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

Optimal. Leaf size=52 \[ \frac {2 b \text {Int}\left (\frac {e^{-(a+b x)^2}}{c+d x},x\right )}{\sqrt {\pi } d}-\frac {\text {erf}(a+b x)}{d (c+d x)} \]

[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)

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Rubi [A]  time = 0.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {Erf}(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[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*} \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) \int \frac {e^{-(a+b x)^2}}{c+d x} \, dx}{d \sqrt {\pi }}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Erf[a + b*x]/(c + d*x)^2,x]

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x + a\right )}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (b x +a \right )}{\left (d x +c \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x\right )}}{\sqrt {\pi } d^{2} x e^{\left (a^{2}\right )} + \sqrt {\pi } c d e^{\left (a^{2}\right )}}\,{d x} - \frac {\operatorname {erf}\left (b x + a\right )}{d^{2} x + c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {erf}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(a + b*x)/(c + d*x)^2,x)

[Out]

int(erf(a + b*x)/(c + d*x)^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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