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

Optimal. Leaf size=346 \[ -\frac{4 b^3 e^4 (b c-a d) \text{Gamma}\left (-4,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \text{Gamma}\left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^2 e^3 (b c-a d)^2 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^5}+\frac{b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b e^2 (b c-a d)^3 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{e (b c-a d)^4 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{2 b e (c+d x) (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}-\frac{2 b (c+d x)^2 (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}+\frac{(c+d x) (b c-a d)^4 e^{\frac{e}{c+d x}}}{d^5} \]

[Out]

((b*c - a*d)^4*E^(e/(c + d*x))*(c + d*x))/d^5 - (2*b*(b*c - a*d)^3*e*E^(e/(c + d
*x))*(c + d*x))/d^5 + (b^2*(b*c - a*d)^2*e^2*E^(e/(c + d*x))*(c + d*x))/d^5 - (2
*b*(b*c - a*d)^3*E^(e/(c + d*x))*(c + d*x)^2)/d^5 + (b^2*(b*c - a*d)^2*e*E^(e/(c
+ d*x))*(c + d*x)^2)/d^5 + (2*b^2*(b*c - a*d)^2*E^(e/(c + d*x))*(c + d*x)^3)/d^
5 - ((b*c - a*d)^4*e*ExpIntegralEi[e/(c + d*x)])/d^5 + (2*b*(b*c - a*d)^3*e^2*Ex
pIntegralEi[e/(c + d*x)])/d^5 - (b^2*(b*c - a*d)^2*e^3*ExpIntegralEi[e/(c + d*x)
])/d^5 - (b^4*e^5*Gamma[-5, -(e/(c + d*x))])/d^5 - (4*b^3*(b*c - a*d)*e^4*Gamma[
-4, -(e/(c + d*x))])/d^5

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Rubi [A]  time = 0.613073, antiderivative size = 346, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{4 b^3 e^4 (b c-a d) \text{Gamma}\left (-4,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^4 e^5 \text{Gamma}\left (-5,-\frac{e}{c+d x}\right )}{d^5}-\frac{b^2 e^3 (b c-a d)^2 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^5}+\frac{b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac{e}{c+d x}}}{d^5}+\frac{2 b e^2 (b c-a d)^3 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{e (b c-a d)^4 \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{d^5}-\frac{2 b e (c+d x) (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}-\frac{2 b (c+d x)^2 (b c-a d)^3 e^{\frac{e}{c+d x}}}{d^5}+\frac{(c+d x) (b c-a d)^4 e^{\frac{e}{c+d x}}}{d^5} \]

Antiderivative was successfully verified.

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

[Out]

((b*c - a*d)^4*E^(e/(c + d*x))*(c + d*x))/d^5 - (2*b*(b*c - a*d)^3*e*E^(e/(c + d
*x))*(c + d*x))/d^5 + (b^2*(b*c - a*d)^2*e^2*E^(e/(c + d*x))*(c + d*x))/d^5 - (2
*b*(b*c - a*d)^3*E^(e/(c + d*x))*(c + d*x)^2)/d^5 + (b^2*(b*c - a*d)^2*e*E^(e/(c
+ d*x))*(c + d*x)^2)/d^5 + (2*b^2*(b*c - a*d)^2*E^(e/(c + d*x))*(c + d*x)^3)/d^
5 - ((b*c - a*d)^4*e*ExpIntegralEi[e/(c + d*x)])/d^5 + (2*b*(b*c - a*d)^3*e^2*Ex
pIntegralEi[e/(c + d*x)])/d^5 - (b^2*(b*c - a*d)^2*e^3*ExpIntegralEi[e/(c + d*x)
])/d^5 - (b^4*e^5*Gamma[-5, -(e/(c + d*x))])/d^5 - (4*b^3*(b*c - a*d)*e^4*Gamma[
-4, -(e/(c + d*x))])/d^5

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Rubi in Sympy [A]  time = 81.9882, size = 311, normalized size = 0.9 \[ - \frac{b^{4} e^{5} \Gamma{\left (-5,- \frac{e}{c + d x} \right )}}{d^{5}} + \frac{4 b^{3} e^{4} \left (a d - b c\right ) \Gamma{\left (-4,- \frac{e}{c + d x} \right )}}{d^{5}} - \frac{b^{2} e^{3} \left (a d - b c\right )^{2} \operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{d^{5}} + \frac{b^{2} e^{2} \left (c + d x\right ) \left (a d - b c\right )^{2} e^{\frac{e}{c + d x}}}{d^{5}} + \frac{b^{2} e \left (c + d x\right )^{2} \left (a d - b c\right )^{2} e^{\frac{e}{c + d x}}}{d^{5}} + \frac{2 b^{2} \left (c + d x\right )^{3} \left (a d - b c\right )^{2} e^{\frac{e}{c + d x}}}{d^{5}} - \frac{2 b e^{2} \left (a d - b c\right )^{3} \operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{d^{5}} + \frac{2 b e \left (c + d x\right ) \left (a d - b c\right )^{3} e^{\frac{e}{c + d x}}}{d^{5}} + \frac{2 b \left (c + d x\right )^{2} \left (a d - b c\right )^{3} e^{\frac{e}{c + d x}}}{d^{5}} - \frac{e \left (a d - b c\right )^{4} \operatorname{Ei}{\left (\frac{e}{c + d x} \right )}}{d^{5}} + \frac{\left (c + d x\right ) \left (a d - b c\right )^{4} e^{\frac{e}{c + d x}}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**4*e**5*Gamma(-5, -e/(c + d*x))/d**5 + 4*b**3*e**4*(a*d - b*c)*Gamma(-4, -e/(
c + d*x))/d**5 - b**2*e**3*(a*d - b*c)**2*Ei(e/(c + d*x))/d**5 + b**2*e**2*(c +
d*x)*(a*d - b*c)**2*exp(e/(c + d*x))/d**5 + b**2*e*(c + d*x)**2*(a*d - b*c)**2*e
xp(e/(c + d*x))/d**5 + 2*b**2*(c + d*x)**3*(a*d - b*c)**2*exp(e/(c + d*x))/d**5
- 2*b*e**2*(a*d - b*c)**3*Ei(e/(c + d*x))/d**5 + 2*b*e*(c + d*x)*(a*d - b*c)**3*
exp(e/(c + d*x))/d**5 + 2*b*(c + d*x)**2*(a*d - b*c)**3*exp(e/(c + d*x))/d**5 -
e*(a*d - b*c)**4*Ei(e/(c + d*x))/d**5 + (c + d*x)*(a*d - b*c)**4*exp(e/(c + d*x)
)/d**5

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Mathematica [A]  time = 0.632384, size = 468, normalized size = 1.35 \[ \frac{d x e^{\frac{e}{c+d x}} \left (120 a^4 d^4+240 a^3 b d^3 (d x+e)+120 a^2 b^2 d^2 \left (-4 c e+2 d^2 x^2+d e x+e^2\right )+20 a b^3 d \left (18 c^2 e-2 c e (3 d x+5 e)+6 d^3 x^3+2 d^2 e x^2+d e^2 x+e^3\right )+b^4 \left (-96 c^3 e+2 c^2 e (18 d x+43 e)-2 c e \left (8 d^2 x^2+7 d e x+9 e^2\right )+24 d^4 x^4+6 d^3 e x^3+2 d^2 e^2 x^2+d e^3 x+e^4\right )\right )-e \left (120 a^4 d^4-240 a^3 b d^3 (2 c-e)+120 a^2 b^2 d^2 \left (6 c^2-6 c e+e^2\right )-20 a b^3 d \left (24 c^3-36 c^2 e+12 c e^2-e^3\right )+b^4 \left (120 c^4-240 c^3 e+120 c^2 e^2-20 c e^3+e^4\right )\right ) \text{ExpIntegralEi}\left (\frac{e}{c+d x}\right )}{120 d^5}+\frac{c e^{\frac{e}{c+d x}} \left (120 a^4 d^4-240 a^3 b d^3 (c-e)+120 a^2 b^2 d^2 \left (2 c^2-5 c e+e^2\right )-20 a b^3 d \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )+b^4 \left (24 c^4-154 c^3 e+102 c^2 e^2-19 c e^3+e^4\right )\right )}{120 d^5} \]

Antiderivative was successfully verified.

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

[Out]

(c*(120*a^4*d^4 - 240*a^3*b*d^3*(c - e) + 120*a^2*b^2*d^2*(2*c^2 - 5*c*e + e^2)
- 20*a*b^3*d*(6*c^3 - 26*c^2*e + 11*c*e^2 - e^3) + b^4*(24*c^4 - 154*c^3*e + 102
*c^2*e^2 - 19*c*e^3 + e^4))*E^(e/(c + d*x)))/(120*d^5) + (d*E^(e/(c + d*x))*x*(1
20*a^4*d^4 + 240*a^3*b*d^3*(e + d*x) + 120*a^2*b^2*d^2*(-4*c*e + e^2 + d*e*x + 2
*d^2*x^2) + 20*a*b^3*d*(18*c^2*e + e^3 + d*e^2*x + 2*d^2*e*x^2 + 6*d^3*x^3 - 2*c
*e*(5*e + 3*d*x)) + b^4*(-96*c^3*e + e^4 + d*e^3*x + 2*d^2*e^2*x^2 + 6*d^3*e*x^3
+ 24*d^4*x^4 + 2*c^2*e*(43*e + 18*d*x) - 2*c*e*(9*e^2 + 7*d*e*x + 8*d^2*x^2)))
- e*(120*a^4*d^4 - 240*a^3*b*d^3*(2*c - e) + 120*a^2*b^2*d^2*(6*c^2 - 6*c*e + e^
2) - 20*a*b^3*d*(24*c^3 - 36*c^2*e + 12*c*e^2 - e^3) + b^4*(120*c^4 - 240*c^3*e
+ 120*c^2*e^2 - 20*c*e^3 + e^4))*ExpIntegralEi[e/(c + d*x)])/(120*d^5)

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Maple [B]  time = 0.016, size = 1146, normalized size = 3.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(e/(d*x+c))*(b*x+a)^4,x)

[Out]

-1/d*e*(a^4*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+b^4/d^4*e^4*(-1/5*(d*x+
c)^5/e^5*exp(e/(d*x+c))-1/20*(d*x+c)^4/e^4*exp(e/(d*x+c))-1/60*(d*x+c)^3/e^3*exp
(e/(d*x+c))-1/120*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/120*(d*x+c)/e*exp(e/(d*x+c))-1/
120*Ei(1,-e/(d*x+c)))+b^4/d^4*c^4*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+4
*b^3/d^3*e^3*a*(-1/4*(d*x+c)^4/e^4*exp(e/(d*x+c))-1/12*(d*x+c)^3/e^3*exp(e/(d*x+
c))-1/24*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/24*(d*x+c)/e*exp(e/(d*x+c))-1/24*Ei(1,-e
/(d*x+c)))-4*b^4/d^4*e^3*c*(-1/4*(d*x+c)^4/e^4*exp(e/(d*x+c))-1/12*(d*x+c)^3/e^3
*exp(e/(d*x+c))-1/24*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/24*(d*x+c)/e*exp(e/(d*x+c))-
1/24*Ei(1,-e/(d*x+c)))+6*b^2/d^2*e^2*a^2*(-1/3*(d*x+c)^3/e^3*exp(e/(d*x+c))-1/6*
exp(e/(d*x+c))*(d*x+c)^2/e^2-1/6*(d*x+c)/e*exp(e/(d*x+c))-1/6*Ei(1,-e/(d*x+c)))+
6*b^4/d^4*e^2*c^2*(-1/3*(d*x+c)^3/e^3*exp(e/(d*x+c))-1/6*exp(e/(d*x+c))*(d*x+c)^
2/e^2-1/6*(d*x+c)/e*exp(e/(d*x+c))-1/6*Ei(1,-e/(d*x+c)))+4*b/d*e*a^3*(-1/2*exp(e
/(d*x+c))*(d*x+c)^2/e^2-1/2*(d*x+c)/e*exp(e/(d*x+c))-1/2*Ei(1,-e/(d*x+c)))-4*b^4
/d^4*e*c^3*(-1/2*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/2*(d*x+c)/e*exp(e/(d*x+c))-1/2*E
i(1,-e/(d*x+c)))-4*b/d*c*a^3*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+6*b^2/
d^2*c^2*a^2*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))-4*b^3/d^3*c^3*a*(-(d*x+
c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))-12*b^3/d^3*e^2*c*a*(-1/3*(d*x+c)^3/e^3*exp
(e/(d*x+c))-1/6*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/6*(d*x+c)/e*exp(e/(d*x+c))-1/6*Ei
(1,-e/(d*x+c)))-12*b^2/d^2*e*c*a^2*(-1/2*exp(e/(d*x+c))*(d*x+c)^2/e^2-1/2*(d*x+c
)/e*exp(e/(d*x+c))-1/2*Ei(1,-e/(d*x+c)))+12*b^3/d^3*e*c^2*a*(-1/2*exp(e/(d*x+c))
*(d*x+c)^2/e^2-1/2*(d*x+c)/e*exp(e/(d*x+c))-1/2*Ei(1,-e/(d*x+c))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \frac{{\left (24 \, b^{4} d^{4} x^{5} + 6 \,{\left (20 \, a b^{3} d^{4} + b^{4} d^{3} e\right )} x^{4} + 2 \,{\left (120 \, a^{2} b^{2} d^{4} + 20 \, a b^{3} d^{3} e -{\left (8 \, c d^{2} e - d^{2} e^{2}\right )} b^{4}\right )} x^{3} +{\left (240 \, a^{3} b d^{4} + 120 \, a^{2} b^{2} d^{3} e - 20 \,{\left (6 \, c d^{2} e - d^{2} e^{2}\right )} a b^{3} +{\left (36 \, c^{2} d e - 14 \, c d e^{2} + d e^{3}\right )} b^{4}\right )} x^{2} +{\left (120 \, a^{4} d^{4} + 240 \, a^{3} b d^{3} e - 120 \,{\left (4 \, c d^{2} e - d^{2} e^{2}\right )} a^{2} b^{2} + 20 \,{\left (18 \, c^{2} d e - 10 \, c d e^{2} + d e^{3}\right )} a b^{3} -{\left (96 \, c^{3} e - 86 \, c^{2} e^{2} + 18 \, c e^{3} - e^{4}\right )} b^{4}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{120 \, d^{4}} + \int -\frac{{\left (240 \, a^{3} b c^{2} d^{3} e - 120 \,{\left (4 \, c^{3} d^{2} e - c^{2} d^{2} e^{2}\right )} a^{2} b^{2} + 20 \,{\left (18 \, c^{4} d e - 10 \, c^{3} d e^{2} + c^{2} d e^{3}\right )} a b^{3} -{\left (96 \, c^{5} e - 86 \, c^{4} e^{2} + 18 \, c^{3} e^{3} - c^{2} e^{4}\right )} b^{4} -{\left (120 \, a^{4} d^{5} e - 240 \,{\left (2 \, c d^{4} e - d^{4} e^{2}\right )} a^{3} b + 120 \,{\left (6 \, c^{2} d^{3} e - 6 \, c d^{3} e^{2} + d^{3} e^{3}\right )} a^{2} b^{2} - 20 \,{\left (24 \, c^{3} d^{2} e - 36 \, c^{2} d^{2} e^{2} + 12 \, c d^{2} e^{3} - d^{2} e^{4}\right )} a b^{3} +{\left (120 \, c^{4} d e - 240 \, c^{3} d e^{2} + 120 \, c^{2} d e^{3} - 20 \, c d e^{4} + d e^{5}\right )} b^{4}\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{120 \,{\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/120*(24*b^4*d^4*x^5 + 6*(20*a*b^3*d^4 + b^4*d^3*e)*x^4 + 2*(120*a^2*b^2*d^4 +
20*a*b^3*d^3*e - (8*c*d^2*e - d^2*e^2)*b^4)*x^3 + (240*a^3*b*d^4 + 120*a^2*b^2*d
^3*e - 20*(6*c*d^2*e - d^2*e^2)*a*b^3 + (36*c^2*d*e - 14*c*d*e^2 + d*e^3)*b^4)*x
^2 + (120*a^4*d^4 + 240*a^3*b*d^3*e - 120*(4*c*d^2*e - d^2*e^2)*a^2*b^2 + 20*(18
*c^2*d*e - 10*c*d*e^2 + d*e^3)*a*b^3 - (96*c^3*e - 86*c^2*e^2 + 18*c*e^3 - e^4)*
b^4)*x)*e^(e/(d*x + c))/d^4 + integrate(-1/120*(240*a^3*b*c^2*d^3*e - 120*(4*c^3
*d^2*e - c^2*d^2*e^2)*a^2*b^2 + 20*(18*c^4*d*e - 10*c^3*d*e^2 + c^2*d*e^3)*a*b^3
- (96*c^5*e - 86*c^4*e^2 + 18*c^3*e^3 - c^2*e^4)*b^4 - (120*a^4*d^5*e - 240*(2*
c*d^4*e - d^4*e^2)*a^3*b + 120*(6*c^2*d^3*e - 6*c*d^3*e^2 + d^3*e^3)*a^2*b^2 - 2
0*(24*c^3*d^2*e - 36*c^2*d^2*e^2 + 12*c*d^2*e^3 - d^2*e^4)*a*b^3 + (120*c^4*d*e
- 240*c^3*d*e^2 + 120*c^2*d*e^3 - 20*c*d*e^4 + d*e^5)*b^4)*x)*e^(e/(d*x + c))/(d
^6*x^2 + 2*c*d^5*x + c^2*d^4), x)

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Fricas [A]  time = 0.255135, size = 861, normalized size = 2.49 \[ -\frac{{\left (b^{4} e^{5} - 20 \,{\left (b^{4} c - a b^{3} d\right )} e^{4} + 120 \,{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} e^{3} - 240 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} e^{2} + 120 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} e\right )}{\rm Ei}\left (\frac{e}{d x + c}\right ) -{\left (24 \, b^{4} d^{5} x^{5} + 24 \, b^{4} c^{5} - 120 \, a b^{3} c^{4} d + 240 \, a^{2} b^{2} c^{3} d^{2} - 240 \, a^{3} b c^{2} d^{3} + 120 \, a^{4} c d^{4} + b^{4} c e^{4} + 6 \,{\left (20 \, a b^{3} d^{5} + b^{4} d^{4} e\right )} x^{4} -{\left (19 \, b^{4} c^{2} - 20 \, a b^{3} c d\right )} e^{3} + 2 \,{\left (120 \, a^{2} b^{2} d^{5} + b^{4} d^{3} e^{2} - 4 \,{\left (2 \, b^{4} c d^{3} - 5 \, a b^{3} d^{4}\right )} e\right )} x^{3} + 2 \,{\left (51 \, b^{4} c^{3} - 110 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2}\right )} e^{2} +{\left (240 \, a^{3} b d^{5} + b^{4} d^{2} e^{3} - 2 \,{\left (7 \, b^{4} c d^{2} - 10 \, a b^{3} d^{3}\right )} e^{2} + 12 \,{\left (3 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} e\right )} x^{2} - 2 \,{\left (77 \, b^{4} c^{4} - 260 \, a b^{3} c^{3} d + 300 \, a^{2} b^{2} c^{2} d^{2} - 120 \, a^{3} b c d^{3}\right )} e +{\left (120 \, a^{4} d^{5} + b^{4} d e^{4} - 2 \,{\left (9 \, b^{4} c d - 10 \, a b^{3} d^{2}\right )} e^{3} + 2 \,{\left (43 \, b^{4} c^{2} d - 100 \, a b^{3} c d^{2} + 60 \, a^{2} b^{2} d^{3}\right )} e^{2} - 24 \,{\left (4 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 20 \, a^{2} b^{2} c d^{3} - 10 \, a^{3} b d^{4}\right )} e\right )} x\right )} e^{\left (\frac{e}{d x + c}\right )}}{120 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/120*((b^4*e^5 - 20*(b^4*c - a*b^3*d)*e^4 + 120*(b^4*c^2 - 2*a*b^3*c*d + a^2*b
^2*d^2)*e^3 - 240*(b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*e^2 +
120*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*e)*E
i(e/(d*x + c)) - (24*b^4*d^5*x^5 + 24*b^4*c^5 - 120*a*b^3*c^4*d + 240*a^2*b^2*c^
3*d^2 - 240*a^3*b*c^2*d^3 + 120*a^4*c*d^4 + b^4*c*e^4 + 6*(20*a*b^3*d^5 + b^4*d^
4*e)*x^4 - (19*b^4*c^2 - 20*a*b^3*c*d)*e^3 + 2*(120*a^2*b^2*d^5 + b^4*d^3*e^2 -
4*(2*b^4*c*d^3 - 5*a*b^3*d^4)*e)*x^3 + 2*(51*b^4*c^3 - 110*a*b^3*c^2*d + 60*a^2*
b^2*c*d^2)*e^2 + (240*a^3*b*d^5 + b^4*d^2*e^3 - 2*(7*b^4*c*d^2 - 10*a*b^3*d^3)*e
^2 + 12*(3*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + 10*a^2*b^2*d^4)*e)*x^2 - 2*(77*b^4*c^4
- 260*a*b^3*c^3*d + 300*a^2*b^2*c^2*d^2 - 120*a^3*b*c*d^3)*e + (120*a^4*d^5 + b
^4*d*e^4 - 2*(9*b^4*c*d - 10*a*b^3*d^2)*e^3 + 2*(43*b^4*c^2*d - 100*a*b^3*c*d^2
+ 60*a^2*b^2*d^3)*e^2 - 24*(4*b^4*c^3*d - 15*a*b^3*c^2*d^2 + 20*a^2*b^2*c*d^3 -
10*a^3*b*d^4)*e)*x)*e^(e/(d*x + c)))/d^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (a + b x\right )^{4} e^{\frac{e}{c + d x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((a + b*x)**4*exp(e/(c + d*x)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x + a\right )}^{4} e^{\left (\frac{e}{d x + c}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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