Optimal. Leaf size=81 \[ -\frac{x^4 e^{b^2 x^2}}{5 \sqrt{\pi } b}+\frac{2 x^2 e^{b^2 x^2}}{5 \sqrt{\pi } b^3}-\frac{2 e^{b^2 x^2}}{5 \sqrt{\pi } b^5}+\frac{1}{5} x^5 \text{Erfi}(b x) \]
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Rubi [A] time = 0.0646533, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6363, 2212, 2209} \[ -\frac{x^4 e^{b^2 x^2}}{5 \sqrt{\pi } b}+\frac{2 x^2 e^{b^2 x^2}}{5 \sqrt{\pi } b^3}-\frac{2 e^{b^2 x^2}}{5 \sqrt{\pi } b^5}+\frac{1}{5} x^5 \text{Erfi}(b x) \]
Antiderivative was successfully verified.
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Rule 6363
Rule 2212
Rule 2209
Rubi steps
\begin{align*} \int x^4 \text{erfi}(b x) \, dx &=\frac{1}{5} x^5 \text{erfi}(b x)-\frac{(2 b) \int e^{b^2 x^2} x^5 \, dx}{5 \sqrt{\pi }}\\ &=-\frac{e^{b^2 x^2} x^4}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)+\frac{4 \int e^{b^2 x^2} x^3 \, dx}{5 b \sqrt{\pi }}\\ &=\frac{2 e^{b^2 x^2} x^2}{5 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^4}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)-\frac{4 \int e^{b^2 x^2} x \, dx}{5 b^3 \sqrt{\pi }}\\ &=-\frac{2 e^{b^2 x^2}}{5 b^5 \sqrt{\pi }}+\frac{2 e^{b^2 x^2} x^2}{5 b^3 \sqrt{\pi }}-\frac{e^{b^2 x^2} x^4}{5 b \sqrt{\pi }}+\frac{1}{5} x^5 \text{erfi}(b x)\\ \end{align*}
Mathematica [A] time = 0.0283888, size = 49, normalized size = 0.6 \[ \frac{1}{5} \left (x^5 \text{Erfi}(b x)-\frac{e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right )}{\sqrt{\pi } b^5}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 64, normalized size = 0.8 \begin{align*}{\frac{1}{{b}^{5}} \left ({\frac{{b}^{5}{x}^{5}{\it erfi} \left ( bx \right ) }{5}}-{\frac{2}{5\,\sqrt{\pi }} \left ({\frac{{{\rm e}^{{b}^{2}{x}^{2}}}{b}^{4}{x}^{4}}{2}}-{b}^{2}{x}^{2}{{\rm e}^{{b}^{2}{x}^{2}}}+{{\rm e}^{{b}^{2}{x}^{2}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01852, size = 58, normalized size = 0.72 \begin{align*} \frac{1}{5} \, x^{5} \operatorname{erfi}\left (b x\right ) - \frac{{\left (b^{4} x^{4} - 2 \, b^{2} x^{2} + 2\right )} e^{\left (b^{2} x^{2}\right )}}{5 \, \sqrt{\pi } b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.35194, size = 116, normalized size = 1.43 \begin{align*} \frac{\pi b^{5} x^{5} \operatorname{erfi}\left (b x\right ) - \sqrt{\pi }{\left (b^{4} x^{4} - 2 \, b^{2} x^{2} + 2\right )} e^{\left (b^{2} x^{2}\right )}}{5 \, \pi b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.895, size = 75, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{erfi}{\left (b x \right )}}{5} - \frac{x^{4} e^{b^{2} x^{2}}}{5 \sqrt{\pi } b} + \frac{2 x^{2} e^{b^{2} x^{2}}}{5 \sqrt{\pi } b^{3}} - \frac{2 e^{b^{2} x^{2}}}{5 \sqrt{\pi } b^{5}} & \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^{4} \operatorname{erfi}\left (b x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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