Integrand size = 19, antiderivative size = 118 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {2 e^c x}{b^5 \sqrt {\pi }}+\frac {2 e^c x^3}{3 b^3 \sqrt {\pi }}-\frac {e^c x^5}{5 b \sqrt {\pi }}+\frac {e^{c+b^2 x^2} \text {erf}(b x)}{b^6}-\frac {e^{c+b^2 x^2} x^2 \text {erf}(b x)}{b^4}+\frac {e^{c+b^2 x^2} x^4 \text {erf}(b x)}{2 b^2} \] Output:
-2*exp(c)*x/b^5/Pi^(1/2)+2/3*exp(c)*x^3/b^3/Pi^(1/2)-1/5*exp(c)*x^5/b/Pi^( 1/2)+exp(b^2*x^2+c)*erf(b*x)/b^6-exp(b^2*x^2+c)*x^2*erf(b*x)/b^4+1/2*exp(b ^2*x^2+c)*x^4*erf(b*x)/b^2
Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {e^c \left (-60 b x+20 b^3 x^3-6 b^5 x^5+15 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {erf}(b x)\right )}{30 b^6 \sqrt {\pi }} \] Input:
Integrate[E^(c + b^2*x^2)*x^5*Erf[b*x],x]
Output:
(E^c*(-60*b*x + 20*b^3*x^3 - 6*b^5*x^5 + 15*E^(b^2*x^2)*Sqrt[Pi]*(2 - 2*b^ 2*x^2 + b^4*x^4)*Erf[b*x]))/(30*b^6*Sqrt[Pi])
Time = 0.51 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6939, 15, 6939, 15, 6936, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^5 e^{b^2 x^2+c} \text {erf}(b x) \, dx\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle -\frac {2 \int e^{b^2 x^2+c} x^3 \text {erf}(b x)dx}{b^2}-\frac {\int e^c x^4dx}{\sqrt {\pi } b}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {2 \int e^{b^2 x^2+c} x^3 \text {erf}(b x)dx}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6939 |
\(\displaystyle -\frac {2 \left (-\frac {\int e^{b^2 x^2+c} x \text {erf}(b x)dx}{b^2}-\frac {\int e^c x^2dx}{\sqrt {\pi } b}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}\right )}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {2 \left (-\frac {\int e^{b^2 x^2+c} x \text {erf}(b x)dx}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^3}{3 \sqrt {\pi } b}\right )}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 6936 |
\(\displaystyle -\frac {2 \left (-\frac {\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {\int e^cdx}{\sqrt {\pi } b}}{b^2}+\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^3}{3 \sqrt {\pi } b}\right )}{b^2}+\frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {x^4 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {2 \left (\frac {x^2 e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {\frac {e^{b^2 x^2+c} \text {erf}(b x)}{2 b^2}-\frac {e^c x}{\sqrt {\pi } b}}{b^2}-\frac {e^c x^3}{3 \sqrt {\pi } b}\right )}{b^2}-\frac {e^c x^5}{5 \sqrt {\pi } b}\) |
Input:
Int[E^(c + b^2*x^2)*x^5*Erf[b*x],x]
Output:
-1/5*(E^c*x^5)/(b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^4*Erf[b*x])/(2*b^2) - (2* (-1/3*(E^c*x^3)/(b*Sqrt[Pi]) + (E^(c + b^2*x^2)*x^2*Erf[b*x])/(2*b^2) - (- ((E^c*x)/(b*Sqrt[Pi])) + (E^(c + b^2*x^2)*Erf[b*x])/(2*b^2))/b^2))/b^2
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Sim p[E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] - Simp[b/(d*Sqrt[Pi]) Int[E^(-a^ 2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]
Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] : > Simp[x^(m - 1)*E^(c + d*x^2)*(Erf[a + b*x]/(2*d)), x] + (-Simp[(m - 1)/(2 *d) Int[x^(m - 2)*E^(c + d*x^2)*Erf[a + b*x], x], x] - Simp[b/(d*Sqrt[Pi] ) Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[ {a, b, c, d}, x] && IGtQ[m, 1]
Time = 2.76 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {\frac {\operatorname {erf}\left (b x \right ) {\mathrm e}^{c} \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{b^{5}}-\frac {{\mathrm e}^{c} \left (\frac {1}{5} b^{5} x^{5}-\frac {2}{3} b^{3} x^{3}+2 b x \right )}{\sqrt {\pi }\, b^{5}}}{b}\) | \(88\) |
orering | \(\frac {\left (3 b^{6} x^{6}+5 b^{4} x^{4}-10 b^{2} x^{2}+90\right ) {\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right )}{15 b^{6}}-\frac {\left (3 b^{4} x^{4}-10 b^{2} x^{2}+30\right ) \left (2 b^{2} x^{6} {\mathrm e}^{b^{2} x^{2}+c} \operatorname {erf}\left (b x \right )+5 \,{\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erf}\left (b x \right )+\frac {2 \,{\mathrm e}^{b^{2} x^{2}+c} x^{5} {\mathrm e}^{-b^{2} x^{2}} b}{\sqrt {\pi }}\right )}{30 x^{4} b^{6}}\) | \(143\) |
parallelrisch | \(\frac {-6 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{5} b^{5}+15 \,{\mathrm e}^{b^{2} x^{2}+c} x^{4} \operatorname {erf}\left (b x \right ) b^{4} \sqrt {\pi }+20 \,{\mathrm e}^{b^{2} x^{2}+c} {\mathrm e}^{-b^{2} x^{2}} x^{3} b^{3}-30 \,{\mathrm e}^{b^{2} x^{2}+c} x^{2} \operatorname {erf}\left (b x \right ) b^{2} \sqrt {\pi }-60 \,{\mathrm e}^{b^{2} x^{2}+c} x \,{\mathrm e}^{-b^{2} x^{2}} b +30 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}+c} \sqrt {\pi }}{30 b^{6} \sqrt {\pi }}\) | \(156\) |
Input:
int(exp(b^2*x^2+c)*x^5*erf(b*x),x,method=_RETURNVERBOSE)
Output:
(erf(b*x)/b^5*exp(c)*(1/2*exp(b^2*x^2)*b^4*x^4-b^2*x^2*exp(b^2*x^2)+exp(b^ 2*x^2))-1/Pi^(1/2)/b^5*exp(c)*(1/5*b^5*x^5-2/3*b^3*x^3+2*b*x))/b
Time = 0.10 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.63 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {15 \, {\left (2 \, \pi + \pi b^{4} x^{4} - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )} - 2 \, \sqrt {\pi } {\left (3 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 30 \, b x\right )} e^{c}}{30 \, \pi b^{6}} \] Input:
integrate(exp(b^2*x^2+c)*x^5*erf(b*x),x, algorithm="fricas")
Output:
1/30*(15*(2*pi + pi*b^4*x^4 - 2*pi*b^2*x^2)*erf(b*x)*e^(b^2*x^2 + c) - 2*s qrt(pi)*(3*b^5*x^5 - 10*b^3*x^3 + 30*b*x)*e^c)/(pi*b^6)
Time = 35.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.01 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\begin {cases} - \frac {x^{5} e^{c}}{5 \sqrt {\pi } b} + \frac {x^{4} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{2 b^{2}} + \frac {2 x^{3} e^{c}}{3 \sqrt {\pi } b^{3}} - \frac {x^{2} e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{b^{4}} - \frac {2 x e^{c}}{\sqrt {\pi } b^{5}} + \frac {e^{c} e^{b^{2} x^{2}} \operatorname {erf}{\left (b x \right )}}{b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(exp(b**2*x**2+c)*x**5*erf(b*x),x)
Output:
Piecewise((-x**5*exp(c)/(5*sqrt(pi)*b) + x**4*exp(c)*exp(b**2*x**2)*erf(b* x)/(2*b**2) + 2*x**3*exp(c)/(3*sqrt(pi)*b**3) - x**2*exp(c)*exp(b**2*x**2) *erf(b*x)/b**4 - 2*x*exp(c)/(sqrt(pi)*b**5) + exp(c)*exp(b**2*x**2)*erf(b* x)/b**6, Ne(b, 0)), (0, True))
Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.69 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=-\frac {6 \, b^{5} x^{5} e^{c} - 20 \, b^{3} x^{3} e^{c} - 15 \, {\left (\sqrt {\pi } b^{4} x^{4} e^{c} - 2 \, \sqrt {\pi } b^{2} x^{2} e^{c} + 2 \, \sqrt {\pi } e^{c}\right )} \operatorname {erf}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 60 \, b x e^{c}}{30 \, \sqrt {\pi } b^{6}} \] Input:
integrate(exp(b^2*x^2+c)*x^5*erf(b*x),x, algorithm="maxima")
Output:
-1/30*(6*b^5*x^5*e^c - 20*b^3*x^3*e^c - 15*(sqrt(pi)*b^4*x^4*e^c - 2*sqrt( pi)*b^2*x^2*e^c + 2*sqrt(pi)*e^c)*erf(b*x)*e^(b^2*x^2) + 60*b*x*e^c)/(sqrt (pi)*b^6)
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {1}{2} \, {\left (\frac {c^{2} e^{\left (b^{2} x^{2} + c\right )}}{b^{6}} - \frac {{\left (2 \, b^{2} x^{2} - {\left (b^{2} x^{2} + c\right )}^{2} + 2 \, {\left (b^{2} x^{2} + c\right )} c - 2\right )} e^{\left (b^{2} x^{2} + c\right )}}{b^{6}}\right )} \operatorname {erf}\left (b x\right ) - \frac {3 \, \sqrt {\pi } b^{4} x^{5} e^{c} - 10 \, \sqrt {\pi } b^{2} x^{3} e^{c} + 30 \, \sqrt {\pi } x e^{c}}{15 \, \pi b^{5}} \] Input:
integrate(exp(b^2*x^2+c)*x^5*erf(b*x),x, algorithm="giac")
Output:
1/2*(c^2*e^(b^2*x^2 + c)/b^6 - (2*b^2*x^2 - (b^2*x^2 + c)^2 + 2*(b^2*x^2 + c)*c - 2)*e^(b^2*x^2 + c)/b^6)*erf(b*x) - 1/15*(3*sqrt(pi)*b^4*x^5*e^c - 10*sqrt(pi)*b^2*x^3*e^c + 30*sqrt(pi)*x*e^c)/(pi*b^5)
Time = 3.96 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.77 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\mathrm {erf}\left (b\,x\right )\,\left (\frac {{\mathrm {e}}^{b^2\,x^2+c}}{b^6}+\frac {x^4\,{\mathrm {e}}^{b^2\,x^2+c}}{2\,b^2}-\frac {x^2\,{\mathrm {e}}^{b^2\,x^2+c}}{b^4}\right )-\frac {3\,{\mathrm {e}}^c\,b^4\,x^5-10\,{\mathrm {e}}^c\,b^2\,x^3+30\,{\mathrm {e}}^c\,x}{15\,b^5\,\sqrt {\pi }} \] Input:
int(x^5*exp(c + b^2*x^2)*erf(b*x),x)
Output:
erf(b*x)*(exp(c + b^2*x^2)/b^6 + (x^4*exp(c + b^2*x^2))/(2*b^2) - (x^2*exp (c + b^2*x^2))/b^4) - (30*x*exp(c) - 10*b^2*x^3*exp(c) + 3*b^4*x^5*exp(c)) /(15*b^5*pi^(1/2))
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.83 \[ \int e^{c+b^2 x^2} x^5 \text {erf}(b x) \, dx=\frac {e^{c} \left (15 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{4} \pi \,x^{4}-30 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b^{2} \pi \,x^{2}+30 e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) \pi -6 \sqrt {\pi }\, b^{5} x^{5}+20 \sqrt {\pi }\, b^{3} x^{3}-60 \sqrt {\pi }\, b x \right )}{30 b^{6} \pi } \] Input:
int(exp(b^2*x^2+c)*x^5*erf(b*x),x)
Output:
(e**c*(15*e**(b**2*x**2)*erf(b*x)*b**4*pi*x**4 - 30*e**(b**2*x**2)*erf(b*x )*b**2*pi*x**2 + 30*e**(b**2*x**2)*erf(b*x)*pi - 6*sqrt(pi)*b**5*x**5 + 20 *sqrt(pi)*b**3*x**3 - 60*sqrt(pi)*b*x))/(30*b**6*pi)