Integrand size = 19, antiderivative size = 346 \[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=\frac {(b c-a d)^4 e^{\frac {e}{c+d x}} (c+d x)}{d^5}-\frac {2 b (b c-a d)^3 e e^{\frac {e}{c+d x}} (c+d x)}{d^5}+\frac {b^2 (b c-a d)^2 e^2 e^{\frac {e}{c+d x}} (c+d x)}{d^5}-\frac {2 b (b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)^2}{d^5}+\frac {b^2 (b c-a d)^2 e e^{\frac {e}{c+d x}} (c+d x)^2}{d^5}+\frac {2 b^2 (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^3}{d^5}-\frac {(b c-a d)^4 e \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^5}+\frac {2 b (b c-a d)^3 e^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {b^2 (b c-a d)^2 e^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {b^4 e^5 \Gamma \left (-5,-\frac {e}{c+d x}\right )}{d^5}-\frac {4 b^3 (b c-a d) e^4 \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^5} \]
(-a*d+b*c)^4*exp(e/(d*x+c))*(d*x+c)/d^5-2*b*(-a*d+b*c)^3*e*exp(e/(d*x+c))* (d*x+c)/d^5+b^2*(-a*d+b*c)^2*e^2*exp(e/(d*x+c))*(d*x+c)/d^5-2*b*(-a*d+b*c) ^3*exp(e/(d*x+c))*(d*x+c)^2/d^5+b^2*(-a*d+b*c)^2*e*exp(e/(d*x+c))*(d*x+c)^ 2/d^5+2*b^2*(-a*d+b*c)^2*exp(e/(d*x+c))*(d*x+c)^3/d^5-(-a*d+b*c)^4*e*Ei(e/ (d*x+c))/d^5+2*b*(-a*d+b*c)^3*e^2*Ei(e/(d*x+c))/d^5-b^2*(-a*d+b*c)^2*e^3*E i(e/(d*x+c))/d^5+b^4*(d*x+c)^5*Ei(6,-e/(d*x+c))/d^5-4*b^3*(-a*d+b*c)*(d*x+ c)^4*Ei(5,-e/(d*x+c))/d^5
Time = 0.27 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.35 \[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=\frac {c \left (120 a^4 d^4-240 a^3 b d^3 (c-e)+120 a^2 b^2 d^2 \left (2 c^2-5 c e+e^2\right )-20 a b^3 d \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )+b^4 \left (24 c^4-154 c^3 e+102 c^2 e^2-19 c e^3+e^4\right )\right ) e^{\frac {e}{c+d x}}}{120 d^5}+\frac {d e^{\frac {e}{c+d x}} x \left (120 a^4 d^4+240 a^3 b d^3 (e+d x)+120 a^2 b^2 d^2 \left (-4 c e+e^2+d e x+2 d^2 x^2\right )+20 a b^3 d \left (18 c^2 e+e^3+d e^2 x+2 d^2 e x^2+6 d^3 x^3-2 c e (5 e+3 d x)\right )+b^4 \left (-96 c^3 e+e^4+d e^3 x+2 d^2 e^2 x^2+6 d^3 e x^3+24 d^4 x^4+2 c^2 e (43 e+18 d x)-2 c e \left (9 e^2+7 d e x+8 d^2 x^2\right )\right )\right )-e \left (120 a^4 d^4-240 a^3 b d^3 (2 c-e)+120 a^2 b^2 d^2 \left (6 c^2-6 c e+e^2\right )-20 a b^3 d \left (24 c^3-36 c^2 e+12 c e^2-e^3\right )+b^4 \left (120 c^4-240 c^3 e+120 c^2 e^2-20 c e^3+e^4\right )\right ) \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{120 d^5} \]
(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*(120*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)
Time = 0.66 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2656, 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.
| \(\displaystyle \int (a+b x)^4 e^{\frac {e}{c+d x}} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (-\frac {4 b^3 (c+d x)^3 (b c-a d) e^{\frac {e}{c+d x}}}{d^4}+\frac {6 b^2 (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^4}+\frac {(a d-b c)^4 e^{\frac {e}{c+d x}}}{d^4}-\frac {4 b (c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^4}+\frac {b^4 (c+d x)^4 e^{\frac {e}{c+d x}}}{d^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 b^3 e^4 (b c-a d) \Gamma \left (-4,-\frac {e}{c+d x}\right )}{d^5}-\frac {b^2 e^3 (b c-a d)^2 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^5}+\frac {b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {2 b e^2 (b c-a d)^3 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {e (b c-a d)^4 \operatorname {ExpIntegralEi}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {2 b e (c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^5}-\frac {2 b (c+d x)^2 (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^5}+\frac {(c+d x) (b c-a d)^4 e^{\frac {e}{c+d x}}}{d^5}-\frac {b^4 e^5 \Gamma \left (-5,-\frac {e}{c+d x}\right )}{d^5}\) |
((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*ExpIntegralEi[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
3.5.1.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(1145\) vs. \(2(347)=694\).
Time = 0.52 (sec) , antiderivative size = 1146, normalized size of antiderivative = 3.31
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1146\) |
| default | \(\text {Expression too large to display}\) | \(1146\) |
| risch | \(\text {Expression too large to display}\) | \(1273\) |
| parts | \(\text {Expression too large to display}\) | \(2204\) |
-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*Ei(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*(-...
Time = 0.32 (sec) , antiderivative size = 638, normalized size of antiderivative = 1.84 \[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=-\frac {{\left (b^{4} e^{5} - 20 \, {\left (b^{4} c - a b^{3} d\right )} e^{4} + 120 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} e^{3} - 240 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} e^{2} + 120 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (24 \, b^{4} d^{5} x^{5} + 24 \, b^{4} c^{5} - 120 \, a b^{3} c^{4} d + 240 \, a^{2} b^{2} c^{3} d^{2} - 240 \, a^{3} b c^{2} d^{3} + 120 \, a^{4} c d^{4} + b^{4} c e^{4} + 6 \, {\left (20 \, a b^{3} d^{5} + b^{4} d^{4} e\right )} x^{4} - {\left (19 \, b^{4} c^{2} - 20 \, a b^{3} c d\right )} e^{3} + 2 \, {\left (120 \, a^{2} b^{2} d^{5} + b^{4} d^{3} e^{2} - 4 \, {\left (2 \, b^{4} c d^{3} - 5 \, a b^{3} d^{4}\right )} e\right )} x^{3} + 2 \, {\left (51 \, b^{4} c^{3} - 110 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2}\right )} e^{2} + {\left (240 \, a^{3} b d^{5} + b^{4} d^{2} e^{3} - 2 \, {\left (7 \, b^{4} c d^{2} - 10 \, a b^{3} d^{3}\right )} e^{2} + 12 \, {\left (3 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} e\right )} x^{2} - 2 \, {\left (77 \, b^{4} c^{4} - 260 \, a b^{3} c^{3} d + 300 \, a^{2} b^{2} c^{2} d^{2} - 120 \, a^{3} b c d^{3}\right )} e + {\left (120 \, a^{4} d^{5} + b^{4} d e^{4} - 2 \, {\left (9 \, b^{4} c d - 10 \, a b^{3} d^{2}\right )} e^{3} + 2 \, {\left (43 \, b^{4} c^{2} d - 100 \, a b^{3} c d^{2} + 60 \, a^{2} b^{2} d^{3}\right )} e^{2} - 24 \, {\left (4 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 20 \, a^{2} b^{2} c d^{3} - 10 \, a^{3} b d^{4}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, d^{5}} \]
-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)*Ei(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 - 1 0*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
\[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=\int \left (a + b x\right )^{4} e^{\frac {e}{c + d x}}\, dx \]
\[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=\int { {\left (b x + a\right )}^{4} e^{\left (\frac {e}{d x + c}\right )} \,d x } \]
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 + 1 20*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(-1/1 20*(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 - 20*(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)
Leaf count of result is larger than twice the leaf count of optimal. 1428 vs. \(2 (347) = 694\).
Time = 0.34 (sec) , antiderivative size = 1428, normalized size of antiderivative = 4.13 \[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=\text {Too large to display} \]
-1/120*(120*b^4*c^4*e^7*Ei(e/(d*x + c))/(d*x + c)^5 - 480*a*b^3*c^3*d*e^7* Ei(e/(d*x + c))/(d*x + c)^5 + 720*a^2*b^2*c^2*d^2*e^7*Ei(e/(d*x + c))/(d*x + c)^5 - 480*a^3*b*c*d^3*e^7*Ei(e/(d*x + c))/(d*x + c)^5 + 120*a^4*d^4*e^ 7*Ei(e/(d*x + c))/(d*x + c)^5 - 240*b^4*c^3*e^8*Ei(e/(d*x + c))/(d*x + c)^ 5 + 720*a*b^3*c^2*d*e^8*Ei(e/(d*x + c))/(d*x + c)^5 - 720*a^2*b^2*c*d^2*e^ 8*Ei(e/(d*x + c))/(d*x + c)^5 + 240*a^3*b*d^3*e^8*Ei(e/(d*x + c))/(d*x + c )^5 + 120*b^4*c^2*e^9*Ei(e/(d*x + c))/(d*x + c)^5 - 240*a*b^3*c*d*e^9*Ei(e /(d*x + c))/(d*x + c)^5 + 120*a^2*b^2*d^2*e^9*Ei(e/(d*x + c))/(d*x + c)^5 - 20*b^4*c*e^10*Ei(e/(d*x + c))/(d*x + c)^5 + 20*a*b^3*d*e^10*Ei(e/(d*x + c))/(d*x + c)^5 + b^4*e^11*Ei(e/(d*x + c))/(d*x + c)^5 - 24*b^4*e^6*e^(e/( d*x + c)) + 120*b^4*c*e^6*e^(e/(d*x + c))/(d*x + c) - 240*b^4*c^2*e^6*e^(e /(d*x + c))/(d*x + c)^2 + 240*b^4*c^3*e^6*e^(e/(d*x + c))/(d*x + c)^3 - 12 0*b^4*c^4*e^6*e^(e/(d*x + c))/(d*x + c)^4 - 120*a*b^3*d*e^6*e^(e/(d*x + c) )/(d*x + c) + 480*a*b^3*c*d*e^6*e^(e/(d*x + c))/(d*x + c)^2 - 720*a*b^3*c^ 2*d*e^6*e^(e/(d*x + c))/(d*x + c)^3 + 480*a*b^3*c^3*d*e^6*e^(e/(d*x + c))/ (d*x + c)^4 - 240*a^2*b^2*d^2*e^6*e^(e/(d*x + c))/(d*x + c)^2 + 720*a^2*b^ 2*c*d^2*e^6*e^(e/(d*x + c))/(d*x + c)^3 - 720*a^2*b^2*c^2*d^2*e^6*e^(e/(d* x + c))/(d*x + c)^4 - 240*a^3*b*d^3*e^6*e^(e/(d*x + c))/(d*x + c)^3 + 480* a^3*b*c*d^3*e^6*e^(e/(d*x + c))/(d*x + c)^4 - 120*a^4*d^4*e^6*e^(e/(d*x + c))/(d*x + c)^4 - 6*b^4*e^7*e^(e/(d*x + c))/(d*x + c) + 40*b^4*c*e^7*e^...
Timed out. \[ \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx=\int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^4 \,d x \]