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

Optimal. Leaf size=557 \[ \frac{b^4 (b c-a d)^4 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{24 d^9}+\frac{2 b^4 (b c-a d)^3 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{3 d^8}+\frac{3 b^4 (b c-a d)^2 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^7}+\frac{4 b^4 (b c-a d) e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{b^4 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^3 e^{-a-b x} (b c-a d)^4}{24 d^8 (c+d x)}+\frac{2 b^3 e^{-a-b x} (b c-a d)^3}{3 d^7 (c+d x)}+\frac{3 b^3 e^{-a-b x} (b c-a d)^2}{d^6 (c+d x)}+\frac{4 b^3 e^{-a-b x} (b c-a d)}{d^5 (c+d x)}-\frac{b^2 e^{-a-b x} (b c-a d)^4}{24 d^7 (c+d x)^2}-\frac{2 b^2 e^{-a-b x} (b c-a d)^3}{3 d^6 (c+d x)^2}-\frac{3 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)^2}+\frac{b e^{-a-b x} (b c-a d)^4}{12 d^6 (c+d x)^3}+\frac{4 b e^{-a-b x} (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac{e^{-a-b x} (b c-a d)^4}{4 d^5 (c+d x)^4} \]

[Out]

-((b*c - a*d)^4*E^(-a - b*x))/(4*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)*E
xpIntegralEi[-((b*(c + d*x))/d)])/(3*d^8) + (b^4*(b*c - a*d)^4*E^(-a + (b*c)/d)*
ExpIntegralEi[-((b*(c + d*x))/d)])/(24*d^9)

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Rubi [A]  time = 1.15225, antiderivative size = 557, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ \frac{b^4 (b c-a d)^4 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{24 d^9}+\frac{2 b^4 (b c-a d)^3 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{3 d^8}+\frac{3 b^4 (b c-a d)^2 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^7}+\frac{4 b^4 (b c-a d) e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{b^4 e^{\frac{b c}{d}-a} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^3 e^{-a-b x} (b c-a d)^4}{24 d^8 (c+d x)}+\frac{2 b^3 e^{-a-b x} (b c-a d)^3}{3 d^7 (c+d x)}+\frac{3 b^3 e^{-a-b x} (b c-a d)^2}{d^6 (c+d x)}+\frac{4 b^3 e^{-a-b x} (b c-a d)}{d^5 (c+d x)}-\frac{b^2 e^{-a-b x} (b c-a d)^4}{24 d^7 (c+d x)^2}-\frac{2 b^2 e^{-a-b x} (b c-a d)^3}{3 d^6 (c+d x)^2}-\frac{3 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)^2}+\frac{b e^{-a-b x} (b c-a d)^4}{12 d^6 (c+d x)^3}+\frac{4 b e^{-a-b x} (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac{e^{-a-b x} (b c-a d)^4}{4 d^5 (c+d x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-((b*c - a*d)^4*E^(-a - b*x))/(4*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)*E
xpIntegralEi[-((b*(c + d*x))/d)])/(3*d^8) + (b^4*(b*c - a*d)^4*E^(-a + (b*c)/d)*
ExpIntegralEi[-((b*(c + d*x))/d)])/(24*d^9)

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Rubi in Sympy [A]  time = 112.574, size = 493, normalized size = 0.89 \[ \frac{b^{4} e^{- a} e^{\frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{d^{5}} - \frac{4 b^{4} \left (a d - b c\right ) e^{- a + \frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{d^{6}} + \frac{3 b^{4} \left (a d - b c\right )^{2} e^{- a} e^{\frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{d^{7}} - \frac{2 b^{4} \left (a d - b c\right )^{3} e^{- a + \frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{3 d^{8}} + \frac{b^{4} \left (a d - b c\right )^{4} e^{- a} e^{\frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{24 d^{9}} - \frac{4 b^{3} \left (a d - b c\right ) e^{- a - b x}}{d^{5} \left (c + d x\right )} + \frac{3 b^{3} \left (a d - b c\right )^{2} e^{- a - b x}}{d^{6} \left (c + d x\right )} - \frac{2 b^{3} \left (a d - b c\right )^{3} e^{- a - b x}}{3 d^{7} \left (c + d x\right )} + \frac{b^{3} \left (a d - b c\right )^{4} e^{- a - b x}}{24 d^{8} \left (c + d x\right )} - \frac{3 b^{2} \left (a d - b c\right )^{2} e^{- a - b x}}{d^{5} \left (c + d x\right )^{2}} + \frac{2 b^{2} \left (a d - b c\right )^{3} e^{- a - b x}}{3 d^{6} \left (c + d x\right )^{2}} - \frac{b^{2} \left (a d - b c\right )^{4} e^{- a - b x}}{24 d^{7} \left (c + d x\right )^{2}} - \frac{4 b \left (a d - b c\right )^{3} e^{- a - b x}}{3 d^{5} \left (c + d x\right )^{3}} + \frac{b \left (a d - b c\right )^{4} e^{- a - b x}}{12 d^{6} \left (c + d x\right )^{3}} - \frac{\left (a d - b c\right )^{4} e^{- a - b x}}{4 d^{5} \left (c + d x\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(-b*x-a)*(b*x+a)**4/(d*x+c)**5,x)

[Out]

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

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Mathematica [A]  time = 0.707048, size = 669, normalized size = 1.2 \[ \frac{e^{-a} \left (a^4 b^4 d^4 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )-4 a^3 b^5 c d^3 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )-16 a^3 b^4 d^4 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+6 a^2 b^6 c^2 d^2 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+48 a^2 b^5 c d^3 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+72 a^2 b^4 d^4 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+\frac{d e^{-b x} \left (-b^2 d (c+d x)^2 \left (\left (a^2-16 a+72\right ) d^2-2 (a-8) b c d+b^2 c^2\right ) (b c-a d)^2+b^3 (c+d x)^3 \left (6 \left (a^2-8 a+12\right ) b^2 c^2 d^2-4 \left (a^3-12 a^2+36 a-24\right ) b c d^3+a \left (a^3-16 a^2+72 a-96\right ) d^4-4 (a-4) b^3 c^3 d+b^4 c^4\right )-6 d^3 (b c-a d)^4+2 b d^2 (c+d x) (b c-(a-16) d) (b c-a d)^3\right )}{(c+d x)^4}-4 a b^7 c^3 d e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )-48 a b^6 c^2 d^2 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )-144 a b^5 c d^3 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )-96 a b^4 d^4 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+b^8 c^4 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+16 b^7 c^3 d e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+72 b^6 c^2 d^2 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+96 b^5 c d^3 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )+24 b^4 d^4 e^{\frac{b c}{d}} \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )\right )}{24 d^9} \]

Antiderivative was successfully verified.

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

[Out]

((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)*ExpIntegralEi[-((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)*E
xpIntegralEi[-((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)*
ExpIntegralEi[-((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)] - 9
6*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)*ExpIntegra
lEi[-((b*(c + d*x))/d)] + a^4*b^4*d^4*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/
d)])/(24*d^9*E^a)

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Maple [A]  time = 0.019, size = 596, normalized size = 1.1 \[ -{\frac{1}{b} \left ({\frac{{b}^{5}}{{d}^{5}}{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) }+4\,{\frac{ \left ( ad-cb \right ){b}^{5}}{{d}^{6}} \left ( -{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) \right ) }-{\frac{ \left ( ad-cb \right ) ^{4}{b}^{5}}{{d}^{9}} \left ( -{\frac{{{\rm e}^{-bx-a}}}{4} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-4}}-{\frac{{{\rm e}^{-bx-a}}}{12} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-3}}-{\frac{{{\rm e}^{-bx-a}}}{24} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-2}}-{\frac{{{\rm e}^{-bx-a}}}{24} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-{\frac{1}{24}{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) } \right ) }-6\,{\frac{ \left ( ad-cb \right ) ^{2}{b}^{5}}{{d}^{7}} \left ( -1/2\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-2}}-1/2\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-1/2\,{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) \right ) }+4\,{\frac{ \left ( ad-cb \right ) ^{3}{b}^{5}}{{d}^{8}} \left ( -1/3\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-3}}-1/6\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-2}}-1/6\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-1/6\,{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/b*(b^5/d^5*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d)+4*(a*d-b*c)/d^6*b^5*(-ex
p(-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))-(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))-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))+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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{{\left (b^{3} d^{2} x^{4} +{\left (4 \, a b^{2} d^{2} - b^{2} d^{2}\right )} x^{3} +{\left (6 \, a^{2} b d^{2} + 5 \, b^{2} c d - 8 \, a b d^{2} + 2 \, b d^{2}\right )} x^{2} +{\left (4 \, a^{3} d^{2} - 5 \, b^{2} c^{2} - 18 \, a^{2} d^{2} - 20 \, b c d + 4 \,{\left (5 \, b c d + 6 \, d^{2}\right )} a - 6 \, d^{2}\right )} x\right )} e^{\left (-b x\right )}}{d^{7} x^{5} e^{a} + 5 \, c d^{6} x^{4} e^{a} + 10 \, c^{2} d^{5} x^{3} e^{a} + 10 \, c^{3} d^{4} x^{2} e^{a} + 5 \, c^{4} d^{3} x e^{a} + c^{5} d^{2} e^{a}} - \frac{a^{4} e^{\left (-a + \frac{b c}{d}\right )} exp_integral_e\left (5, \frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{4} d} - \int -\frac{{\left (4 \, a^{3} c d^{2} - 5 \, b^{2} c^{3} - 18 \, a^{2} c d^{2} - 20 \, b c^{2} d - 6 \, c d^{2} + 4 \,{\left (5 \, b c^{2} d + 6 \, c d^{2}\right )} a +{\left (5 \, b^{3} c^{3} - 16 \, a^{3} d^{3} + 50 \, b^{2} c^{2} d + 90 \, b c d^{2} + 6 \,{\left (5 \, b c d^{2} + 12 \, d^{3}\right )} a^{2} + 24 \, d^{3} - 4 \,{\left (5 \, b^{2} c^{2} d + 30 \, b c d^{2} + 24 \, d^{3}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{8} x^{6} e^{a} + 6 \, c d^{7} x^{5} e^{a} + 15 \, c^{2} d^{6} x^{4} e^{a} + 20 \, c^{3} d^{5} x^{3} e^{a} + 15 \, c^{4} d^{4} x^{2} e^{a} + 6 \, c^{5} d^{3} x e^{a} + c^{6} d^{2} e^{a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(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)*e
xp_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)

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Fricas [A]  time = 0.269128, size = 1463, normalized size = 2.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 + 24)*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 + 36*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 - 1
2*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 - 70*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*d^8)*x^2 + (3*b^7*c^6*d^2 - 2*(6*a - 23)*b^
6*c^5*d^3 + 2*(9*a^2 - 68*a + 93)*b^5*c^4*d^4 - 4*(3*a^3 - 33*a^2 + 86*a - 44)*b
^4*c^3*d^5 + (3*a^4 - 40*a^3 + 132*a^2 - 96*a)*b^3*c^2*d^6 - 2*(a^4 - 12*a^3 + 2
4*a^2)*b^2*c*d^7 + 2*(a^4 - 16*a^3)*b*d^8)*x)*e^(-b*x - a))/(d^13*x^4 + 4*c*d^12
*x^3 + 6*c^2*d^11*x^2 + 4*c^3*d^10*x + c^4*d^9)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \mathit{undef} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

undef