\(\int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx\) [20]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=-\frac {\text {erf}(a+b x)}{d (c+d x)}+\frac {2 b \text {Int}\left (\frac {e^{-(a+b x)^2}}{c+d x},x\right )}{d \sqrt {\pi }} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 14, 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 = {} \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx \]

[In]

Int[Erf[a + b*x]/(c + d*x)^2,x]

[Out]

-(Erf[a + b*x]/(d*(c + d*x))) + (2*b*Defer[Int][1/(E^(a + b*x)^2*(c + d*x)), x])/(d*Sqrt[Pi])

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

Mathematica [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx \]

[In]

Integrate[Erf[a + b*x]/(c + d*x)^2,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.22 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 10.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\operatorname {erf}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 66, normalized size of antiderivative = 4.71 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.38 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx=\int \frac {\mathrm {erf}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(erf(a + b*x)/(c + d*x)^2,x)

[Out]

int(erf(a + b*x)/(c + d*x)^2, x)