Optimal. Leaf size=70 \[ \frac{x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{2 \sqrt{\pi } b}-\frac{x e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}+\frac{x^2}{2 \sqrt{\pi } b} \]
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Rubi [A] time = 0.0563927, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6387, 6378, 30} \[ \frac{x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{2 \sqrt{\pi } b}-\frac{x e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}+\frac{x^2}{2 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6387
Rule 6378
Rule 30
Rubi steps
\begin{align*} \int e^{-b^2 x^2} x^2 \text{erfi}(b x) \, dx &=-\frac{e^{-b^2 x^2} x \text{erfi}(b x)}{2 b^2}+\frac{\int e^{-b^2 x^2} \text{erfi}(b x) \, dx}{2 b^2}+\frac{\int x \, dx}{b \sqrt{\pi }}\\ &=\frac{x^2}{2 b \sqrt{\pi }}-\frac{e^{-b^2 x^2} x \text{erfi}(b x)}{2 b^2}+\frac{x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{2 b \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0140755, size = 36, normalized size = 0.51 \[ \frac{x^2 \left (1-\text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{1}{2},2\right \},-b^2 x^2\right )\right )}{2 \sqrt{\pi } b} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}{\it erfi} \left ( bx \right ) }{{{\rm e}^{{b}^{2}{x}^{2}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{2} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} \operatorname{erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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