\(\int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx\) [89]

Optimal result

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

[Out]

-1/2*exp(d*x^2+c)*erf(b*x+a)/x^2-b*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))*(b^2-d)^(1/2)-b*exp
(-a^2+c-2*a*b*x-(b^2-d)*x^2)/x/Pi^(1/2)-2*a*b^2*Unintegrable(exp(-a^2+c-2*a*b*x+(-b^2+d)*x^2)/x,x)/Pi^(1/2)+d*
Unintegrable(exp(d*x^2+c)*erf(b*x+a)/x,x)

Rubi [N/A]

Not integrable

Time = 0.36 (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 = {} \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=\int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx \]

[In]

Int[(E^(c + d*x^2)*Erf[a + b*x])/x^3,x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=\int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx \]

[In]

Integrate[(E^(c + d*x^2)*Erf[a + b*x])/x^3,x]

[Out]

Integrate[(E^(c + d*x^2)*Erf[a + b*x])/x^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erf}\left (b x +a \right )}{x^{3}}d x\]

[In]

int(exp(d*x^2+c)*erf(b*x+a)/x^3,x)

[Out]

int(exp(d*x^2+c)*erf(b*x+a)/x^3,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x+a)/x^3,x, algorithm="fricas")

[Out]

integral(erf(b*x + a)*e^(d*x^2 + c)/x^3, x)

Sympy [N/A]

Not integrable

Time = 26.88 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erf}{\left (a + b x \right )}}{x^{3}}\, dx \]

[In]

integrate(exp(d*x**2+c)*erf(b*x+a)/x**3,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erf(a + b*x)/x**3, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x+a)/x^3,x, algorithm="maxima")

[Out]

integrate(erf(b*x + a)*e^(d*x^2 + c)/x^3, x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]

[In]

integrate(exp(d*x^2+c)*erf(b*x+a)/x^3,x, algorithm="giac")

[Out]

integrate(erf(b*x + a)*e^(d*x^2 + c)/x^3, x)

Mupad [N/A]

Not integrable

Time = 5.55 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^3} \,d x \]

[In]

int((erf(a + b*x)*exp(c + d*x^2))/x^3,x)

[Out]

int((erf(a + b*x)*exp(c + d*x^2))/x^3, x)