Optimal. Leaf size=119 \[ \frac{3 e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{4 \sqrt{\pi } b^3}+\frac{x^3 e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{3 x e^{b^2 x^2+c} \text{Erf}(b x)}{4 b^4}+\frac{3 e^c x^2}{4 \sqrt{\pi } b^3}-\frac{e^c x^4}{4 \sqrt{\pi } b} \]
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Rubi [A] time = 0.113313, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {6385, 6376, 12, 30} \[ \frac{3 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{4 \sqrt{\pi } b^3}+\frac{x^3 e^{b^2 x^2+c} \text{Erf}(b x)}{2 b^2}-\frac{3 x e^{b^2 x^2+c} \text{Erf}(b x)}{4 b^4}+\frac{3 e^c x^2}{4 \sqrt{\pi } b^3}-\frac{e^c x^4}{4 \sqrt{\pi } b} \]
Antiderivative was successfully verified.
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Rule 6385
Rule 6376
Rule 12
Rule 30
Rubi steps
\begin{align*} \int e^{c+b^2 x^2} x^4 \text{erf}(b x) \, dx &=\frac{e^{c+b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}-\frac{3 \int e^{c+b^2 x^2} x^2 \text{erf}(b x) \, dx}{2 b^2}-\frac{\int e^c x^3 \, dx}{b \sqrt{\pi }}\\ &=-\frac{3 e^{c+b^2 x^2} x \text{erf}(b x)}{4 b^4}+\frac{e^{c+b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{3 \int e^{c+b^2 x^2} \text{erf}(b x) \, dx}{4 b^4}+\frac{3 \int e^c x \, dx}{2 b^3 \sqrt{\pi }}-\frac{e^c \int x^3 \, dx}{b \sqrt{\pi }}\\ &=-\frac{e^c x^4}{4 b \sqrt{\pi }}-\frac{3 e^{c+b^2 x^2} x \text{erf}(b x)}{4 b^4}+\frac{e^{c+b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{3 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt{\pi }}+\frac{\left (3 e^c\right ) \int x \, dx}{2 b^3 \sqrt{\pi }}\\ &=\frac{3 e^c x^2}{4 b^3 \sqrt{\pi }}-\frac{e^c x^4}{4 b \sqrt{\pi }}-\frac{3 e^{c+b^2 x^2} x \text{erf}(b x)}{4 b^4}+\frac{e^{c+b^2 x^2} x^3 \text{erf}(b x)}{2 b^2}+\frac{3 e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{4 b^3 \sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.311147, size = 100, normalized size = 0.84 \[ \frac{e^c \left (-6 b^2 x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},-b^2 x^2\right )+2 \sqrt{\pi } b x e^{b^2 x^2} \left (2 b^2 x^2-3\right ) \text{Erf}(b x)-2 b^4 x^4+6 b^2 x^2+3 \pi \text{Erf}(b x) \text{Erfi}(b x)\right )}{8 \sqrt{\pi } b^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.119, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{x}^{4}{\it Erf} \left ( bx \right ) \, 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^{4} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\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^{4} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\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^{4} \operatorname{erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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