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

Optimal. Leaf size=107 \[ \frac {2 b^2 (b c-a d) \text {Int}\left (\frac {e^{-(a+b x)^2}}{c+d x},x\right )}{\sqrt {\pi } d^3}-\frac {b^2 \text {erf}(a+b x)}{d^3}-\frac {b e^{-(a+b x)^2}}{\sqrt {\pi } d^2 (c+d x)}-\frac {\text {erf}(a+b x)}{2 d (c+d x)^2} \]

[Out]

-b^2*erf(b*x+a)/d^3-1/2*erf(b*x+a)/d/(d*x+c)^2-b/d^2/exp((b*x+a)^2)/(d*x+c)/Pi^(1/2)+2*b^2*(-a*d+b*c)*Unintegr
able(1/exp((b*x+a)^2)/(d*x+c),x)/d^3/Pi^(1/2)

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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.

[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*}

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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.

[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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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.

[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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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.

[In]

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

[Out]

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

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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.

[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 = 0.00, 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.

[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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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.

[In]

int(erf(a + b*x)/(c + d*x)^3,x)

[Out]

int(erf(a + b*x)/(c + d*x)^3, x)

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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.

[In]

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

[Out]

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

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