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