3.124 \(\int \frac{\text{Erfc}(a+b x)}{(c+d x)^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac{2 b^2 (b c-a d) \text{Unintegrable}\left (\frac{e^{-(a+b x)^2}}{c+d x},x\right )}{\sqrt{\pi } d^3}+\frac{b^2 \text{Erf}(a+b x)}{d^3}+\frac{b e^{-(a+b x)^2}}{\sqrt{\pi } d^2 (c+d x)}-\frac{\text{Erfc}(a+b x)}{2 d (c+d x)^2} \]

[Out]

b/(d^2*E^(a + b*x)^2*Sqrt[Pi]*(c + d*x)) + (b^2*Erf[a + b*x])/d^3 - Erfc[a + b*x]/(2*d*(c + d*x)^2) - (2*b^2*(
b*c - a*d)*Unintegrable[1/(E^(a + b*x)^2*(c + d*x)), x])/(d^3*Sqrt[Pi])

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Rubi [A]  time = 0.0792894, 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 = {} \[ \int \frac{\text{Erfc}(a+b x)}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Erfc[a + b*x]/(c + d*x)^3,x]

[Out]

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

Rubi steps

\begin{align*} \int \frac{\text{erfc}(a+b x)}{(c+d x)^3} \, dx &=-\frac{\text{erfc}(a+b x)}{2 d (c+d x)^2}-\frac{b \int \frac{e^{-(a+b x)^2}}{(c+d x)^2} \, dx}{d \sqrt{\pi }}\\ &=\frac{b e^{-(a+b x)^2}}{d^2 \sqrt{\pi } (c+d x)}-\frac{\text{erfc}(a+b x)}{2 d (c+d x)^2}+\frac{\left (2 b^3\right ) \int e^{-(a+b x)^2} \, dx}{d^3 \sqrt{\pi }}-\frac{\left (2 b^2 (b c-a d)\right ) \int \frac{e^{-(a+b x)^2}}{c+d x} \, dx}{d^3 \sqrt{\pi }}\\ &=\frac{b e^{-(a+b x)^2}}{d^2 \sqrt{\pi } (c+d x)}+\frac{b^2 \text{erf}(a+b x)}{d^3}-\frac{\text{erfc}(a+b x)}{2 d (c+d x)^2}-\frac{\left (2 b^2 (b c-a d)\right ) \int \frac{e^{-(a+b x)^2}}{c+d x} \, dx}{d^3 \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 0.959073, size = 0, normalized size = 0. \[ \int \frac{\text{Erfc}(a+b x)}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.399, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it erfc} \left ( bx+a \right ) }{ \left ( dx+c \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(b*x+a)/(d*x+c)^3,x)

[Out]

int(erfc(b*x+a)/(d*x+c)^3,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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x + a)/(d*x + c)^3, 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^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(erf(b*x + a) - 1)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erfc(b*x+a)/(d*x+c)**3,x)

[Out]

Timed out

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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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x + a)/(d*x + c)^3, x)