Optimal. Leaf size=185 \[ \frac{3 \text{Unintegrable}\left (\text{Erf}(b x) e^{c+d x^2},x\right )}{4 d^2}-\frac{3 b e^{c-x^2 \left (b^2-d\right )}}{4 \sqrt{\pi } d^2 \left (b^2-d\right )}+\frac{b x^2 e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )}+\frac{b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt{\pi } d \left (b^2-d\right )^2}-\frac{3 x \text{Erf}(b x) e^{c+d x^2}}{4 d^2}+\frac{x^3 \text{Erf}(b x) e^{c+d x^2}}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.243807, 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 = {} \[ \int e^{c+d x^2} x^4 \text{Erf}(b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int e^{c+d x^2} x^4 \text{erf}(b x) \, dx &=\frac{e^{c+d x^2} x^3 \text{erf}(b x)}{2 d}-\frac{3 \int e^{c+d x^2} x^2 \text{erf}(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 \left (b^2-d\right ) d \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erf}(b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erf}(b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erf}(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}{\left (b^2-d\right ) d \sqrt{\pi }}\\ &=-\frac{3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt{\pi }}+\frac{b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt{\pi }}+\frac{b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt{\pi }}-\frac{3 e^{c+d x^2} x \text{erf}(b x)}{4 d^2}+\frac{e^{c+d x^2} x^3 \text{erf}(b x)}{2 d}+\frac{3 \int e^{c+d x^2} \text{erf}(b x) \, dx}{4 d^2}\\ \end{align*}
Mathematica [A] time = 0.289221, size = 0, normalized size = 0. \[ \int e^{c+d x^2} x^4 \text{Erf}(b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.142, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{d{x}^{2}+c}}{x}^{4}{\it Erf} \left ( bx \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erf}\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]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{4} \operatorname{erf}\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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{erf}\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]
[Out]