\(\int e^{c+d x^2} x^4 \text {erf}(b x) \, dx\) [59]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=-\frac {3 b e^{c-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {b e^{c-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erf}(b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erf}(b x)}{2 d}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {erf}(b x),x\right )}{4 d^2} \]

[Out]

-3/4*exp(d*x^2+c)*x*erf(b*x)/d^2+1/2*exp(d*x^2+c)*x^3*erf(b*x)/d-3/4*b*exp(c-(b^2-d)*x^2)/(b^2-d)/d^2/Pi^(1/2)
+1/2*b*exp(c-(b^2-d)*x^2)/(b^2-d)^2/d/Pi^(1/2)+1/2*b*exp(c-(b^2-d)*x^2)*x^2/(b^2-d)/d/Pi^(1/2)+3/4*Unintegrabl
e(exp(d*x^2+c)*erf(b*x),x)/d^2

Rubi [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=\int e^{c+d x^2} x^4 \text {erf}(b x) \, dx \]

[In]

Int[E^(c + d*x^2)*x^4*Erf[b*x],x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \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 [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=\int e^{c+d x^2} x^4 \text {erf}(b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.08 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94

\[\int {\mathrm e}^{d \,x^{2}+c} x^{4} \operatorname {erf}\left (b x \right )d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 42.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=e^{c} \int x^{4} e^{d x^{2}} \operatorname {erf}{\left (b x \right )}\, dx \]

[In]

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

[Out]

exp(c)*Integral(x**4*exp(d*x**2)*erf(b*x), x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=\int { x^{4} \operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.96 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int e^{c+d x^2} x^4 \text {erf}(b x) \, dx=\int x^4\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right ) \,d x \]

[In]

int(x^4*exp(c + d*x^2)*erf(b*x),x)

[Out]

int(x^4*exp(c + d*x^2)*erf(b*x), x)