3.265 \(\int e^{c+d x^2} x^4 \text{Erfi}(b x) \, dx\)

Optimal. Leaf size=170 \[ \frac{3 \text{Unintegrable}\left (\text{Erfi}(b x) e^{c+d x^2},x\right )}{4 d^2}+\frac{3 b e^{x^2 \left (b^2+d\right )+c}}{4 \sqrt{\pi } d^2 \left (b^2+d\right )}-\frac{b x^2 e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )}+\frac{b e^{x^2 \left (b^2+d\right )+c}}{2 \sqrt{\pi } d \left (b^2+d\right )^2}-\frac{3 x \text{Erfi}(b x) e^{c+d x^2}}{4 d^2}+\frac{x^3 \text{Erfi}(b x) e^{c+d x^2}}{2 d} \]

[Out]

(b*E^(c + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) + (3*b*E^(c + (b^2 + d)*x^2))/(4*d^2*(b^2 + d)*Sqrt[Pi])
- (b*E^(c + (b^2 + d)*x^2)*x^2)/(2*d*(b^2 + d)*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfi[b*x])/(4*d^2) + (E^(c + d*x
^2)*x^3*Erfi[b*x])/(2*d) + (3*Unintegrable[E^(c + d*x^2)*Erfi[b*x], x])/(4*d^2)

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Rubi [A]  time = 0.234193, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int e^{c+d x^2} x^4 \text{Erfi}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Int[E^(c + d*x^2)*x^4*Erfi[b*x],x]

[Out]

(b*E^(c + (b^2 + d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) + (3*b*E^(c + (b^2 + d)*x^2))/(4*d^2*(b^2 + d)*Sqrt[Pi])
- (b*E^(c + (b^2 + d)*x^2)*x^2)/(2*d*(b^2 + d)*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfi[b*x])/(4*d^2) + (E^(c + d*x
^2)*x^3*Erfi[b*x])/(2*d) + (3*Defer[Int][E^(c + d*x^2)*Erfi[b*x], x])/(4*d^2)

Rubi steps

\begin{align*} \int e^{c+d x^2} x^4 \text{erfi}(b x) \, dx &=\frac{e^{c+d x^2} x^3 \text{erfi}(b x)}{2 d}-\frac{3 \int e^{c+d x^2} x^2 \text{erfi}(b x) \, dx}{2 d}-\frac{b \int e^{c+\left (b^2+d\right ) x^2} x^3 \, dx}{d \sqrt{\pi }}\\ &=-\frac{b e^{c+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erfi}(b x) \, dx}{4 d^2}+\frac{(3 b) \int e^{c+\left (b^2+d\right ) x^2} x \, dx}{2 d^2 \sqrt{\pi }}+\frac{b \int e^{c+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right ) \sqrt{\pi }}\\ &=\frac{b e^{c+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt{\pi }}+\frac{3 b e^{c+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt{\pi }}-\frac{b e^{c+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erfi}(b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erfi}(b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erfi}(b x) \, dx}{4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.268197, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^4 \text{Erfi}(b x) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(c + d*x^2)*x^4*Erfi[b*x],x]

[Out]

Integrate[E^(c + d*x^2)*x^4*Erfi[b*x], x]

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Maple [A]  time = 0.11, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{4}{\it erfi} \left ( bx \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*x^4*erfi(b*x),x)

[Out]

int(exp(d*x^2+c)*x^4*erfi(b*x),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^4*erfi(b*x),x, algorithm="maxima")

[Out]

integrate(x^4*erfi(b*x)*e^(d*x^2 + c), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^4*erfi(b*x),x, algorithm="fricas")

[Out]

integral(x^4*erfi(b*x)*e^(d*x^2 + c), x)

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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.

[In]

integrate(exp(d*x**2+c)*x**4*erfi(b*x),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*x^4*erfi(b*x),x, algorithm="giac")

[Out]

integrate(x^4*erfi(b*x)*e^(d*x^2 + c), x)