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

Optimal. Leaf size=106 \[ \frac{2 b^2 (b c-a d) \text{Unintegrable}\left (\frac{e^{-(a+b x)^2}}{c+d x},x\right )}{\sqrt{\pi } d^3}-\frac{b^2 \text{Erf}(a+b x)}{d^3}-\frac{b e^{-(a+b x)^2}}{\sqrt{\pi } d^2 (c+d x)}-\frac{\text{Erf}(a+b x)}{2 d (c+d x)^2} \]

[Out]

-(b/(d^2*E^(a + b*x)^2*Sqrt[Pi]*(c + d*x))) - (b^2*Erf[a + b*x])/d^3 - Erf[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 = 0.0839015, 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 = {} \[ \int \frac{\text{Erf}(a+b x)}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Erf[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 - Erf[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{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*}

Mathematica [A]  time = 0.530478, size = 0, normalized size = 0. \[ \int \frac{\text{Erf}(a+b x)}{(c+d x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Erf[a + b*x]/(c + d*x)^3,x]

[Out]

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

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Maple [A]  time = 0.426, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Erf} \left ( bx+a \right ) }{ \left ( dx+c \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*integrate(e^(-b^2*x^2 - 2*a*b*x)/(sqrt(pi)*d^3*x^2*e^(a^2) + 2*sqrt(pi)*c*d^2*x*e^(a^2) + sqrt(pi)*c^2*d*e^(
a^2)), x) - 1/2*erf(b*x + a)/(d^3*x^2 + 2*c*d^2*x + c^2*d)

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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 )}{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(erf(b*x+a)/(d*x+c)^3,x, algorithm="fricas")

[Out]

integral(erf(b*x + a)/(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(erf(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{erf}\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(erf(b*x+a)/(d*x+c)^3,x, algorithm="giac")

[Out]

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