Optimal. Leaf size=148 \[ \frac{15 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )}{8 \sqrt{\pi } b^5}-\frac{x^5 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{5 x^3 e^{-b^2 x^2} \text{Erfi}(b x)}{4 b^4}-\frac{15 x e^{-b^2 x^2} \text{Erfi}(b x)}{8 b^6}+\frac{5 x^4}{8 \sqrt{\pi } b^3}+\frac{15 x^2}{8 \sqrt{\pi } b^5}+\frac{x^6}{6 \sqrt{\pi } b} \]
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Rubi [A] time = 0.141605, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6387, 6378, 30} \[ \frac{15 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{8 \sqrt{\pi } b^5}-\frac{x^5 e^{-b^2 x^2} \text{Erfi}(b x)}{2 b^2}-\frac{5 x^3 e^{-b^2 x^2} \text{Erfi}(b x)}{4 b^4}-\frac{15 x e^{-b^2 x^2} \text{Erfi}(b x)}{8 b^6}+\frac{5 x^4}{8 \sqrt{\pi } b^3}+\frac{15 x^2}{8 \sqrt{\pi } b^5}+\frac{x^6}{6 \sqrt{\pi } b} \]
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^6 \text{erfi}(b x) \, dx &=-\frac{e^{-b^2 x^2} x^5 \text{erfi}(b x)}{2 b^2}+\frac{5 \int e^{-b^2 x^2} x^4 \text{erfi}(b x) \, dx}{2 b^2}+\frac{\int x^5 \, dx}{b \sqrt{\pi }}\\ &=\frac{x^6}{6 b \sqrt{\pi }}-\frac{5 e^{-b^2 x^2} x^3 \text{erfi}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^5 \text{erfi}(b x)}{2 b^2}+\frac{15 \int e^{-b^2 x^2} x^2 \text{erfi}(b x) \, dx}{4 b^4}+\frac{5 \int x^3 \, dx}{2 b^3 \sqrt{\pi }}\\ &=\frac{5 x^4}{8 b^3 \sqrt{\pi }}+\frac{x^6}{6 b \sqrt{\pi }}-\frac{15 e^{-b^2 x^2} x \text{erfi}(b x)}{8 b^6}-\frac{5 e^{-b^2 x^2} x^3 \text{erfi}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^5 \text{erfi}(b x)}{2 b^2}+\frac{15 \int e^{-b^2 x^2} \text{erfi}(b x) \, dx}{8 b^6}+\frac{15 \int x \, dx}{4 b^5 \sqrt{\pi }}\\ &=\frac{15 x^2}{8 b^5 \sqrt{\pi }}+\frac{5 x^4}{8 b^3 \sqrt{\pi }}+\frac{x^6}{6 b \sqrt{\pi }}-\frac{15 e^{-b^2 x^2} x \text{erfi}(b x)}{8 b^6}-\frac{5 e^{-b^2 x^2} x^3 \text{erfi}(b x)}{4 b^4}-\frac{e^{-b^2 x^2} x^5 \text{erfi}(b x)}{2 b^2}+\frac{15 x^2 \, _2F_2\left (1,1;\frac{3}{2},2;-b^2 x^2\right )}{8 b^5 \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.0234395, size = 52, normalized size = 0.35 \[ \frac{x^2 \left (-9 \text{HypergeometricPFQ}\left (\{1,1\},\left \{-\frac{3}{2},2\right \},-b^2 x^2\right )+4 b^4 x^4+3 b^2 x^2+9\right )}{24 \sqrt{\pi } b^5} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.43, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}{\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^{6} \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^{6} \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^{6} \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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