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

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\text{Erf}(a+b x)^2}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable[Erf[a + b*x]^2/(c + d*x)^2, x]

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Rubi [A]  time = 0.0222502, 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 = {} \[ \int \frac{\text{Erf}(a+b x)^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Erf[a + b*x]^2/(c + d*x)^2,x]

[Out]

Defer[Int][Erf[a + b*x]^2/(c + d*x)^2, x]

Rubi steps

\begin{align*} \int \frac{\text{erf}(a+b x)^2}{(c+d x)^2} \, dx &=\int \frac{\text{erf}(a+b x)^2}{(c+d x)^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.105703, size = 0, normalized size = 0. \[ \int \frac{\text{Erf}(a+b x)^2}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.365, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\it Erf} \left ( bx+a \right ) \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, b \int \frac{\operatorname{erf}\left (b x + a\right ) e^{\left (-b^{2} x^{2} - 2 \, a b x\right )}}{\sqrt{\pi } d^{2} x e^{\left (a^{2}\right )} + \sqrt{\pi } c d e^{\left (a^{2}\right )}}\,{d x} - \frac{\operatorname{erf}\left (b x + a\right )^{2}}{d^{2} x + c d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*b*integrate(erf(b*x + a)*e^(-b^2*x^2 - 2*a*b*x)/(sqrt(pi)*d^2*x*e^(a^2) + sqrt(pi)*c*d*e^(a^2)), x) - erf(b*
x + a)^2/(d^2*x + c*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 )^{2}}{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(erf(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(erf(b*x + a)^2/(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{erf}^{2}{\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(erf(b*x+a)**2/(d*x+c)**2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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