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

Optimal. Leaf size=51 \[ -\frac{2 b \text{Unintegrable}\left (\frac{e^{-(a+b x)^2}}{c+d x},x\right )}{\sqrt{\pi } d}-\frac{\text{Erfc}(a+b x)}{d (c+d x)} \]

[Out]

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

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Rubi [A]  time = 0.038967, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{Erfc}(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Erfc[a + b*x]/(c + d*x)^2,x]

[Out]

-(Erfc[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{erfc}(a+b x)}{(c+d x)^2} \, dx &=-\frac{\text{erfc}(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 [A]  time = 0.488978, size = 0, normalized size = 0. \[ \int \frac{\text{Erfc}(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.375, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfc} \left ( bx+a \right ) }{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\operatorname{erf}\left (b x + a\right ) - 1}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x + a\right )}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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