\(\int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx\) [193]

Optimal result

Integrand size = 19, antiderivative size = 19 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d^2}-\frac {a^3 b^4 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{7/2} d}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{5/2} d}-\frac {3 e^{c+d x^2} x \text {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {erfc}(a+b x),x\right )}{4 d^2} \] Output:

3/4*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)/d^2/Pi^(1/2)-1/2*a^2*b^3*exp 
(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)^3/d/Pi^(1/2)-1/2*b*exp(-a^2+c-2*a*b*x 
-(b^2-d)*x^2)/(b^2-d)^2/d/Pi^(1/2)+1/2*a*b^2*exp(-a^2+c-2*a*b*x-(b^2-d)*x^ 
2)*x/(b^2-d)^2/d/Pi^(1/2)-1/2*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)*x^2/(b^2-d 
)/d/Pi^(1/2)+3/4*a*b^2*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1 
/2))/(b^2-d)^(3/2)/d^2-1/2*a^3*b^4*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x 
)/(b^2-d)^(1/2))/(b^2-d)^(7/2)/d-3/4*a*b^2*exp(c+a^2*d/(b^2-d))*erf((a*b+( 
b^2-d)*x)/(b^2-d)^(1/2))/(b^2-d)^(5/2)/d-3/4*exp(d*x^2+c)*x*erfc(b*x+a)/d^ 
2+1/2*exp(d*x^2+c)*x^3*erfc(b*x+a)/d+3/4*Defer(Int)(exp(d*x^2+c)*erfc(b*x+ 
a),x)/d^2
 

Mathematica [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx \] Input:

Integrate[E^(c + d*x^2)*x^4*Erfc[a + b*x],x]
 

Output:

Integrate[E^(c + d*x^2)*x^4*Erfc[a + b*x], x]
 

Rubi [N/A]

Not integrable

Time = 3.82 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}

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.

\(\displaystyle \int x^4 e^{c+d x^2} \text {erfc}(a+b x) \, dx\)

\(\Big \downarrow \) 6940

\(\displaystyle \frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^3dx}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2671

\(\displaystyle \frac {b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle \frac {b \left (\frac {-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {b \left (\frac {-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2671

\(\displaystyle \frac {b \left (-\frac {a b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{2 \left (b^2-d\right )}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{2 \left (b^2-d\right )}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 6940

\(\displaystyle -\frac {3 \left (\frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2670

\(\displaystyle -\frac {3 \left (\frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2664

\(\displaystyle -\frac {3 \left (\frac {b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {3 \left (-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

\(\Big \downarrow \) 6934

\(\displaystyle -\frac {3 \left (-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\)

Input:

Int[E^(c + d*x^2)*x^4*Erfc[a + b*x],x]
 

Output:

$Aborted
 

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95

\[\int {\mathrm e}^{d \,x^{2}+c} x^{4} \operatorname {erfc}\left (b x +a \right )d x\]

Input:

int(exp(d*x^2+c)*x^4*erfc(b*x+a),x)
 

Output:

int(exp(d*x^2+c)*x^4*erfc(b*x+a),x)
 

Fricas [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

integrate(exp(d*x^2+c)*x^4*erfc(b*x+a),x, algorithm="fricas")
 

Output:

integral(-(x^4*erf(b*x + a) - x^4)*e^(d*x^2 + c), x)
 

Sympy [F(-1)]

Timed out. \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\text {Timed out} \] Input:

integrate(exp(d*x**2+c)*x**4*erfc(b*x+a),x)
 

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

integrate(exp(d*x^2+c)*x^4*erfc(b*x+a),x, algorithm="maxima")
 

Output:

integrate(x^4*erfc(b*x + a)*e^(d*x^2 + c), x)
 

Giac [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \] Input:

integrate(exp(d*x^2+c)*x^4*erfc(b*x+a),x, algorithm="giac")
 

Output:

integrate(x^4*erfc(b*x + a)*e^(d*x^2 + c), x)
 

Mupad [N/A]

Not integrable

Time = 4.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int x^4\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \] Input:

int(x^4*erfc(a + b*x)*exp(c + d*x^2),x)
 

Output:

int(x^4*erfc(a + b*x)*exp(c + d*x^2), x)
 

Reduce [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 4.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\frac {e^{c} \left (-3 \sqrt {\pi }\, \mathrm {erf}\left (\sqrt {d}\, i x \right ) i +4 e^{d \,x^{2}} \sqrt {d}\, d \,x^{3}-6 e^{d \,x^{2}} \sqrt {d}\, x -8 \sqrt {d}\, \left (\int e^{d \,x^{2}} \mathrm {erf}\left (b x +a \right ) x^{4}d x \right ) d^{2}\right )}{8 \sqrt {d}\, d^{2}} \] Input:

int(exp(d*x^2+c)*x^4*erfc(b*x+a),x)
 

Output:

(e**c*( - 3*sqrt(pi)*erf(sqrt(d)*i*x)*i + 4*e**(d*x**2)*sqrt(d)*d*x**3 - 6 
*e**(d*x**2)*sqrt(d)*x - 8*sqrt(d)*int(e**(d*x**2)*erf(a + b*x)*x**4,x)*d* 
*2))/(8*sqrt(d)*d**2)