∫ e e ______ c+dx (a + bx)4 dx [401]

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

(-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*Ei(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)*(

________________________________________________________________________________________

Rubi [A]
time = 0.25, antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2258, 2237, 2241, 2245, 2250} \begin {gather*} -\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 {Ei}\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 {Ei}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {e (b c-a d)^4 \text {Ei}\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} \end {gather*}

((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)*

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(c + d*x)*(F^(a + b*(c + d*x)^n)/d), x]

- Dist[b*n*Log[F], Int[(c + d*x)^n*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2/n]

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[F^a*(ExpIntegralEi[

b*(c + d*x)^n*Log[F]]/(f*n)), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(c + d*x)^(m

+ 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))), x] - Dist[b*n*(Log[F]/(m + 1)), Int[(c + d*x)^(m + n)*F^(a + b*(c +

d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[-4, (m + 1)/n, 5] && IntegerQ[n

] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +

f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre

eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*

x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

\begin {align*} \int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx &=\int \left (\frac {(-b c+a d)^4 e^{\frac {e}{c+d x}}}{d^4}-\frac {4 b (b c-a d)^3 e^{\frac {e}{c+d x}} (c+d x)}{d^4}+\frac {6 b^2 (b c-a d)^2 e^{\frac {e}{c+d x}} (c+d x)^2}{d^4}-\frac {4 b^3 (b c-a d) e^{\frac {e}{c+d x}} (c+d x)^3}{d^4}+\frac {b^4 e^{\frac {e}{c+d x}} (c+d x)^4}{d^4}\right ) \, dx\\ &=\frac {b^4 \int e^{\frac {e}{c+d x}} (c+d x)^4 \, dx}{d^4}-\frac {\left (4 b^3 (b c-a d)\right ) \int e^{\frac {e}{c+d x}} (c+d x)^3 \, dx}{d^4}+\frac {\left (6 b^2 (b c-a d)^2\right ) \int e^{\frac {e}{c+d x}} (c+d x)^2 \, dx}{d^4}-\frac {\left (4 b (b c-a d)^3\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^4}+\frac {(b c-a d)^4 \int e^{\frac {e}{c+d x}} \, dx}{d^4}\\ &=\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^{\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^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}+\frac {\left (2 b^2 (b c-a d)^2 e\right ) \int e^{\frac {e}{c+d x}} (c+d x) \, dx}{d^4}-\frac {\left (2 b (b c-a d)^3 e\right ) \int e^{\frac {e}{c+d x}} \, dx}{d^4}+\frac {\left ((b c-a d)^4 e\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^4}\\ &=\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 {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 \text {Ei}\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}+\frac {\left (b^2 (b c-a d)^2 e^2\right ) \int e^{\frac {e}{c+d x}} \, dx}{d^4}-\frac {\left (2 b (b c-a d)^3 e^2\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^4}\\ &=\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 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}+\frac {2 b (b c-a d)^3 e^2 \text {Ei}\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}+\frac {\left (b^2 (b c-a d)^2 e^3\right ) \int \frac {e^{\frac {e}{c+d x}}}{c+d x} \, dx}{d^4}\\ &=\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 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}+\frac {2 b (b c-a d)^3 e^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {b^2 (b c-a d)^2 e^3 \text {Ei}\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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.37, size = 468, normalized size = 1.35 \begin {gather*} \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 ) \text {Ei}\left (\frac {e}{c+d x}\right )}{120 d^5} \end {gather*}

(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/(

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1145\) vs. \(2(347)=694\).
time = 0.12, size = 1146, normalized size = 3.31


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1146\)
default	\(\text {Expression too large to display}\)	\(1146\)
risch	\(\text {Expression too large to display}\)	\(1273\)

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

-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*(-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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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 - 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

________________________________________________________________________________________

Fricas [A]
time = 0.09, size = 626, normalized size = 1.81 \begin {gather*} -\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} + 120 \, a b^{3} d^{5} x^{4} + 240 \, a^{2} b^{2} d^{5} x^{3} + 240 \, a^{3} b d^{5} x^{2} + 120 \, a^{4} d^{5} x + 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} + {\left (b^{4} d x + b^{4} c\right )} e^{4} + {\left (b^{4} d^{2} x^{2} - 19 \, b^{4} c^{2} + 20 \, a b^{3} c d - 2 \, {\left (9 \, b^{4} c d - 10 \, a b^{3} d^{2}\right )} x\right )} e^{3} + 2 \, {\left (b^{4} d^{3} x^{3} + 51 \, b^{4} c^{3} - 110 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2} - {\left (7 \, b^{4} c d^{2} - 10 \, a b^{3} d^{3}\right )} x^{2} + {\left (43 \, b^{4} c^{2} d - 100 \, a b^{3} c d^{2} + 60 \, a^{2} b^{2} d^{3}\right )} x\right )} e^{2} + 2 \, {\left (3 \, b^{4} d^{4} x^{4} - 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} - 4 \, {\left (2 \, b^{4} c d^{3} - 5 \, a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (3 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} x^{2} - 12 \, {\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 )} x\right )} e\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, d^{5}} \end {gather*}

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

-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 + 120*a*b^3*d^5*x^4 + 240*a^2*b^2*d^5*x^3 + 240*a^3*b*d^5

*x^2 + 120*a^4*d^5*x + 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*d*x + b^4*c)*e^4 + (b^4*d^2*x^2 - 19*b^4*c^2 + 20*a*b^3*c*d - 2*(9*b^4*c*d - 10*a*b^3*d^2)*x)*e^3 + 2*(

b^4*d^3*x^3 + 51*b^4*c^3 - 110*a*b^3*c^2*d + 60*a^2*b^2*c*d^2 - (7*b^4*c*d^2 - 10*a*b^3*d^3)*x^2 + (43*b^4*c^2

*d - 100*a*b^3*c*d^2 + 60*a^2*b^2*d^3)*x)*e^2 + 2*(3*b^4*d^4*x^4 - 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 - 4*(2*b^4*c*d^3 - 5*a*b^3*d^4)*x^3 + 6*(3*b^4*c^2*d^2 - 10*a*b^3*c*d^3 + 10*a^2*b^2

*d^4)*x^2 - 12*(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)*x)*e)*e^(e/(d*x + c)))/d^5

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b x\right )^{4} e^{\frac {e}{c + d x}}\, dx \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (358) = 716\).
time = 1.07, size = 1427, normalized size = 4.12 \begin {gather*} -\frac {{\left (\frac {120 \, b^{4} c^{4} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{5}} - \frac {480 \, a b^{3} c^{3} d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{5}} + \frac {720 \, a^{2} b^{2} c^{2} d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{5}} - \frac {480 \, a^{3} b c d^{3} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{5}} + \frac {120 \, a^{4} d^{4} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{7}}{{\left (d x + c\right )}^{5}} - 24 \, b^{4} e^{\left (\frac {e}{d x + c} + 6\right )} + \frac {120 \, b^{4} c e^{\left (\frac {e}{d x + c} + 6\right )}}{d x + c} - \frac {240 \, b^{4} c^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{2}} + \frac {240 \, b^{4} c^{3} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} - \frac {120 \, b^{4} c^{4} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{4}} - \frac {120 \, a b^{3} d e^{\left (\frac {e}{d x + c} + 6\right )}}{d x + c} + \frac {480 \, a b^{3} c d e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{2}} - \frac {720 \, a b^{3} c^{2} d e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} + \frac {480 \, a b^{3} c^{3} d e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{4}} - \frac {240 \, a^{2} b^{2} d^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{2}} + \frac {720 \, a^{2} b^{2} c d^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} - \frac {720 \, a^{2} b^{2} c^{2} d^{2} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{4}} - \frac {240 \, a^{3} b d^{3} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{3}} + \frac {480 \, a^{3} b c d^{3} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{4}} - \frac {120 \, a^{4} d^{4} e^{\left (\frac {e}{d x + c} + 6\right )}}{{\left (d x + c\right )}^{4}} - \frac {240 \, b^{4} c^{3} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{8}}{{\left (d x + c\right )}^{5}} + \frac {720 \, a b^{3} c^{2} d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{8}}{{\left (d x + c\right )}^{5}} - \frac {720 \, a^{2} b^{2} c d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{8}}{{\left (d x + c\right )}^{5}} + \frac {240 \, a^{3} b d^{3} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{8}}{{\left (d x + c\right )}^{5}} - \frac {6 \, b^{4} e^{\left (\frac {e}{d x + c} + 7\right )}}{d x + c} + \frac {40 \, b^{4} c e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{2}} - \frac {120 \, b^{4} c^{2} e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{3}} + \frac {240 \, b^{4} c^{3} e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{4}} - \frac {40 \, a b^{3} d e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{2}} + \frac {240 \, a b^{3} c d e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{3}} - \frac {720 \, a b^{3} c^{2} d e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{4}} - \frac {120 \, a^{2} b^{2} d^{2} e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{3}} + \frac {720 \, a^{2} b^{2} c d^{2} e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{4}} - \frac {240 \, a^{3} b d^{3} e^{\left (\frac {e}{d x + c} + 7\right )}}{{\left (d x + c\right )}^{4}} + \frac {120 \, b^{4} c^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{9}}{{\left (d x + c\right )}^{5}} - \frac {240 \, a b^{3} c d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{9}}{{\left (d x + c\right )}^{5}} + \frac {120 \, a^{2} b^{2} d^{2} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{9}}{{\left (d x + c\right )}^{5}} - \frac {2 \, b^{4} e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{2}} + \frac {20 \, b^{4} c e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{3}} - \frac {120 \, b^{4} c^{2} e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{4}} - \frac {20 \, a b^{3} d e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{3}} + \frac {240 \, a b^{3} c d e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{4}} - \frac {120 \, a^{2} b^{2} d^{2} e^{\left (\frac {e}{d x + c} + 8\right )}}{{\left (d x + c\right )}^{4}} - \frac {20 \, b^{4} c {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{10}}{{\left (d x + c\right )}^{5}} + \frac {20 \, a b^{3} d {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{10}}{{\left (d x + c\right )}^{5}} - \frac {b^{4} e^{\left (\frac {e}{d x + c} + 9\right )}}{{\left (d x + c\right )}^{3}} + \frac {20 \, b^{4} c e^{\left (\frac {e}{d x + c} + 9\right )}}{{\left (d x + c\right )}^{4}} - \frac {20 \, a b^{3} d e^{\left (\frac {e}{d x + c} + 9\right )}}{{\left (d x + c\right )}^{4}} + \frac {b^{4} {\rm Ei}\left (\frac {e}{d x + c}\right ) e^{11}}{{\left (d x + c\right )}^{5}} - \frac {b^{4} e^{\left (\frac {e}{d x + c} + 10\right )}}{{\left (d x + c\right )}^{4}}\right )} {\left (d x + c\right )}^{5} e^{\left (-6\right )}}{120 \, d^{5}} \end {gather*}

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

-1/120*(120*b^4*c^4*Ei(e/(d*x + c))*e^7/(d*x + c)^5 - 480*a*b^3*c^3*d*Ei(e/(d*x + c))*e^7/(d*x + c)^5 + 720*a^

2*b^2*c^2*d^2*Ei(e/(d*x + c))*e^7/(d*x + c)^5 - 480*a^3*b*c*d^3*Ei(e/(d*x + c))*e^7/(d*x + c)^5 + 120*a^4*d^4*

Ei(e/(d*x + c))*e^7/(d*x + c)^5 - 24*b^4*e^(e/(d*x + c) + 6) + 120*b^4*c*e^(e/(d*x + c) + 6)/(d*x + c) - 240*b

^4*c^2*e^(e/(d*x + c) + 6)/(d*x + c)^2 + 240*b^4*c^3*e^(e/(d*x + c) + 6)/(d*x + c)^3 - 120*b^4*c^4*e^(e/(d*x +

c) + 6)/(d*x + c)^4 - 120*a*b^3*d*e^(e/(d*x + c) + 6)/(d*x + c) + 480*a*b^3*c*d*e^(e/(d*x + c) + 6)/(d*x + c)

^2 - 720*a*b^3*c^2*d*e^(e/(d*x + c) + 6)/(d*x + c)^3 + 480*a*b^3*c^3*d*e^(e/(d*x + c) + 6)/(d*x + c)^4 - 240*a

^2*b^2*d^2*e^(e/(d*x + c) + 6)/(d*x + c)^2 + 720*a^2*b^2*c*d^2*e^(e/(d*x + c) + 6)/(d*x + c)^3 - 720*a^2*b^2*c

^2*d^2*e^(e/(d*x + c) + 6)/(d*x + c)^4 - 240*a^3*b*d^3*e^(e/(d*x + c) + 6)/(d*x + c)^3 + 480*a^3*b*c*d^3*e^(e/

(d*x + c) + 6)/(d*x + c)^4 - 120*a^4*d^4*e^(e/(d*x + c) + 6)/(d*x + c)^4 - 240*b^4*c^3*Ei(e/(d*x + c))*e^8/(d*

x + c)^5 + 720*a*b^3*c^2*d*Ei(e/(d*x + c))*e^8/(d*x + c)^5 - 720*a^2*b^2*c*d^2*Ei(e/(d*x + c))*e^8/(d*x + c)^5

+ 240*a^3*b*d^3*Ei(e/(d*x + c))*e^8/(d*x + c)^5 - 6*b^4*e^(e/(d*x + c) + 7)/(d*x + c) + 40*b^4*c*e^(e/(d*x +

c) + 7)/(d*x + c)^2 - 120*b^4*c^2*e^(e/(d*x + c) + 7)/(d*x + c)^3 + 240*b^4*c^3*e^(e/(d*x + c) + 7)/(d*x + c)^

4 - 40*a*b^3*d*e^(e/(d*x + c) + 7)/(d*x + c)^2 + 240*a*b^3*c*d*e^(e/(d*x + c) + 7)/(d*x + c)^3 - 720*a*b^3*c^2

*d*e^(e/(d*x + c) + 7)/(d*x + c)^4 - 120*a^2*b^2*d^2*e^(e/(d*x + c) + 7)/(d*x + c)^3 + 720*a^2*b^2*c*d^2*e^(e/

(d*x + c) + 7)/(d*x + c)^4 - 240*a^3*b*d^3*e^(e/(d*x + c) + 7)/(d*x + c)^4 + 120*b^4*c^2*Ei(e/(d*x + c))*e^9/(

d*x + c)^5 - 240*a*b^3*c*d*Ei(e/(d*x + c))*e^9/(d*x + c)^5 + 120*a^2*b^2*d^2*Ei(e/(d*x + c))*e^9/(d*x + c)^5 -

2*b^4*e^(e/(d*x + c) + 8)/(d*x + c)^2 + 20*b^4*c*e^(e/(d*x + c) + 8)/(d*x + c)^3 - 120*b^4*c^2*e^(e/(d*x + c)

+ 8)/(d*x + c)^4 - 20*a*b^3*d*e^(e/(d*x + c) + 8)/(d*x + c)^3 + 240*a*b^3*c*d*e^(e/(d*x + c) + 8)/(d*x + c)^4

- 120*a^2*b^2*d^2*e^(e/(d*x + c) + 8)/(d*x + c)^4 - 20*b^4*c*Ei(e/(d*x + c))*e^10/(d*x + c)^5 + 20*a*b^3*d*Ei

(e/(d*x + c))*e^10/(d*x + c)^5 - b^4*e^(e/(d*x + c) + 9)/(d*x + c)^3 + 20*b^4*c*e^(e/(d*x + c) + 9)/(d*x + c)^

4 - 20*a*b^3*d*e^(e/(d*x + c) + 9)/(d*x + c)^4 + b^4*Ei(e/(d*x + c))*e^11/(d*x + c)^5 - b^4*e^(e/(d*x + c) + 1

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {e}}^{\frac {e}{c+d\,x}}\,{\left (a+b\,x\right )}^4 \,d x \end {gather*}

________________________________________________________________________________________

3.5.1 \(\int e^{\frac {e}{c+d x}} (a+b x)^4 \, dx\) [401]