\ & = -\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2}-\frac {2 \int e^c \, dx}{b^5 \sqrt {\pi }}+\frac {\left (2 e^c\right ) \int x^2 \, dx}{b^3 \sqrt {\pi }} \\ & = -\frac {2 e^c x}{b^5 \sqrt {\pi }}+\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}-\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {e^c \left (-60 b x+20 b^3 x^3-6 b^5 x^5+15 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erf}(b x)\right )}{30 b^6 \sqrt {\pi }} \]
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Time = 2.55 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}-\frac {{\mathrm e}^{c} \left (\frac {1}{5} b^{5} x^{5}-\frac {2}{3} b^{3} x^{3}+2 b x \right )}{\sqrt {\pi }\, b^{5}}}{b}\) | \(88\) |
parallelrisch | \(\frac {-6 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+15 \,{\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erf}\left (b x \right ) b^{4} \sqrt {\pi }+20 \,{\mathrm e}^{b^{2} x^{2}+c} x^{3} {\mathrm e}^{-b^{2} x^{2}} b^{3}-30 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erf}\left (b x \right ) b^{2} \sqrt {\pi }-60 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +30 \,{\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right ) \sqrt {\pi }}{30 b^{6} \sqrt {\pi }}\) | \(156\) |
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Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {15 \, {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c}}{30 \, \pi b^{6}} \]
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Time = 33.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\begin {cases} - \frac {x^{5} e^{c}}{5 \sqrt {\pi } b} + \frac {x^{4} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {2 x^{3} e^{c}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{b^{4}} - \frac {2 x e^{c}}{\sqrt {\pi } b^{5}} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {6 \, b^{5} x^{5} e^{c} - 20 \, b^{3} x^{3} e^{c} - 15 \, {\left (\sqrt {\pi } b^{4} x^{4} e^{c} - 2 \, \sqrt {\pi } b^{2} x^{2} e^{c} + 2 \, \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 60 \, b x e^{c}}{30 \, \sqrt {\pi } b^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {1}{2} \, {\left (\frac {c^{2} e^{\left (b^{2} x^{2} + c\right )}}{b^{6}} - \frac {{\left (2 \, b^{2} x^{2} - {\left (b^{2} x^{2} + c\right )}^{2} + 2 \, {\left (b^{2} x^{2} + c\right )} c - 2\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{6}}\right )} \operatorname {erf}\left (b x\right ) - \frac {3 \, \sqrt {\pi } b^{4} x^{5} e^{c} - 10 \, \sqrt {\pi } b^{2} x^{3} e^{c} + 30 \, \sqrt {\pi } x e^{c}}{15 \, \pi b^{5}} \]
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Time = 5.44 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{b^6}+\frac {x^4\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{b^4}\right )-\frac {3\,{\mathrm {e}}^c\,b^4\,x^5-10\,{\mathrm {e}}^c\,b^2\,x^3+30\,{\mathrm {e}}^c\,x}{15\,b^5\,\sqrt {\pi }} \]
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