\(\int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx\) [21]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=-\frac {b e^{-(a+b x)^2}}{d^2 \sqrt {\pi } (c+d x)}-\frac {b^2 \text {erf}(a+b x)}{d^3}-\frac {\text {erf}(a+b x)}{2 d (c+d x)^2}+\frac {2 b^2 (b c-a d) \text {Int}\left (\frac {e^{-(a+b x)^2}}{c+d x},x\right )}{d^3 \sqrt {\pi }} \]

[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)

Rubi [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx \]

[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*} \text {integral}& = -\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 [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx \]

[In]

Integrate[Erf[a + b*x]/(c + d*x)^3,x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.23 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {\operatorname {erf}\left (b x +a \right )}{\left (d x +c \right )^{3}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.71 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{3}} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 72.58 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int \frac {\operatorname {erf}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 94, normalized size of antiderivative = 6.71 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{3}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int { \frac {\operatorname {erf}\left (b x + a\right )}{{\left (d x + c\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {\text {erf}(a+b x)}{(c+d x)^3} \, dx=\int \frac {\mathrm {erf}\left (a+b\,x\right )}{{\left (c+d\,x\right )}^3} \,d x \]

[In]

int(erf(a + b*x)/(c + d*x)^3,x)

[Out]

int(erf(a + b*x)/(c + d*x)^3, x)