\(\int e^{-a-b x} (c+d x)^3 \, dx\) [111]

Optimal result

Integrand size = 18, antiderivative size = 87 \[ \int e^{-a-b x} (c+d x)^3 \, dx=-\frac {6 d^3 e^{-a-b x}}{b^4}-\frac {6 d^2 e^{-a-b x} (c+d x)}{b^3}-\frac {3 d e^{-a-b x} (c+d x)^2}{b^2}-\frac {e^{-a-b x} (c+d x)^3}{b} \] Output:

-6*d^3*exp(-b*x-a)/b^4-6*d^2*exp(-b*x-a)*(d*x+c)/b^3-3*d*exp(-b*x-a)*(d*x+ 
c)^2/b^2-exp(-b*x-a)*(d*x+c)^3/b
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.64 \[ \int e^{-a-b x} (c+d x)^3 \, dx=\frac {e^{-a-b x} \left (-6 d^3-6 b d^2 (c+d x)-3 b^2 d (c+d x)^2-b^3 (c+d x)^3\right )}{b^4} \] Input:

Integrate[E^(-a - b*x)*(c + d*x)^3,x]
 

Output:

(E^(-a - b*x)*(-6*d^3 - 6*b*d^2*(c + d*x) - 3*b^2*d*(c + d*x)^2 - b^3*(c + 
 d*x)^3))/b^4
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2607, 2607, 2607, 2624}

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.

\(\displaystyle \int e^{-a-b x} (c+d x)^3 \, dx\)

\(\Big \downarrow \) 2607

\(\displaystyle \frac {3 d \int e^{-a-b x} (c+d x)^2dx}{b}-\frac {e^{-a-b x} (c+d x)^3}{b}\)

\(\Big \downarrow \) 2607

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

\(\Big \downarrow \) 2607

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

\(\Big \downarrow \) 2624

\(\displaystyle \frac {3 d \left (\frac {2 d \left (-\frac {d e^{-a-b x}}{b^2}-\frac {e^{-a-b x} (c+d x)}{b}\right )}{b}-\frac {e^{-a-b x} (c+d x)^2}{b}\right )}{b}-\frac {e^{-a-b x} (c+d x)^3}{b}\)

Input:

Int[E^(-a - b*x)*(c + d*x)^3,x]
 

Output:

-((E^(-a - b*x)*(c + d*x)^3)/b) + (3*d*(-((E^(-a - b*x)*(c + d*x)^2)/b) + 
(2*d*(-((d*E^(-a - b*x))/b^2) - (E^(-a - b*x)*(c + d*x))/b))/b))/b
 

Defintions of rubi rules used

rule 2607

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), 
x] - Simp[d*(m/(f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ 
n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* 
m] &&  !TrueQ[$UseGamma]
 

rule 2624

Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; 
FreeQ[{F, n}, x] && LinearQ[v, x]
 

Maple [A] (verified)

Time = 0.47 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.20

method	result	size
gosper	\(-\frac {\left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 d^{2} c b +6 d^{3}\right ) {\mathrm e}^{-b x -a}}{b^{4}}\)	\(104\)
risch	\(-\frac {\left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 d^{2} c b +6 d^{3}\right ) {\mathrm e}^{-b x -a}}{b^{4}}\)	\(104\)
orering	\(-\frac {\left (b^{3} d^{3} x^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +3 b^{2} d^{3} x^{2}+b^{3} c^{3}+6 b^{2} c \,d^{2} x +3 b^{2} c^{2} d +6 b \,d^{3} x +6 d^{2} c b +6 d^{3}\right ) {\mathrm e}^{-b x -a}}{b^{4}}\)	\(104\)
norman	\(-\frac {\left (b^{3} c^{3}+3 b^{2} c^{2} d +6 d^{2} c b +6 d^{3}\right ) {\mathrm e}^{-b x -a}}{b^{4}}-\frac {d^{3} x^{3} {\mathrm e}^{-b x -a}}{b}-\frac {3 d \left (b^{2} c^{2}+2 b c d +2 d^{2}\right ) x \,{\mathrm e}^{-b x -a}}{b^{3}}-\frac {3 d^{2} \left (c b +d \right ) x^{2} {\mathrm e}^{-b x -a}}{b^{2}}\)	\(124\)
meijerg	\(\frac {{\mathrm e}^{-a} d^{3} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{4}}+\frac {3 \,{\mathrm e}^{-a} d^{2} c \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {3 \,{\mathrm e}^{-a} d \,c^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} c^{3} \left (1-{\mathrm e}^{-b x}\right )}{b}\)	\(128\)
parallelrisch	\(-\frac {x^{3} {\mathrm e}^{-b x -a} d^{3} b^{3}+3 x^{2} {\mathrm e}^{-b x -a} b^{3} c \,d^{2}+3 x^{2} {\mathrm e}^{-b x -a} b^{2} d^{3}+3 x \,{\mathrm e}^{-b x -a} b^{3} c^{2} d +6 x \,{\mathrm e}^{-b x -a} b^{2} c \,d^{2}+{\mathrm e}^{-b x -a} b^{3} c^{3}+6 x \,{\mathrm e}^{-b x -a} b \,d^{3}+3 \,{\mathrm e}^{-b x -a} b^{2} c^{2} d +6 \,{\mathrm e}^{-b x -a} b c \,d^{2}+6 \,{\mathrm e}^{-b x -a} d^{3}}{b^{4}}\)	\(185\)
parts	\(-\frac {d^{3} x^{3} {\mathrm e}^{-b x -a}}{b}-\frac {3 \,{\mathrm e}^{-b x -a} d^{2} c \,x^{2}}{b}-\frac {3 \,{\mathrm e}^{-b x -a} d \,c^{2} x}{b}-\frac {{\mathrm e}^{-b x -a} c^{3}}{b}-\frac {3 d \left ({\mathrm e}^{-b x -a} c^{2}+\frac {{\mathrm e}^{-b x -a} d^{2} a^{2}}{b^{2}}+\frac {d^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 \,{\mathrm e}^{-b x -a} d a c}{b}-\frac {2 d c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}\right )}{b^{2}}\)	\(269\)
derivativedivides	\(-\frac {{\mathrm e}^{-b x -a} c^{3}-\frac {{\mathrm e}^{-b x -a} d^{3} a^{3}}{b^{3}}-\frac {d^{3} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}-\frac {3 \,{\mathrm e}^{-b x -a} d a \,c^{2}}{b}-\frac {3 d \,c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {3 \,{\mathrm e}^{-b x -a} d^{2} a^{2} c}{b^{2}}+\frac {3 d^{2} c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {3 d^{3} a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}+\frac {6 d^{2} a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}}{b}\)	\(400\)
default	\(-\frac {{\mathrm e}^{-b x -a} c^{3}-\frac {{\mathrm e}^{-b x -a} d^{3} a^{3}}{b^{3}}-\frac {d^{3} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}-\frac {3 \,{\mathrm e}^{-b x -a} d a \,c^{2}}{b}-\frac {3 d \,c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {3 \,{\mathrm e}^{-b x -a} d^{2} a^{2} c}{b^{2}}+\frac {3 d^{2} c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {3 d^{3} a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{3}}-\frac {3 d^{3} a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}+\frac {6 d^{2} a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}}{b}\)	\(400\)

Input:

int(exp(-b*x-a)*(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

-(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+3*b^2*d^3*x^2+b^3*c^3+6*b^2*c* 
d^2*x+3*b^2*c^2*d+6*b*d^3*x+6*b*c*d^2+6*d^3)*exp(-b*x-a)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int e^{-a-b x} (c+d x)^3 \, dx=-\frac {{\left (b^{3} d^{3} x^{3} + b^{3} c^{3} + 3 \, b^{2} c^{2} d + 6 \, b c d^{2} + 6 \, d^{3} + 3 \, {\left (b^{3} c d^{2} + b^{2} d^{3}\right )} x^{2} + 3 \, {\left (b^{3} c^{2} d + 2 \, b^{2} c d^{2} + 2 \, b d^{3}\right )} x\right )} e^{\left (-b x - a\right )}}{b^{4}} \] Input:

integrate(exp(-b*x-a)*(d*x+c)^3,x, algorithm="fricas")
 

Output:

-(b^3*d^3*x^3 + b^3*c^3 + 3*b^2*c^2*d + 6*b*c*d^2 + 6*d^3 + 3*(b^3*c*d^2 + 
 b^2*d^3)*x^2 + 3*(b^3*c^2*d + 2*b^2*c*d^2 + 2*b*d^3)*x)*e^(-b*x - a)/b^4
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.68 \[ \int e^{-a-b x} (c+d x)^3 \, dx=\begin {cases} \frac {\left (- b^{3} c^{3} - 3 b^{3} c^{2} d x - 3 b^{3} c d^{2} x^{2} - b^{3} d^{3} x^{3} - 3 b^{2} c^{2} d - 6 b^{2} c d^{2} x - 3 b^{2} d^{3} x^{2} - 6 b c d^{2} - 6 b d^{3} x - 6 d^{3}\right ) e^{- a - b x}}{b^{4}} & \text {for}\: b^{4} \neq 0 \\c^{3} x + \frac {3 c^{2} d x^{2}}{2} + c d^{2} x^{3} + \frac {d^{3} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:

integrate(exp(-b*x-a)*(d*x+c)**3,x)
 

Output:

Piecewise(((-b**3*c**3 - 3*b**3*c**2*d*x - 3*b**3*c*d**2*x**2 - b**3*d**3* 
x**3 - 3*b**2*c**2*d - 6*b**2*c*d**2*x - 3*b**2*d**3*x**2 - 6*b*c*d**2 - 6 
*b*d**3*x - 6*d**3)*exp(-a - b*x)/b**4, Ne(b**4, 0)), (c**3*x + 3*c**2*d*x 
**2/2 + c*d**2*x**3 + d**3*x**4/4, True))
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.26 \[ \int e^{-a-b x} (c+d x)^3 \, dx=-\frac {c^{3} e^{\left (-b x - a\right )}}{b} - \frac {3 \, {\left (b x + 1\right )} c^{2} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {3 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} c d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} d^{3} e^{\left (-b x - a\right )}}{b^{4}} \] Input:

integrate(exp(-b*x-a)*(d*x+c)^3,x, algorithm="maxima")
 

Output:

-c^3*e^(-b*x - a)/b - 3*(b*x + 1)*c^2*d*e^(-b*x - a)/b^2 - 3*(b^2*x^2 + 2* 
b*x + 2)*c*d^2*e^(-b*x - a)/b^3 - (b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*d^3*e^ 
(-b*x - a)/b^4
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.18 \[ \int e^{-a-b x} (c+d x)^3 \, dx=-\frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + 3 \, b^{2} d^{3} x^{2} + b^{3} c^{3} + 6 \, b^{2} c d^{2} x + 3 \, b^{2} c^{2} d + 6 \, b d^{3} x + 6 \, b c d^{2} + 6 \, d^{3}\right )} e^{\left (-b x - a\right )}}{b^{4}} \] Input:

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

Output:

-(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + 3*b^2*d^3*x^2 + b^3*c^3 
+ 6*b^2*c*d^2*x + 3*b^2*c^2*d + 6*b*d^3*x + 6*b*c*d^2 + 6*d^3)*e^(-b*x - a 
)/b^4
 

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.18 \[ \int e^{-a-b x} (c+d x)^3 \, dx=-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (b^3\,c^3+3\,b^3\,c^2\,d\,x+3\,b^3\,c\,d^2\,x^2+b^3\,d^3\,x^3+3\,b^2\,c^2\,d+6\,b^2\,c\,d^2\,x+3\,b^2\,d^3\,x^2+6\,b\,c\,d^2+6\,b\,d^3\,x+6\,d^3\right )}{b^4} \] Input:

int(exp(- a - b*x)*(c + d*x)^3,x)
 

Output:

-(exp(- a - b*x)*(6*d^3 + b^3*c^3 + 3*b^2*c^2*d + 3*b^2*d^3*x^2 + b^3*d^3* 
x^3 + 6*b*c*d^2 + 6*b*d^3*x + 3*b^3*c*d^2*x^2 + 6*b^2*c*d^2*x + 3*b^3*c^2* 
d*x))/b^4
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.20 \[ \int e^{-a-b x} (c+d x)^3 \, dx=\frac {-b^{3} d^{3} x^{3}-3 b^{3} c \,d^{2} x^{2}-3 b^{3} c^{2} d x -3 b^{2} d^{3} x^{2}-b^{3} c^{3}-6 b^{2} c \,d^{2} x -3 b^{2} c^{2} d -6 b \,d^{3} x -6 b c \,d^{2}-6 d^{3}}{e^{b x +a} b^{4}} \] Input:

int(exp(-b*x-a)*(d*x+c)^3,x)
 

Output:

( - b**3*c**3 - 3*b**3*c**2*d*x - 3*b**3*c*d**2*x**2 - b**3*d**3*x**3 - 3* 
b**2*c**2*d - 6*b**2*c*d**2*x - 3*b**2*d**3*x**2 - 6*b*c*d**2 - 6*b*d**3*x 
 - 6*d**3)/(e**(a + b*x)*b**4)