Optimal. Leaf size=107 \[ \frac {2 b^2 (b c-a d) \text {Int}\left (\frac {e^{-(a+b x)^2}}{c+d x},x\right )}{\sqrt {\pi } d^3}-\frac {b^2 \text {erf}(a+b x)}{d^3}-\frac {b e^{-(a+b x)^2}}{\sqrt {\pi } d^2 (c+d x)}-\frac {\text {erf}(a+b x)}{2 d (c+d x)^2} \]
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Rubi [A] time = 0.08, 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 = {} \[ \int \frac {\text {Erf}(a+b x)}{(c+d x)^3} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx &=-\frac {\text {erf}(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 {erf}(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 {erf}(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*}
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Mathematica [A] time = 0.58, size = 0, normalized size = 0.00 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x + a\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\erf \left (b x +a \right )}{\left (d x +c \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ b \int \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x\right )}}{\sqrt {\pi } d^{3} x^{2} e^{\left (a^{2}\right )} + 2 \, \sqrt {\pi } c d^{2} x e^{\left (a^{2}\right )} + \sqrt {\pi } c^{2} d e^{\left (a^{2}\right )}}\,{d x} - \frac {\operatorname {erf}\left (b x + a\right )}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {erf}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erf}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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