3.72 \(\int e^{c+b^2 x^2} \text{Erf}(b x) \, dx\)

Optimal. Leaf size=29 \[ \frac{b e^c x^2 \text{HypergeometricPFQ}\left (\{1,1\},\left \{\frac{3}{2},2\right \},b^2 x^2\right )}{\sqrt{\pi }} \]

[Out]

(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi]

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Rubi [A]  time = 0.0176664, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {6376} \[ \frac{b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{\sqrt{\pi }} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)*Erf[b*x],x]

[Out]

(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/Sqrt[Pi]

Rule 6376

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[(b*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2},
b^2*x^2])/Sqrt[Pi], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rubi steps

\begin{align*} \int e^{c+b^2 x^2} \text{erf}(b x) \, dx &=\frac{b e^c x^2 \, _2F_2\left (1,1;\frac{3}{2},2;b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}

Mathematica [F]  time = 0.022814, size = 0, normalized size = 0. \[ \int e^{c+b^2 x^2} \text{Erf}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(c + b^2*x^2)*Erf[b*x],x]

[Out]

Integrate[E^(c + b^2*x^2)*Erf[b*x], x]

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it Erf} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(b^2*x^2+c)*erf(b*x),x)

[Out]

int(exp(b^2*x^2+c)*erf(b*x),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.

[In]

integrate(exp(b^2*x^2+c)*erf(b*x),x, algorithm="maxima")

[Out]

integrate(erf(b*x)*e^(b^2*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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.

[In]

integrate(exp(b^2*x^2+c)*erf(b*x),x, algorithm="fricas")

[Out]

integral(erf(b*x)*e^(b^2*x^2 + c), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b**2*x**2+c)*erf(b*x),x)

[Out]

Exception raised: AttributeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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.

[In]

integrate(exp(b^2*x^2+c)*erf(b*x),x, algorithm="giac")

[Out]

integrate(erf(b*x)*e^(b^2*x^2 + c), x)