Optimal. Leaf size=93 \[ \frac{5 \text{Erfi}(b x)}{16 b^6}-\frac{x^5 e^{b^2 x^2}}{6 \sqrt{\pi } b}+\frac{5 x^3 e^{b^2 x^2}}{12 \sqrt{\pi } b^3}-\frac{5 x e^{b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erfi}(b x) \]
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Rubi [A] time = 0.0823144, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2212, 2204} \[ \frac{5 \text{Erfi}(b x)}{16 b^6}-\frac{x^5 e^{b^2 x^2}}{6 \sqrt{\pi } b}+\frac{5 x^3 e^{b^2 x^2}}{12 \sqrt{\pi } b^3}-\frac{5 x e^{b^2 x^2}}{8 \sqrt{\pi } b^5}+\frac{1}{6} x^6 \text{Erfi}(b x) \]
Antiderivative was successfully verified.
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Rule 6363
Rule 2212
Rule 2204
Rubi steps
\begin{align*} \int x^5 \text{erfi}(b x) \, dx &=\frac{1}{6} x^6 \text{erfi}(b x)-\frac{b \int e^{b^2 x^2} x^6 \, dx}{3 \sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)+\frac{5 \int e^{b^2 x^2} x^4 \, dx}{6 b \sqrt{\pi }}\\ &=\frac{5 e^{b^2 x^2} x^3}{12 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)-\frac{5 \int e^{b^2 x^2} x^2 \, dx}{4 b^3 \sqrt{\pi }}\\ &=-\frac{5 e^{b^2 x^2} x}{8 b^5 \sqrt{\pi }}+\frac{5 e^{b^2 x^2} x^3}{12 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{1}{6} x^6 \text{erfi}(b x)+\frac{5 \int e^{b^2 x^2} \, dx}{8 b^5 \sqrt{\pi }}\\ &=-\frac{5 e^{b^2 x^2} x}{8 b^5 \sqrt{\pi }}+\frac{5 e^{b^2 x^2} x^3}{12 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^5}{6 b \sqrt{\pi }}+\frac{5 \text{erfi}(b x)}{16 b^6}+\frac{1}{6} x^6 \text{erfi}(b x)\\ \end{align*}
Mathematica [A] time = 0.0264514, size = 64, normalized size = 0.69 \[ \frac{\sqrt{\pi } \left (8 b^6 x^6+15\right ) \text{Erfi}(b x)-2 b x e^{b^2 x^2} \left (4 b^4 x^4-10 b^2 x^2+15\right )}{48 \sqrt{\pi } b^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 77, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{6}} \left ({\frac{{b}^{6}{x}^{6}{\it erfi} \left ( bx \right ) }{6}}-{\frac{1}{3\,\sqrt{\pi }} \left ({\frac{{b}^{5}{x}^{5}{{\rm e}^{{b}^{2}{x}^{2}}}}{2}}-{\frac{5\,{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{3}{x}^{3}}{4}}+{\frac{15\,{{\rm e}^{{b}^{2}{x}^{2}}}bx}{8}}-{\frac{15\,\sqrt{\pi }{\it erfi} \left ( bx \right ) }{16}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.04066, size = 85, normalized size = 0.91 \begin{align*} \frac{1}{6} \, x^{6} \operatorname{erfi}\left (b x\right ) - \frac{b{\left (\frac{2 \,{\left (4 \, b^{4} x^{5} - 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (b^{2} x^{2}\right )}}{b^{6}} + \frac{15 i \, \sqrt{\pi } \operatorname{erf}\left (i \, b x\right )}{b^{7}}\right )}}{48 \, \sqrt{\pi }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.39547, size = 149, normalized size = 1.6 \begin{align*} -\frac{2 \, \sqrt{\pi }{\left (4 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (b^{2} x^{2}\right )} -{\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname{erfi}\left (b x\right )}{48 \, \pi b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.70205, size = 88, normalized size = 0.95 \begin{align*} \begin{cases} \frac{x^{6} \operatorname{erfi}{\left (b x \right )}}{6} - \frac{x^{5} e^{b^{2} x^{2}}}{6 \sqrt{\pi } b} + \frac{5 x^{3} e^{b^{2} x^{2}}}{12 \sqrt{\pi } b^{3}} - \frac{5 x e^{b^{2} x^{2}}}{8 \sqrt{\pi } b^{5}} + \frac{5 \operatorname{erfi}{\left (b x \right )}}{16 b^{6}} & \text{for}\: b \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{5} \operatorname{erfi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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