\(\int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx\) [298]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=-\frac {a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt {\pi }}+\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt {\pi }}+\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2}{2 d \left (b^2+d\right ) \sqrt {\pi }}-\frac {3 e^{c+d x^2} x \text {erfi}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}+\frac {a^3 b^4 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d \left (b^2+d\right )^{7/2}}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{5/2}}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d^2 \left (b^2+d\right )^{3/2}}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {erfi}(a+b x),x\right )}{4 d^2} \]

[Out]

-3/4*exp(d*x^2+c)*x*erfi(b*x+a)/d^2+1/2*exp(d*x^2+c)*x^3*erfi(b*x+a)/d+1/2*a^3*b^4*exp(c+a^2*d/(b^2+d))*erfi((
a*b+(b^2+d)*x)/(b^2+d)^(1/2))/d/(b^2+d)^(7/2)-3/4*a*b^2*exp(c+a^2*d/(b^2+d))*erfi((a*b+(b^2+d)*x)/(b^2+d)^(1/2
))/d/(b^2+d)^(5/2)-3/4*a*b^2*exp(c+a^2*d/(b^2+d))*erfi((a*b+(b^2+d)*x)/(b^2+d)^(1/2))/d^2/(b^2+d)^(3/2)-1/2*a^
2*b^3*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/d/(b^2+d)^3/Pi^(1/2)+1/2*b*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/d/(b^2+d)^2/Pi^
(1/2)+3/4*b*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/d^2/(b^2+d)/Pi^(1/2)+1/2*a*b^2*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)*x/d/(
b^2+d)^2/Pi^(1/2)-1/2*b*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)*x^2/d/(b^2+d)/Pi^(1/2)+3/4*Unintegrable(exp(d*x^2+c)*er
fi(b*x+a),x)/d^2

Rubi [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 19, 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 = {} \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx \]

[In]

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

[Out]

-1/2*(a^2*b^3*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(d*(b^2 + d)^3*Sqrt[Pi]) + (b*E^(a^2 + c + 2*a*b*x + (b^2
+ d)*x^2))/(2*d*(b^2 + d)^2*Sqrt[Pi]) + (3*b*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2))/(4*d^2*(b^2 + d)*Sqrt[Pi]
) + (a*b^2*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2)*x)/(2*d*(b^2 + d)^2*Sqrt[Pi]) - (b*E^(a^2 + c + 2*a*b*x + (b^
2 + d)*x^2)*x^2)/(2*d*(b^2 + d)*Sqrt[Pi]) - (3*E^(c + d*x^2)*x*Erfi[a + b*x])/(4*d^2) + (E^(c + d*x^2)*x^3*Erf
i[a + b*x])/(2*d) + (a^3*b^4*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(2*d*(b^2 + d)
^(7/2)) - (3*a*b^2*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d*(b^2 + d)^(5/2)) -
(3*a*b^2*E^(c + (a^2*d)/(b^2 + d))*Erfi[(a*b + (b^2 + d)*x)/Sqrt[b^2 + d]])/(4*d^2*(b^2 + d)^(3/2)) + (3*Defer
[Int][E^(c + d*x^2)*Erfi[a + b*x], x])/(4*d^2)

Rubi steps \begin{align*} \text {integral}& = \frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}-\frac {3 \int e^{c+d x^2} x^2 \text {erfi}(a+b x) \, dx}{2 d}-\frac {b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^3 \, dx}{d \sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\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}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfi}(a+b x) \, dx}{4 d^2}+\frac {(3 b) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{2 d^2 \sqrt {\pi }}+\frac {b \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right ) \sqrt {\pi }}+\frac {\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x^2 \, dx}{d \left (b^2+d\right ) \sqrt {\pi }} \\ & = \frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt {\pi }}+\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\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}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfi}(a+b x) \, dx}{4 d^2}-\frac {\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {\left (a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {\left (a^2 b^3\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x \, dx}{d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {\left (3 a b^2\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{2 d^2 \left (b^2+d\right ) \sqrt {\pi }} \\ & = -\frac {a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt {\pi }}+\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt {\pi }}+\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\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}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}+\frac {3 \int e^{c+d x^2} \text {erfi}(a+b x) \, dx}{4 d^2}+\frac {\left (a^3 b^4\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{d \left (b^2+d\right )^3 \sqrt {\pi }}-\frac {\left (a b^2 e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {\left (a b^2 e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {\left (3 a b^2 e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{2 d^2 \left (b^2+d\right ) \sqrt {\pi }} \\ & = -\frac {a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt {\pi }}+\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt {\pi }}+\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\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}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{5/2}}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d^2 \left (b^2+d\right )^{3/2}}+\frac {3 \int e^{c+d x^2} \text {erfi}(a+b x) \, dx}{4 d^2}+\frac {\left (a^3 b^4 e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{d \left (b^2+d\right )^3 \sqrt {\pi }} \\ & = -\frac {a^2 b^3 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^3 \sqrt {\pi }}+\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}+\frac {3 b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{4 d^2 \left (b^2+d\right ) \sqrt {\pi }}+\frac {a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} x}{2 d \left (b^2+d\right )^2 \sqrt {\pi }}-\frac {b e^{a^2+c+2 a b x+\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}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfi}(a+b x)}{2 d}+\frac {a^3 b^4 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{2 d \left (b^2+d\right )^{7/2}}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d \left (b^2+d\right )^{5/2}}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )}{4 d^2 \left (b^2+d\right )^{3/2}}+\frac {3 \int e^{c+d x^2} \text {erfi}(a+b x) \, dx}{4 d^2} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.07 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.89 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfi}(a+b x) \, dx=\int x^4\,\mathrm {erfi}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]

[In]

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

[Out]

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