\(\int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx\) [121]

Optimal result

Integrand size = 25, antiderivative size = 557 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=-\frac {(b c-a d)^4 e^{-a-b x}}{4 d^5 (c+d x)^4}+\frac {4 b (b c-a d)^3 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {b (b c-a d)^4 e^{-a-b x}}{12 d^6 (c+d x)^3}-\frac {3 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)^2}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{3 d^6 (c+d x)^2}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{24 d^7 (c+d x)^2}+\frac {4 b^3 (b c-a d) e^{-a-b x}}{d^5 (c+d x)}+\frac {3 b^3 (b c-a d)^2 e^{-a-b x}}{d^6 (c+d x)}+\frac {2 b^3 (b c-a d)^3 e^{-a-b x}}{3 d^7 (c+d x)}+\frac {b^3 (b c-a d)^4 e^{-a-b x}}{24 d^8 (c+d x)}+\frac {b^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {4 b^4 (b c-a d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {3 b^4 (b c-a d)^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^7}+\frac {2 b^4 (b c-a d)^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{3 d^8}+\frac {b^4 (b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{24 d^9} \] Output:

-1/4*(-a*d+b*c)^4*exp(-b*x-a)/d^5/(d*x+c)^4+4/3*b*(-a*d+b*c)^3*exp(-b*x-a) 
/d^5/(d*x+c)^3+1/12*b*(-a*d+b*c)^4*exp(-b*x-a)/d^6/(d*x+c)^3-3*b^2*(-a*d+b 
*c)^2*exp(-b*x-a)/d^5/(d*x+c)^2-2/3*b^2*(-a*d+b*c)^3*exp(-b*x-a)/d^6/(d*x+ 
c)^2-1/24*b^2*(-a*d+b*c)^4*exp(-b*x-a)/d^7/(d*x+c)^2+4*b^3*(-a*d+b*c)*exp( 
-b*x-a)/d^5/(d*x+c)+3*b^3*(-a*d+b*c)^2*exp(-b*x-a)/d^6/(d*x+c)+2/3*b^3*(-a 
*d+b*c)^3*exp(-b*x-a)/d^7/(d*x+c)+1/24*b^3*(-a*d+b*c)^4*exp(-b*x-a)/d^8/(d 
*x+c)+b^4*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^5+4*b^4*(-a*d+b*c)*exp(-a+b*c/d 
)*Ei(-b*(d*x+c)/d)/d^6+3*b^4*(-a*d+b*c)^2*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d 
^7+2/3*b^4*(-a*d+b*c)^3*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^8+1/24*b^4*(-a*d+ 
b*c)^4*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^9
 

Mathematica [A] (verified)

Time = 3.00 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\frac {e^{-a} \left (\frac {d e^{-b x} \left (-6 d^3 (b c-a d)^4+2 b d^2 (b c-(-16+a) d) (b c-a d)^3 (c+d x)-b^2 d (b c-a d)^2 \left (b^2 c^2-2 (-8+a) b c d+\left (72-16 a+a^2\right ) d^2\right ) (c+d x)^2+b^3 \left (b^4 c^4-4 (-4+a) b^3 c^3 d+6 \left (12-8 a+a^2\right ) b^2 c^2 d^2-4 \left (-24+36 a-12 a^2+a^3\right ) b c d^3+a \left (-96+72 a-16 a^2+a^3\right ) d^4\right ) (c+d x)^3\right )}{(c+d x)^4}+b^8 c^4 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+16 b^7 c^3 d e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-4 a b^7 c^3 d e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+72 b^6 c^2 d^2 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-48 a b^6 c^2 d^2 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+6 a^2 b^6 c^2 d^2 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+96 b^5 c d^3 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-144 a b^5 c d^3 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+48 a^2 b^5 c d^3 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-4 a^3 b^5 c d^3 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+24 b^4 d^4 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-96 a b^4 d^4 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+72 a^2 b^4 d^4 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )-16 a^3 b^4 d^4 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )+a^4 b^4 d^4 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{24 d^9} \] Input:

Integrate[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^5,x]
 

Output:

((d*(-6*d^3*(b*c - a*d)^4 + 2*b*d^2*(b*c - (-16 + a)*d)*(b*c - a*d)^3*(c + 
 d*x) - b^2*d*(b*c - a*d)^2*(b^2*c^2 - 2*(-8 + a)*b*c*d + (72 - 16*a + a^2 
)*d^2)*(c + d*x)^2 + b^3*(b^4*c^4 - 4*(-4 + a)*b^3*c^3*d + 6*(12 - 8*a + a 
^2)*b^2*c^2*d^2 - 4*(-24 + 36*a - 12*a^2 + a^3)*b*c*d^3 + a*(-96 + 72*a - 
16*a^2 + a^3)*d^4)*(c + d*x)^3))/(E^(b*x)*(c + d*x)^4) + b^8*c^4*E^((b*c)/ 
d)*ExpIntegralEi[-((b*(c + d*x))/d)] + 16*b^7*c^3*d*E^((b*c)/d)*ExpIntegra 
lEi[-((b*(c + d*x))/d)] - 4*a*b^7*c^3*d*E^((b*c)/d)*ExpIntegralEi[-((b*(c 
+ d*x))/d)] + 72*b^6*c^2*d^2*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] 
 - 48*a*b^6*c^2*d^2*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] + 6*a^2* 
b^6*c^2*d^2*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] + 96*b^5*c*d^3*E 
^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] - 144*a*b^5*c*d^3*E^((b*c)/d) 
*ExpIntegralEi[-((b*(c + d*x))/d)] + 48*a^2*b^5*c*d^3*E^((b*c)/d)*ExpInteg 
ralEi[-((b*(c + d*x))/d)] - 4*a^3*b^5*c*d^3*E^((b*c)/d)*ExpIntegralEi[-((b 
*(c + d*x))/d)] + 24*b^4*d^4*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] 
 - 96*a*b^4*d^4*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] + 72*a^2*b^4 
*d^4*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] - 16*a^3*b^4*d^4*E^((b* 
c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)] + a^4*b^4*d^4*E^((b*c)/d)*ExpInteg 
ralEi[-((b*(c + d*x))/d)])/(24*d^9*E^a)
 

Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2629, 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.

\(\displaystyle \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx\)

\(\Big \downarrow \) 2629

\(\displaystyle \int \left (\frac {b^4 e^{-a-b x}}{d^4 (c+d x)}-\frac {4 b^3 e^{-a-b x} (b c-a d)}{d^4 (c+d x)^2}+\frac {6 b^2 e^{-a-b x} (b c-a d)^2}{d^4 (c+d x)^3}-\frac {4 b e^{-a-b x} (b c-a d)^3}{d^4 (c+d x)^4}+\frac {e^{-a-b x} (a d-b c)^4}{d^4 (c+d x)^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b^4 (b c-a d)^4 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{24 d^9}+\frac {2 b^4 (b c-a d)^3 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{3 d^8}+\frac {3 b^4 (b c-a d)^2 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^7}+\frac {4 b^4 (b c-a d) e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {b^4 e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {b^3 e^{-a-b x} (b c-a d)^4}{24 d^8 (c+d x)}+\frac {2 b^3 e^{-a-b x} (b c-a d)^3}{3 d^7 (c+d x)}+\frac {3 b^3 e^{-a-b x} (b c-a d)^2}{d^6 (c+d x)}+\frac {4 b^3 e^{-a-b x} (b c-a d)}{d^5 (c+d x)}-\frac {b^2 e^{-a-b x} (b c-a d)^4}{24 d^7 (c+d x)^2}-\frac {2 b^2 e^{-a-b x} (b c-a d)^3}{3 d^6 (c+d x)^2}-\frac {3 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)^2}+\frac {b e^{-a-b x} (b c-a d)^4}{12 d^6 (c+d x)^3}+\frac {4 b e^{-a-b x} (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac {e^{-a-b x} (b c-a d)^4}{4 d^5 (c+d x)^4}\)

Input:

Int[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^5,x]
 

Output:

-1/4*((b*c - a*d)^4*E^(-a - b*x))/(d^5*(c + d*x)^4) + (4*b*(b*c - a*d)^3*E 
^(-a - b*x))/(3*d^5*(c + d*x)^3) + (b*(b*c - a*d)^4*E^(-a - b*x))/(12*d^6* 
(c + d*x)^3) - (3*b^2*(b*c - a*d)^2*E^(-a - b*x))/(d^5*(c + d*x)^2) - (2*b 
^2*(b*c - a*d)^3*E^(-a - b*x))/(3*d^6*(c + d*x)^2) - (b^2*(b*c - a*d)^4*E^ 
(-a - b*x))/(24*d^7*(c + d*x)^2) + (4*b^3*(b*c - a*d)*E^(-a - b*x))/(d^5*( 
c + d*x)) + (3*b^3*(b*c - a*d)^2*E^(-a - b*x))/(d^6*(c + d*x)) + (2*b^3*(b 
*c - a*d)^3*E^(-a - b*x))/(3*d^7*(c + d*x)) + (b^3*(b*c - a*d)^4*E^(-a - b 
*x))/(24*d^8*(c + d*x)) + (b^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d* 
x))/d)])/d^5 + (4*b^4*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + 
 d*x))/d)])/d^6 + (3*b^4*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b 
*(c + d*x))/d)])/d^7 + (2*b^4*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi 
[-((b*(c + d*x))/d)])/(3*d^8) + (b^4*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpInt 
egralEi[-((b*(c + d*x))/d)])/(24*d^9)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2629

Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte 
grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ 
Px, x] && LinearQ[v, x] &&  !TrueQ[$UseGamma]
 

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(-\frac {\frac {4 \left (d a -b c \right ) b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {d a -b c}{d}}-{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )\right )}{d^{6}}+\frac {4 \left (d a -b c \right )^{3} b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {d a -b c}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {d a -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {d a -b c}{d}\right )}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{6}\right )}{d^{8}}-\frac {\left (d a -b c \right )^{4} b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {d a -b c}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {d a -b c}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {d a -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {d a -b c}{d}\right )}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{24}\right )}{d^{9}}+\frac {b^{5} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{d^{5}}-\frac {6 \left (d a -b c \right )^{2} b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {d a -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {d a -b c}{d}\right )}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{2}\right )}{d^{7}}}{b}\)	\(596\)
default	\(-\frac {\frac {4 \left (d a -b c \right ) b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {d a -b c}{d}}-{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )\right )}{d^{6}}+\frac {4 \left (d a -b c \right )^{3} b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {d a -b c}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {d a -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {d a -b c}{d}\right )}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{6}\right )}{d^{8}}-\frac {\left (d a -b c \right )^{4} b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {d a -b c}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {d a -b c}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {d a -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {d a -b c}{d}\right )}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{24}\right )}{d^{9}}+\frac {b^{5} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{d^{5}}-\frac {6 \left (d a -b c \right )^{2} b^{5} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {d a -b c}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {d a -b c}{d}\right )}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{2}\right )}{d^{7}}}{b}\)	\(596\)
risch	\(\text {Expression too large to display}\)	\(2054\)

Input:

int(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^5,x,method=_RETURNVERBOSE)
 

Output:

-1/b*(4*(a*d-b*c)/d^6*b^5*(-exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-exp(-(a*d-b*c 
)/d)*Ei(1,b*x+a-(a*d-b*c)/d))+4*(a*d-b*c)^3/d^8*b^5*(-1/3*exp(-b*x-a)/(-b* 
x-a+(a*d-b*c)/d)^3-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/ 
(-b*x-a+(a*d-b*c)/d)-1/6*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))-(a*d-b 
*c)^4/d^9*b^5*(-1/4*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^4-1/12*exp(-b*x-a)/(- 
b*x-a+(a*d-b*c)/d)^3-1/24*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/24*exp(-b*x 
-a)/(-b*x-a+(a*d-b*c)/d)-1/24*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))+b 
^5/d^5*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)-6*(a*d-b*c)^2/d^7*b^5*(-1 
/2*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/2*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d) 
-1/2*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (524) = 1048\).

Time = 0.10 (sec) , antiderivative size = 1084, normalized size of antiderivative = 1.95 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^5,x, algorithm="fricas")
 

Output:

1/24*((b^8*c^8 - 4*(a - 4)*b^7*c^7*d + 6*(a^2 - 8*a + 12)*b^6*c^6*d^2 - 4* 
(a^3 - 12*a^2 + 36*a - 24)*b^5*c^5*d^3 + (a^4 - 16*a^3 + 72*a^2 - 96*a + 2 
4)*b^4*c^4*d^4 + (b^8*c^4*d^4 - 4*(a - 4)*b^7*c^3*d^5 + 6*(a^2 - 8*a + 12) 
*b^6*c^2*d^6 - 4*(a^3 - 12*a^2 + 36*a - 24)*b^5*c*d^7 + (a^4 - 16*a^3 + 72 
*a^2 - 96*a + 24)*b^4*d^8)*x^4 + 4*(b^8*c^5*d^3 - 4*(a - 4)*b^7*c^4*d^4 + 
6*(a^2 - 8*a + 12)*b^6*c^3*d^5 - 4*(a^3 - 12*a^2 + 36*a - 24)*b^5*c^2*d^6 
+ (a^4 - 16*a^3 + 72*a^2 - 96*a + 24)*b^4*c*d^7)*x^3 + 6*(b^8*c^6*d^2 - 4* 
(a - 4)*b^7*c^5*d^3 + 6*(a^2 - 8*a + 12)*b^6*c^4*d^4 - 4*(a^3 - 12*a^2 + 3 
6*a - 24)*b^5*c^3*d^5 + (a^4 - 16*a^3 + 72*a^2 - 96*a + 24)*b^4*c^2*d^6)*x 
^2 + 4*(b^8*c^7*d - 4*(a - 4)*b^7*c^6*d^2 + 6*(a^2 - 8*a + 12)*b^6*c^5*d^3 
 - 4*(a^3 - 12*a^2 + 36*a - 24)*b^5*c^4*d^4 + (a^4 - 16*a^3 + 72*a^2 - 96* 
a + 24)*b^4*c^3*d^5)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) + (b^7*c^7* 
d - (4*a - 15)*b^6*c^6*d^2 + 2*(3*a^2 - 22*a + 29)*b^5*c^5*d^3 - 2*(2*a^3 
- 21*a^2 + 52*a - 25)*b^4*c^4*d^4 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*c^3 
*d^5 - 6*a^4*d^8 - (a^4 - 8*a^3 + 12*a^2)*b^2*c^2*d^6 + 2*(a^4 - 4*a^3)*b* 
c*d^7 + (b^7*c^4*d^4 - 4*(a - 4)*b^6*c^3*d^5 + 6*(a^2 - 8*a + 12)*b^5*c^2* 
d^6 - 4*(a^3 - 12*a^2 + 36*a - 24)*b^4*c*d^7 + (a^4 - 16*a^3 + 72*a^2 - 96 
*a)*b^3*d^8)*x^3 + (3*b^7*c^5*d^3 - (12*a - 47)*b^6*c^4*d^4 + 2*(9*a^2 - 7 
0*a + 100)*b^5*c^3*d^5 - 6*(2*a^3 - 23*a^2 + 64*a - 36)*b^4*c^2*d^6 + (3*a 
^4 - 44*a^3 + 168*a^2 - 144*a)*b^3*c*d^7 - (a^4 - 16*a^3 + 72*a^2)*b^2*...
 

Sympy [F]

\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\left (\int \frac {a^{4}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx\right ) e^{- a} \] Input:

integrate(exp(-b*x-a)*(b*x+a)**4/(d*x+c)**5,x)
                                                                                    
                                                                                    
 

Output:

(Integral(a**4/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 10*c**3*d**2*x**2*ex 
p(b*x) + 10*c**2*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*e 
xp(b*x)), x) + Integral(b**4*x**4/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 1 
0*c**3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp 
(b*x) + d**5*x**5*exp(b*x)), x) + Integral(4*a*b**3*x**3/(c**5*exp(b*x) + 
5*c**4*d*x*exp(b*x) + 10*c**3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*exp(b 
*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*exp(b*x)), x) + Integral(6*a**2*b 
**2*x**2/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 10*c**3*d**2*x**2*exp(b*x) 
 + 10*c**2*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*exp(b*x 
)), x) + Integral(4*a**3*b*x/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 10*c** 
3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp(b*x) 
 + d**5*x**5*exp(b*x)), x))*exp(-a)
 

Maxima [F]

\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\int { \frac {{\left (b x + a\right )}^{4} e^{\left (-b x - a\right )}}{{\left (d x + c\right )}^{5}} \,d x } \] Input:

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^5,x, algorithm="maxima")
 

Output:

-(b^3*d^2*x^4 + (4*a*b^2*d^2 - b^2*d^2)*x^3 + (6*a^2*b*d^2 + 5*b^2*c*d - 8 
*a*b*d^2 + 2*b*d^2)*x^2 + (4*a^3*d^2 - 5*b^2*c^2 - 18*a^2*d^2 - 20*b*c*d + 
 4*(5*b*c*d + 6*d^2)*a - 6*d^2)*x)*e^(-b*x)/(d^7*x^5*e^a + 5*c*d^6*x^4*e^a 
 + 10*c^2*d^5*x^3*e^a + 10*c^3*d^4*x^2*e^a + 5*c^4*d^3*x*e^a + c^5*d^2*e^a 
) - a^4*e^(-a + b*c/d)*exp_integral_e(5, (d*x + c)*b/d)/((d*x + c)^4*d) - 
integrate(-(4*a^3*c*d^2 - 5*b^2*c^3 - 18*a^2*c*d^2 - 20*b*c^2*d - 6*c*d^2 
+ 4*(5*b*c^2*d + 6*c*d^2)*a + (5*b^3*c^3 - 16*a^3*d^3 + 50*b^2*c^2*d + 90* 
b*c*d^2 + 6*(5*b*c*d^2 + 12*d^3)*a^2 + 24*d^3 - 4*(5*b^2*c^2*d + 30*b*c*d^ 
2 + 24*d^3)*a)*x)*e^(-b*x)/(d^8*x^6*e^a + 6*c*d^7*x^5*e^a + 15*c^2*d^6*x^4 
*e^a + 20*c^3*d^5*x^3*e^a + 15*c^4*d^4*x^2*e^a + 6*c^5*d^3*x*e^a + c^6*d^2 
*e^a), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 16988 vs. \(2 (524) = 1048\).

Time = 0.24 (sec) , antiderivative size = 16988, normalized size of antiderivative = 30.50 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^5,x, algorithm="giac")
 

Output:

1/24*((d*x + c)^4*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^9*c^4*Ei(-((d*x 
+ c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) 
 + 4*(d*x + c)^3*(b - b*c/(d*x + c) + a*d/(d*x + c))^3*b^10*c^5*Ei(-((d*x 
+ c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) 
 + 6*(d*x + c)^2*(b - b*c/(d*x + c) + a*d/(d*x + c))^2*b^11*c^6*Ei(-((d*x 
+ c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) 
 + 4*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^12*c^7*Ei(-((d*x + c) 
*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + b 
^13*c^8*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) 
*e^((b*c - a*d)/d) - 4*(d*x + c)^4*a*(b - b*c/(d*x + c) + a*d/(d*x + c))^4 
*b^8*c^3*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d) 
/d)*e^((b*c - a*d)/d) - 20*(d*x + c)^3*a*(b - b*c/(d*x + c) + a*d/(d*x + c 
))^3*b^9*c^4*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - 
a*d)/d)*e^((b*c - a*d)/d) - 36*(d*x + c)^2*a*(b - b*c/(d*x + c) + a*d/(d*x 
 + c))^2*b^10*c^5*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b 
*c - a*d)/d)*e^((b*c - a*d)/d) - 28*(d*x + c)*a*(b - b*c/(d*x + c) + a*d/( 
d*x + c))*b^11*c^6*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + 
b*c - a*d)/d)*e^((b*c - a*d)/d) - 8*a*b^12*c^7*d*Ei(-((d*x + c)*(b - b*c/( 
d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 6*(d*x + c)^ 
4*a^2*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^7*c^2*d^2*Ei(-((d*x + c)*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^5} \,d x \] Input:

int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x)^5,x)
 

Output:

int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x)^5, x)
 

Reduce [F]

\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx=\text {too large to display} \] Input:

int(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^5,x)
 

Output:

( - e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)* 
b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x 
**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x 
 + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c 
*d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a**4*b**2*c**5*d**4 - 4*e**(b*x)*int 
(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 
 + 10*e**(b*x)*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b 
*c*d**5*x**5 + 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c 
**3*d**3*x**2 + 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e 
**(b*x)*d**6*x**5),x)*a**4*b**2*c**4*d**5*x - 6*e**(b*x)*int(x/(e**(b*x)*b 
*c**6 + 5*e**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x) 
*b*c**3*d**3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d**5*x**5 + 
 4*e**(b*x)*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 
+ 40*e**(b*x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x)*d**6*x 
**5),x)*a**4*b**2*c**3*d**6*x**2 - 4*e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e 
**(b*x)*b*c**5*d*x + 10*e**(b*x)*b*c**4*d**2*x**2 + 10*e**(b*x)*b*c**3*d** 
3*x**3 + 5*e**(b*x)*b*c**2*d**4*x**4 + e**(b*x)*b*c*d**5*x**5 + 4*e**(b*x) 
*c**5*d + 20*e**(b*x)*c**4*d**2*x + 40*e**(b*x)*c**3*d**3*x**2 + 40*e**(b* 
x)*c**2*d**4*x**3 + 20*e**(b*x)*c*d**5*x**4 + 4*e**(b*x)*d**6*x**5),x)*a** 
4*b**2*c**2*d**7*x**3 - e**(b*x)*int(x/(e**(b*x)*b*c**6 + 5*e**(b*x)*b*...