Integrand size = 14, antiderivative size = 292 \[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=-\frac {d^2 (b c-a d) e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {(b c-a d)^3 e^{-(a+b x)^2}}{b^4 \sqrt {\pi }}-\frac {3 d^3 e^{-(a+b x)^2} (a+b x)}{8 b^4 \sqrt {\pi }}-\frac {3 d (b c-a d)^2 e^{-(a+b x)^2} (a+b x)}{2 b^4 \sqrt {\pi }}-\frac {d^2 (b c-a d) e^{-(a+b x)^2} (a+b x)^2}{b^4 \sqrt {\pi }}-\frac {d^3 e^{-(a+b x)^2} (a+b x)^3}{4 b^4 \sqrt {\pi }}+\frac {3 d^3 \text {erf}(a+b x)}{16 b^4}+\frac {3 d (b c-a d)^2 \text {erf}(a+b x)}{4 b^4}+\frac {(b c-a d)^4 \text {erf}(a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d} \] Output:
-d^2*(-a*d+b*c)/b^4/exp((b*x+a)^2)/Pi^(1/2)-(-a*d+b*c)^3/b^4/exp((b*x+a)^2 )/Pi^(1/2)-3/8*d^3*(b*x+a)/b^4/exp((b*x+a)^2)/Pi^(1/2)-3/2*d*(-a*d+b*c)^2* (b*x+a)/b^4/exp((b*x+a)^2)/Pi^(1/2)-d^2*(-a*d+b*c)*(b*x+a)^2/b^4/exp((b*x+ a)^2)/Pi^(1/2)-1/4*d^3*(b*x+a)^3/b^4/exp((b*x+a)^2)/Pi^(1/2)+3/16*d^3*erf( b*x+a)/b^4+3/4*d*(-a*d+b*c)^2*erf(b*x+a)/b^4+1/4*(-a*d+b*c)^4*erf(b*x+a)/b ^4/d+1/4*(d*x+c)^4*erfc(b*x+a)/d
Time = 0.24 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.92 \[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=\frac {e^{-(a+b x)^2} \left (2 a \left (5+2 a^2\right ) d^3-2 b d^2 \left (8 \left (1+a^2\right ) c+\left (3+2 a^2\right ) d x\right )+4 a b^2 d \left (6 c^2+4 c d x+d^2 x^2\right )-4 b^3 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )+\left (-16 a^3 b c d^2+4 a^4 d^3-8 a \left (2 b^3 c^3+3 b c d^2\right )+12 a^2 \left (2 b^2 c^2 d+d^3\right )+3 \left (4 b^2 c^2 d+d^3\right )\right ) e^{(a+b x)^2} \sqrt {\pi } \text {erf}(a+b x)+4 b^4 e^{(a+b x)^2} \sqrt {\pi } x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \text {erfc}(a+b x)\right )}{16 b^4 \sqrt {\pi }} \] Input:
Integrate[(c + d*x)^3*Erfc[a + b*x],x]
Output:
(2*a*(5 + 2*a^2)*d^3 - 2*b*d^2*(8*(1 + a^2)*c + (3 + 2*a^2)*d*x) + 4*a*b^2 *d*(6*c^2 + 4*c*d*x + d^2*x^2) - 4*b^3*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + (-16*a^3*b*c*d^2 + 4*a^4*d^3 - 8*a*(2*b^3*c^3 + 3*b*c*d^2) + 12 *a^2*(2*b^2*c^2*d + d^3) + 3*(4*b^2*c^2*d + d^3))*E^(a + b*x)^2*Sqrt[Pi]*E rf[a + b*x] + 4*b^4*E^(a + b*x)^2*Sqrt[Pi]*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2* x^2 + d^3*x^3)*Erfc[a + b*x])/(16*b^4*E^(a + b*x)^2*Sqrt[Pi])
Time = 0.59 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6916, 2656, 2009}
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.
| \(\displaystyle \int (c+d x)^3 \text {erfc}(a+b x) \, dx\) |
| \(\Big \downarrow \) 6916 |
| \(\displaystyle \frac {b \int e^{-(a+b x)^2} (c+d x)^4dx}{2 \sqrt {\pi } d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \frac {b \int \left (\frac {e^{-(a+b x)^2} (b c-a d)^4}{b^4}+\frac {4 d e^{-(a+b x)^2} (a+b x) (b c-a d)^3}{b^4}+\frac {6 d^2 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)^2}{b^4}+\frac {4 d^3 e^{-(a+b x)^2} (a+b x)^3 (b c-a d)}{b^4}+\frac {d^4 e^{-(a+b x)^2} (a+b x)^4}{b^4}\right )dx}{2 \sqrt {\pi } d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (-\frac {2 d^3 e^{-(a+b x)^2} (b c-a d)}{b^5}-\frac {2 d^3 e^{-(a+b x)^2} (a+b x)^2 (b c-a d)}{b^5}+\frac {3 \sqrt {\pi } d^2 (b c-a d)^2 \text {erf}(a+b x)}{2 b^5}-\frac {3 d^2 e^{-(a+b x)^2} (a+b x) (b c-a d)^2}{b^5}+\frac {\sqrt {\pi } (b c-a d)^4 \text {erf}(a+b x)}{2 b^5}-\frac {2 d e^{-(a+b x)^2} (b c-a d)^3}{b^5}+\frac {3 \sqrt {\pi } d^4 \text {erf}(a+b x)}{8 b^5}-\frac {d^4 e^{-(a+b x)^2} (a+b x)^3}{2 b^5}-\frac {3 d^4 e^{-(a+b x)^2} (a+b x)}{4 b^5}\right )}{2 \sqrt {\pi } d}+\frac {(c+d x)^4 \text {erfc}(a+b x)}{4 d}\) |
Input:
Int[(c + d*x)^3*Erfc[a + b*x],x]
Output:
(b*((-2*d^3*(b*c - a*d))/(b^5*E^(a + b*x)^2) - (2*d*(b*c - a*d)^3)/(b^5*E^ (a + b*x)^2) - (3*d^4*(a + b*x))/(4*b^5*E^(a + b*x)^2) - (3*d^2*(b*c - a*d )^2*(a + b*x))/(b^5*E^(a + b*x)^2) - (2*d^3*(b*c - a*d)*(a + b*x)^2)/(b^5* E^(a + b*x)^2) - (d^4*(a + b*x)^3)/(2*b^5*E^(a + b*x)^2) + (3*d^4*Sqrt[Pi] *Erf[a + b*x])/(8*b^5) + (3*d^2*(b*c - a*d)^2*Sqrt[Pi]*Erf[a + b*x])/(2*b^ 5) + ((b*c - a*d)^4*Sqrt[Pi]*Erf[a + b*x])/(2*b^5)))/(2*d*Sqrt[Pi]) + ((c + d*x)^4*Erfc[a + b*x])/(4*d)
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Int[Erfc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[ (c + d*x)^(m + 1)*(Erfc[a + b*x]/(d*(m + 1))), x] + Simp[2*(b/(Sqrt[Pi]*d*( m + 1))) Int[(c + d*x)^(m + 1)/E^(a + b*x)^2, x], x] /; FreeQ[{a, b, c, d , m}, x] && NeQ[m, -1]
Time = 0.73 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.65
| method | result | size |
| parallelrisch | \(\frac {16 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} a \,b^{2} c \,d^{2}+4 d^{3} \operatorname {erfc}\left (b x +a \right ) x^{4} \sqrt {\pi }\, b^{4}+16 x \,\operatorname {erfc}\left (b x +a \right ) c^{3} \sqrt {\pi }\, b^{4}+16 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a \,b^{3} c^{3}-12 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) b^{2} c^{2} d +4 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{3} d^{3}-16 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b^{3} c^{3}+10 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a \,d^{3}+16 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a^{3} b c \,d^{2}-24 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a^{2} b^{2} c^{2} d +24 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a b c \,d^{2}+16 d^{2} c \,\operatorname {erfc}\left (b x +a \right ) x^{3} \sqrt {\pi }\, b^{4}+24 c^{2} d \,\operatorname {erfc}\left (b x +a \right ) x^{2} \sqrt {\pi }\, b^{4}-3 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) d^{3}-4 d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}} x^{3} b^{3}-6 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} b \,d^{3}-16 \,{\mathrm e}^{-\left (b x +a \right )^{2}} b c \,d^{2}-4 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{2} b \,d^{3}-24 x \,{\mathrm e}^{-\left (b x +a \right )^{2}} b^{3} c^{2} d +4 x^{2} {\mathrm e}^{-\left (b x +a \right )^{2}} a \,b^{2} d^{3}-16 x^{2} {\mathrm e}^{-\left (b x +a \right )^{2}} b^{3} c \,d^{2}-16 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a^{2} b c \,d^{2}+24 \,{\mathrm e}^{-\left (b x +a \right )^{2}} a \,b^{2} c^{2} d -4 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a^{4} d^{3}-12 \sqrt {\pi }\, \operatorname {erfc}\left (b x +a \right ) a^{2} d^{3}}{16 \sqrt {\pi }\, b^{4}}\) | \(483\) |
| parts | \(\frac {\operatorname {erfc}\left (b x +a \right ) d^{3} x^{4}}{4}+\operatorname {erfc}\left (b x +a \right ) d^{2} c \,x^{3}+\frac {3 \,\operatorname {erfc}\left (b x +a \right ) d \,c^{2} x^{2}}{2}+\operatorname {erfc}\left (b x +a \right ) c^{3} x +\frac {\operatorname {erfc}\left (b x +a \right ) c^{4}}{4 d}+\frac {b \left (\frac {c^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2 b}+{\mathrm e}^{-a^{2}} d^{4} \left (-\frac {x^{3} {\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )}{b}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}}{b^{2}}\right )}{b}+\frac {-\frac {3 x \,{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{4 b^{2}}-\frac {3 a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{2 b}+\frac {3 \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{8 b^{3}}}{b^{2}}\right )+4 \,{\mathrm e}^{-a^{2}} c \,d^{3} \left (-\frac {x^{2} {\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )}{b}+\frac {-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}}{b^{2}}\right )+6 \,{\mathrm e}^{-a^{2}} c^{2} d^{2} \left (-\frac {x \,{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )}{b}+\frac {\sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{4 b^{3}}\right )+4 \,{\mathrm e}^{-a^{2}} c^{3} d \left (-\frac {{\mathrm e}^{-b^{2} x^{2}-2 b x a}}{2 b^{2}}-\frac {a \sqrt {\pi }\, {\mathrm e}^{a^{2}} \operatorname {erf}\left (b x +a \right )}{2 b^{2}}\right )\right )}{2 d \sqrt {\pi }}\) | \(712\) |
| derivativedivides | \(\frac {\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) a^{4}}{4 b^{3}}-\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a^{3} c}{b^{2}}-\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) a^{3} \left (b x +a \right )}{b^{3}}+\frac {3 d \,\operatorname {erfc}\left (b x +a \right ) a^{2} c^{2}}{2 b}+\frac {3 d^{2} \operatorname {erfc}\left (b x +a \right ) a^{2} c \left (b x +a \right )}{b^{2}}+\frac {3 d^{3} \operatorname {erfc}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2 b^{3}}-\operatorname {erfc}\left (b x +a \right ) a \,c^{3}-\frac {3 d \,\operatorname {erfc}\left (b x +a \right ) a \,c^{2} \left (b x +a \right )}{b}-\frac {3 d^{2} \operatorname {erfc}\left (b x +a \right ) a c \left (b x +a \right )^{2}}{b^{2}}-\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) a \left (b x +a \right )^{3}}{b^{3}}+\frac {b \,\operatorname {erfc}\left (b x +a \right ) c^{4}}{4 d}+\operatorname {erfc}\left (b x +a \right ) c^{3} \left (b x +a \right )+\frac {3 d \,\operatorname {erfc}\left (b x +a \right ) c^{2} \left (b x +a \right )^{2}}{2 b}+\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) c \left (b x +a \right )^{3}}{b^{2}}+\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) \left (b x +a \right )^{4}}{4 b^{3}}+\frac {\frac {a^{4} d^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{3}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{8}\right )-4 a \,d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 a^{2} d^{4} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+2 a^{3} d^{4} {\mathrm e}^{-\left (b x +a \right )^{2}}-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+4 b c \,d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 b^{2} c^{2} d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )-2 b^{3} c^{3} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}-6 a^{2} b c \,d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2 \sqrt {\pi }\, b^{3} d}}{b}\) | \(729\) |
| default | \(\frac {\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) a^{4}}{4 b^{3}}-\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) a^{3} c}{b^{2}}-\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) a^{3} \left (b x +a \right )}{b^{3}}+\frac {3 d \,\operatorname {erfc}\left (b x +a \right ) a^{2} c^{2}}{2 b}+\frac {3 d^{2} \operatorname {erfc}\left (b x +a \right ) a^{2} c \left (b x +a \right )}{b^{2}}+\frac {3 d^{3} \operatorname {erfc}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2 b^{3}}-\operatorname {erfc}\left (b x +a \right ) a \,c^{3}-\frac {3 d \,\operatorname {erfc}\left (b x +a \right ) a \,c^{2} \left (b x +a \right )}{b}-\frac {3 d^{2} \operatorname {erfc}\left (b x +a \right ) a c \left (b x +a \right )^{2}}{b^{2}}-\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) a \left (b x +a \right )^{3}}{b^{3}}+\frac {b \,\operatorname {erfc}\left (b x +a \right ) c^{4}}{4 d}+\operatorname {erfc}\left (b x +a \right ) c^{3} \left (b x +a \right )+\frac {3 d \,\operatorname {erfc}\left (b x +a \right ) c^{2} \left (b x +a \right )^{2}}{2 b}+\frac {d^{2} \operatorname {erfc}\left (b x +a \right ) c \left (b x +a \right )^{3}}{b^{2}}+\frac {d^{3} \operatorname {erfc}\left (b x +a \right ) \left (b x +a \right )^{4}}{4 b^{3}}+\frac {\frac {a^{4} d^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+\frac {b^{4} c^{4} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{2}+d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{3}}{2}-\frac {3 \left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{4}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{8}\right )-4 a \,d^{4} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 a^{2} d^{4} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+2 a^{3} d^{4} {\mathrm e}^{-\left (b x +a \right )^{2}}-2 a \,b^{3} c^{3} d \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+3 a^{2} b^{2} c^{2} d^{2} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )-2 a^{3} b c \,d^{3} \sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )+4 b c \,d^{3} \left (-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}} \left (b x +a \right )^{2}}{2}-\frac {{\mathrm e}^{-\left (b x +a \right )^{2}}}{2}\right )+6 b^{2} c^{2} d^{2} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )-2 b^{3} c^{3} d \,{\mathrm e}^{-\left (b x +a \right )^{2}}-12 a b c \,d^{3} \left (-\frac {\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{2}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (b x +a \right )}{4}\right )+6 a \,b^{2} c^{2} d^{2} {\mathrm e}^{-\left (b x +a \right )^{2}}-6 a^{2} b c \,d^{3} {\mathrm e}^{-\left (b x +a \right )^{2}}}{2 \sqrt {\pi }\, b^{3} d}}{b}\) | \(729\) |
Input:
int((d*x+c)^3*erfc(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/16*(16*x*exp(-(b*x+a)^2)*a*b^2*c*d^2+4*d^3*erfc(b*x+a)*x^4*Pi^(1/2)*b^4+ 16*x*erfc(b*x+a)*c^3*Pi^(1/2)*b^4+16*Pi^(1/2)*erfc(b*x+a)*a*b^3*c^3-12*Pi^ (1/2)*erfc(b*x+a)*b^2*c^2*d+4*exp(-(b*x+a)^2)*a^3*d^3-16*exp(-(b*x+a)^2)*b ^3*c^3+10*exp(-(b*x+a)^2)*a*d^3+16*Pi^(1/2)*erfc(b*x+a)*a^3*b*c*d^2-24*Pi^ (1/2)*erfc(b*x+a)*a^2*b^2*c^2*d+24*Pi^(1/2)*erfc(b*x+a)*a*b*c*d^2+16*d^2*c *erfc(b*x+a)*x^3*Pi^(1/2)*b^4+24*c^2*d*erfc(b*x+a)*x^2*Pi^(1/2)*b^4-3*Pi^( 1/2)*erfc(b*x+a)*d^3-4*d^3*exp(-(b*x+a)^2)*x^3*b^3-6*x*exp(-(b*x+a)^2)*b*d ^3-16*exp(-(b*x+a)^2)*b*c*d^2-4*x*exp(-(b*x+a)^2)*a^2*b*d^3-24*x*exp(-(b*x +a)^2)*b^3*c^2*d+4*x^2*exp(-(b*x+a)^2)*a*b^2*d^3-16*x^2*exp(-(b*x+a)^2)*b^ 3*c*d^2-16*exp(-(b*x+a)^2)*a^2*b*c*d^2+24*exp(-(b*x+a)^2)*a*b^2*c^2*d-4*Pi ^(1/2)*erfc(b*x+a)*a^4*d^3-12*Pi^(1/2)*erfc(b*x+a)*a^2*d^3)/Pi^(1/2)/b^4
Time = 0.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.08 \[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=\frac {4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x - 2 \, \sqrt {\pi } {\left (2 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 12 \, a b^{2} c^{2} d + 8 \, {\left (a^{2} + 1\right )} b c d^{2} - {\left (2 \, a^{3} + 5 \, a\right )} d^{3} + 2 \, {\left (4 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} + {\left (12 \, b^{3} c^{2} d - 8 \, a b^{2} c d^{2} + {\left (2 \, a^{2} + 3\right )} b d^{3}\right )} x\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - {\left (4 \, \pi b^{4} d^{3} x^{4} + 16 \, \pi b^{4} c d^{2} x^{3} + 24 \, \pi b^{4} c^{2} d x^{2} + 16 \, \pi b^{4} c^{3} x + \pi {\left (16 \, a b^{3} c^{3} - 12 \, {\left (2 \, a^{2} + 1\right )} b^{2} c^{2} d + 8 \, {\left (2 \, a^{3} + 3 \, a\right )} b c d^{2} - {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} d^{3}\right )}\right )} \operatorname {erf}\left (b x + a\right )}{16 \, \pi b^{4}} \] Input:
integrate((d*x+c)^3*erfc(b*x+a),x, algorithm="fricas")
Output:
1/16*(4*pi*b^4*d^3*x^4 + 16*pi*b^4*c*d^2*x^3 + 24*pi*b^4*c^2*d*x^2 + 16*pi *b^4*c^3*x - 2*sqrt(pi)*(2*b^3*d^3*x^3 + 8*b^3*c^3 - 12*a*b^2*c^2*d + 8*(a ^2 + 1)*b*c*d^2 - (2*a^3 + 5*a)*d^3 + 2*(4*b^3*c*d^2 - a*b^2*d^3)*x^2 + (1 2*b^3*c^2*d - 8*a*b^2*c*d^2 + (2*a^2 + 3)*b*d^3)*x)*e^(-b^2*x^2 - 2*a*b*x - a^2) - (4*pi*b^4*d^3*x^4 + 16*pi*b^4*c*d^2*x^3 + 24*pi*b^4*c^2*d*x^2 + 1 6*pi*b^4*c^3*x + pi*(16*a*b^3*c^3 - 12*(2*a^2 + 1)*b^2*c^2*d + 8*(2*a^3 + 3*a)*b*c*d^2 - (4*a^4 + 12*a^2 + 3)*d^3))*erf(b*x + a))/(pi*b^4)
Leaf count of result is larger than twice the leaf count of optimal. 746 vs. \(2 (258) = 516\).
Time = 1.87 (sec) , antiderivative size = 746, normalized size of antiderivative = 2.55 \[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)**3*erfc(b*x+a),x)
Output:
Piecewise((-a**4*d**3*erfc(a + b*x)/(4*b**4) + a**3*c*d**2*erfc(a + b*x)/b **3 + a**3*d**3*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(4*sqrt(pi)*b**4) - 3*a**2*c**2*d*erfc(a + b*x)/(2*b**2) - a**2*c*d**2*exp(-a**2)*exp(-b**2 *x**2)*exp(-2*a*b*x)/(sqrt(pi)*b**3) - a**2*d**3*x*exp(-a**2)*exp(-b**2*x* *2)*exp(-2*a*b*x)/(4*sqrt(pi)*b**3) - 3*a**2*d**3*erfc(a + b*x)/(4*b**4) + a*c**3*erfc(a + b*x)/b + 3*a*c**2*d*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b *x)/(2*sqrt(pi)*b**2) + a*c*d**2*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x )/(sqrt(pi)*b**2) + a*d**3*x**2*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/( 4*sqrt(pi)*b**2) + 3*a*c*d**2*erfc(a + b*x)/(2*b**3) + 5*a*d**3*exp(-a**2) *exp(-b**2*x**2)*exp(-2*a*b*x)/(8*sqrt(pi)*b**4) + c**3*x*erfc(a + b*x) + 3*c**2*d*x**2*erfc(a + b*x)/2 + c*d**2*x**3*erfc(a + b*x) + d**3*x**4*erfc (a + b*x)/4 - c**3*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi)*b) - 3*c**2*d*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(2*sqrt(pi)*b) - c*d* *2*x**2*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi)*b) - d**3*x**3* exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(4*sqrt(pi)*b) - 3*c**2*d*erfc(a + b*x)/(4*b**2) - c*d**2*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(sqrt(pi )*b**3) - 3*d**3*x*exp(-a**2)*exp(-b**2*x**2)*exp(-2*a*b*x)/(8*sqrt(pi)*b* *3) - 3*d**3*erfc(a + b*x)/(16*b**4), Ne(b, 0)), ((c**3*x + 3*c**2*d*x**2/ 2 + c*d**2*x**3 + d**3*x**4/4)*erfc(a), True))
\[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \operatorname {erfc}\left (b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^3*erfc(b*x+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^3*erfc(b*x + a), x)
Time = 0.27 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.49 \[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=\frac {1}{4} \, d^{3} x^{4} + c d^{2} x^{3} + \frac {3}{2} \, c^{2} d x^{2} - {\left (x \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } a \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi }}\right )} c^{3} - \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (2 \, a^{2} + 1\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (b {\left (x + \frac {a}{b}\right )} - 2 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b}\right )} c^{2} d - \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {erf}\left (b x + a\right ) - \frac {\frac {\sqrt {\pi } {\left (2 \, a^{3} + 3 \, a\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} - \frac {2 \, {\left (b^{2} {\left (x + \frac {a}{b}\right )}^{2} - 3 \, a b {\left (x + \frac {a}{b}\right )} + 3 \, a^{2} + 1\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b^{2}}\right )} c d^{2} - \frac {1}{16} \, {\left (4 \, x^{4} \operatorname {erf}\left (b x + a\right ) + \frac {\frac {\sqrt {\pi } {\left (4 \, a^{4} + 12 \, a^{2} + 3\right )} \operatorname {erf}\left (-b {\left (x + \frac {a}{b}\right )}\right )}{b} + \frac {2 \, {\left (2 \, b^{3} {\left (x + \frac {a}{b}\right )}^{3} - 8 \, a b^{2} {\left (x + \frac {a}{b}\right )}^{2} + 12 \, a^{2} b {\left (x + \frac {a}{b}\right )} - 8 \, a^{3} + 3 \, b {\left (x + \frac {a}{b}\right )} - 8 \, a\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{b}}{\sqrt {\pi } b^{3}}\right )} d^{3} + c^{3} x \] Input:
integrate((d*x+c)^3*erfc(b*x+a),x, algorithm="giac")
Output:
1/4*d^3*x^4 + c*d^2*x^3 + 3/2*c^2*d*x^2 - (x*erf(b*x + a) - (sqrt(pi)*a*er f(-b*(x + a/b))/b - e^(-b^2*x^2 - 2*a*b*x - a^2)/b)/sqrt(pi))*c^3 - 3/4*(2 *x^2*erf(b*x + a) + (sqrt(pi)*(2*a^2 + 1)*erf(-b*(x + a/b))/b + 2*(b*(x + a/b) - 2*a)*e^(-b^2*x^2 - 2*a*b*x - a^2)/b)/(sqrt(pi)*b))*c^2*d - 1/2*(2*x ^3*erf(b*x + a) - (sqrt(pi)*(2*a^3 + 3*a)*erf(-b*(x + a/b))/b - 2*(b^2*(x + a/b)^2 - 3*a*b*(x + a/b) + 3*a^2 + 1)*e^(-b^2*x^2 - 2*a*b*x - a^2)/b)/(s qrt(pi)*b^2))*c*d^2 - 1/16*(4*x^4*erf(b*x + a) + (sqrt(pi)*(4*a^4 + 12*a^2 + 3)*erf(-b*(x + a/b))/b + 2*(2*b^3*(x + a/b)^3 - 8*a*b^2*(x + a/b)^2 + 1 2*a^2*b*(x + a/b) - 8*a^3 + 3*b*(x + a/b) - 8*a)*e^(-b^2*x^2 - 2*a*b*x - a ^2)/b)/(sqrt(pi)*b^3))*d^3 + c^3*x
Time = 3.96 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.21 \[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=\frac {d^3\,x^4\,\mathrm {erfc}\left (a+b\,x\right )}{4}-\frac {\mathrm {erfc}\left (a+b\,x\right )\,\left (b^2\,\left (\frac {3\,d\,a^2\,c^2}{2}+\frac {3\,d\,c^2}{4}\right )-b\,\left (c\,a^3\,d^2+\frac {3\,c\,a\,d^2}{2}\right )+\frac {3\,d^3}{16}+\frac {3\,a^2\,d^3}{4}+\frac {a^4\,d^3}{4}-a\,b^3\,c^3\right )}{b^4}+c^3\,x\,\mathrm {erfc}\left (a+b\,x\right )+\frac {{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^3\,d^3-8\,a^2\,b\,c\,d^2+12\,a\,b^2\,c^2\,d+5\,a\,d^3-8\,b^3\,c^3-8\,b\,c\,d^2\right )}{8\,b^4\,\sqrt {\pi }}+\frac {3\,c^2\,d\,x^2\,\mathrm {erfc}\left (a+b\,x\right )}{2}+c\,d^2\,x^3\,\mathrm {erfc}\left (a+b\,x\right )-\frac {x\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (2\,a^2\,d^3-8\,a\,b\,c\,d^2+12\,b^2\,c^2\,d+3\,d^3\right )}{8\,b^3\,\sqrt {\pi }}-\frac {d^3\,x^3\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}}{4\,b\,\sqrt {\pi }}+\frac {x^2\,{\mathrm {e}}^{-a^2-2\,a\,b\,x-b^2\,x^2}\,\left (a\,d^3-4\,b\,c\,d^2\right )}{4\,b^2\,\sqrt {\pi }} \] Input:
int(erfc(a + b*x)*(c + d*x)^3,x)
Output:
(d^3*x^4*erfc(a + b*x))/4 - (erfc(a + b*x)*(b^2*((3*c^2*d)/4 + (3*a^2*c^2* d)/2) - b*(a^3*c*d^2 + (3*a*c*d^2)/2) + (3*d^3)/16 + (3*a^2*d^3)/4 + (a^4* d^3)/4 - a*b^3*c^3))/b^4 + c^3*x*erfc(a + b*x) + (exp(- a^2 - b^2*x^2 - 2* a*b*x)*(5*a*d^3 + 2*a^3*d^3 - 8*b^3*c^3 - 8*b*c*d^2 + 12*a*b^2*c^2*d - 8*a ^2*b*c*d^2))/(8*b^4*pi^(1/2)) + (3*c^2*d*x^2*erfc(a + b*x))/2 + c*d^2*x^3* erfc(a + b*x) - (x*exp(- a^2 - b^2*x^2 - 2*a*b*x)*(3*d^3 + 2*a^2*d^3 + 12* b^2*c^2*d - 8*a*b*c*d^2))/(8*b^3*pi^(1/2)) - (d^3*x^3*exp(- a^2 - b^2*x^2 - 2*a*b*x))/(4*b*pi^(1/2)) + (x^2*exp(- a^2 - b^2*x^2 - 2*a*b*x)*(a*d^3 - 4*b*c*d^2))/(4*b^2*pi^(1/2))
\[ \int (c+d x)^3 \text {erfc}(a+b x) \, dx=\frac {-4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \mathrm {erf}\left (b x +a \right ) a \,c^{3}-4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \mathrm {erf}\left (b x +a \right ) b \,c^{3} x -4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right ) x^{3}d x \right ) b \,d^{3}-12 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right ) x^{2}d x \right ) b c \,d^{2}-12 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right ) x d x \right ) b \,c^{2} d +4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b \,c^{3} x +6 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b \,c^{2} d \,x^{2}+4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b c \,d^{2} x^{3}+\sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b \,d^{3} x^{4}-4 c^{3}}{4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b} \] Input:
int((d*x+c)^3*erfc(b*x+a),x)
Output:
( - 4*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*erf(a + b*x)*a*c**3 - 4*sqr t(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*erf(a + b*x)*b*c**3*x - 4*sqrt(pi)*e **(a**2 + 2*a*b*x + b**2*x**2)*int(erf(a + b*x)*x**3,x)*b*d**3 - 12*sqrt(p i)*e**(a**2 + 2*a*b*x + b**2*x**2)*int(erf(a + b*x)*x**2,x)*b*c*d**2 - 12* sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*int(erf(a + b*x)*x,x)*b*c**2*d + 4*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b*c**3*x + 6*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b*c**2*d*x**2 + 4*sqrt(pi)*e**(a**2 + 2*a*b*x + b** 2*x**2)*b*c*d**2*x**3 + sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b*d**3*x* *4 - 4*c**3)/(4*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b)