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

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

[Out]

(-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)*(
d*x+c)^4*Ei(5,-e/(d*x+c))/d^5

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Rubi [A]  time = 0.36, 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 = {2226, 2206, 2210, 2214, 2218} \[ -\frac {4 b^3 e^4 (b c-a d) \text {Gamma}\left (-4,-\frac {e}{c+d x}\right )}{d^5}-\frac {b^4 e^5 \text {Gamma}\left (-5,-\frac {e}{c+d x}\right )}{d^5}-\frac {b^2 e^3 (b c-a d)^2 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}+\frac {b^2 e^2 (c+d x) (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {b^2 e (c+d x)^2 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {2 b^2 (c+d x)^3 (b c-a d)^2 e^{\frac {e}{c+d x}}}{d^5}+\frac {2 b e^2 (b c-a d)^3 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {e (b c-a d)^4 \text {Ei}\left (\frac {e}{c+d x}\right )}{d^5}-\frac {2 b e (c+d x) (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^5}-\frac {2 b (c+d x)^2 (b c-a d)^3 e^{\frac {e}{c+d x}}}{d^5}+\frac {(c+d x) (b c-a d)^4 e^{\frac {e}{c+d x}}}{d^5} \]

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*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

Rule 2206

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]
&& ILtQ[n, 0]

Rule 2210

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]

Rule 2214

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]))

Rule 2218

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 2226

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]

Rubi steps

\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*}

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Mathematica [A]  time = 0.50, size = 468, normalized size = 1.35 \[ \frac {d x e^{\frac {e}{c+d x}} \left (120 a^4 d^4+240 a^3 b d^3 (d x+e)+120 a^2 b^2 d^2 \left (-4 c e+2 d^2 x^2+d e x+e^2\right )+20 a b^3 d \left (18 c^2 e-2 c e (3 d x+5 e)+6 d^3 x^3+2 d^2 e x^2+d e^2 x+e^3\right )+b^4 \left (-96 c^3 e+2 c^2 e (18 d x+43 e)-2 c e \left (8 d^2 x^2+7 d e x+9 e^2\right )+24 d^4 x^4+6 d^3 e x^3+2 d^2 e^2 x^2+d e^3 x+e^4\right )\right )-e \left (120 a^4 d^4-240 a^3 b d^3 (2 c-e)+120 a^2 b^2 d^2 \left (6 c^2-6 c e+e^2\right )-20 a b^3 d \left (24 c^3-36 c^2 e+12 c e^2-e^3\right )+b^4 \left (120 c^4-240 c^3 e+120 c^2 e^2-20 c e^3+e^4\right )\right ) \text {Ei}\left (\frac {e}{c+d x}\right )}{120 d^5}+\frac {c e^{\frac {e}{c+d x}} \left (120 a^4 d^4-240 a^3 b d^3 (c-e)+120 a^2 b^2 d^2 \left (2 c^2-5 c e+e^2\right )-20 a b^3 d \left (6 c^3-26 c^2 e+11 c e^2-e^3\right )+b^4 \left (24 c^4-154 c^3 e+102 c^2 e^2-19 c e^3+e^4\right )\right )}{120 d^5} \]

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*(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)

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fricas [B]  time = 0.45, size = 638, normalized size = 1.84 \[ -\frac {{\left (b^{4} e^{5} - 20 \, {\left (b^{4} c - a b^{3} d\right )} e^{4} + 120 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} e^{3} - 240 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} e^{2} + 120 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} e\right )} {\rm Ei}\left (\frac {e}{d x + c}\right ) - {\left (24 \, b^{4} d^{5} x^{5} + 24 \, b^{4} c^{5} - 120 \, a b^{3} c^{4} d + 240 \, a^{2} b^{2} c^{3} d^{2} - 240 \, a^{3} b c^{2} d^{3} + 120 \, a^{4} c d^{4} + b^{4} c e^{4} + 6 \, {\left (20 \, a b^{3} d^{5} + b^{4} d^{4} e\right )} x^{4} - {\left (19 \, b^{4} c^{2} - 20 \, a b^{3} c d\right )} e^{3} + 2 \, {\left (120 \, a^{2} b^{2} d^{5} + b^{4} d^{3} e^{2} - 4 \, {\left (2 \, b^{4} c d^{3} - 5 \, a b^{3} d^{4}\right )} e\right )} x^{3} + 2 \, {\left (51 \, b^{4} c^{3} - 110 \, a b^{3} c^{2} d + 60 \, a^{2} b^{2} c d^{2}\right )} e^{2} + {\left (240 \, a^{3} b d^{5} + b^{4} d^{2} e^{3} - 2 \, {\left (7 \, b^{4} c d^{2} - 10 \, a b^{3} d^{3}\right )} e^{2} + 12 \, {\left (3 \, b^{4} c^{2} d^{2} - 10 \, a b^{3} c d^{3} + 10 \, a^{2} b^{2} d^{4}\right )} e\right )} x^{2} - 2 \, {\left (77 \, b^{4} c^{4} - 260 \, a b^{3} c^{3} d + 300 \, a^{2} b^{2} c^{2} d^{2} - 120 \, a^{3} b c d^{3}\right )} e + {\left (120 \, a^{4} d^{5} + b^{4} d e^{4} - 2 \, {\left (9 \, b^{4} c d - 10 \, a b^{3} d^{2}\right )} e^{3} + 2 \, {\left (43 \, b^{4} c^{2} d - 100 \, a b^{3} c d^{2} + 60 \, a^{2} b^{2} d^{3}\right )} e^{2} - 24 \, {\left (4 \, b^{4} c^{3} d - 15 \, a b^{3} c^{2} d^{2} + 20 \, a^{2} b^{2} c d^{3} - 10 \, a^{3} b d^{4}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))*(b*x+a)^4,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)*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 - 24
0*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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giac [B]  time = 0.75, size = 1427, normalized size = 4.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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
0)/(d*x + c)^4)*(d*x + c)^5*e^(-6)/d^5

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maple [B]  time = 0.02, size = 1146, normalized size = 3.31 \[ \text {result too large to display} \]

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*c^4/d^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*c/d*a^3*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))+6*b^2*c^2/d^2*a^
2*(-(d*x+c)/e*exp(e/(d*x+c))-Ei(1,-e/(d*x+c)))-4*b^3*c^3/d^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 = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (24 \, b^{4} d^{4} x^{5} + 6 \, {\left (20 \, a b^{3} d^{4} + b^{4} d^{3} e\right )} x^{4} + 2 \, {\left (120 \, a^{2} b^{2} d^{4} + 20 \, a b^{3} d^{3} e - {\left (8 \, c d^{2} e - d^{2} e^{2}\right )} b^{4}\right )} x^{3} + {\left (240 \, a^{3} b d^{4} + 120 \, a^{2} b^{2} d^{3} e - 20 \, {\left (6 \, c d^{2} e - d^{2} e^{2}\right )} a b^{3} + {\left (36 \, c^{2} d e - 14 \, c d e^{2} + d e^{3}\right )} b^{4}\right )} x^{2} + {\left (120 \, a^{4} d^{4} + 240 \, a^{3} b d^{3} e - 120 \, {\left (4 \, c d^{2} e - d^{2} e^{2}\right )} a^{2} b^{2} + 20 \, {\left (18 \, c^{2} d e - 10 \, c d e^{2} + d e^{3}\right )} a b^{3} - {\left (96 \, c^{3} e - 86 \, c^{2} e^{2} + 18 \, c e^{3} - e^{4}\right )} b^{4}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, d^{4}} + \int -\frac {{\left (240 \, a^{3} b c^{2} d^{3} e - 120 \, {\left (4 \, c^{3} d^{2} e - c^{2} d^{2} e^{2}\right )} a^{2} b^{2} + 20 \, {\left (18 \, c^{4} d e - 10 \, c^{3} d e^{2} + c^{2} d e^{3}\right )} a b^{3} - {\left (96 \, c^{5} e - 86 \, c^{4} e^{2} + 18 \, c^{3} e^{3} - c^{2} e^{4}\right )} b^{4} - {\left (120 \, a^{4} d^{5} e - 240 \, {\left (2 \, c d^{4} e - d^{4} e^{2}\right )} a^{3} b + 120 \, {\left (6 \, c^{2} d^{3} e - 6 \, c d^{3} e^{2} + d^{3} e^{3}\right )} a^{2} b^{2} - 20 \, {\left (24 \, c^{3} d^{2} e - 36 \, c^{2} d^{2} e^{2} + 12 \, c d^{2} e^{3} - d^{2} e^{4}\right )} a b^{3} + {\left (120 \, c^{4} d e - 240 \, c^{3} d e^{2} + 120 \, c^{2} d e^{3} - 20 \, c d e^{4} + d e^{5}\right )} b^{4}\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{120 \, {\left (d^{6} x^{2} + 2 \, c d^{5} x + c^{2} d^{4}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x))*(a + b*x)^4,x)

[Out]

int(exp(e/(c + d*x))*(a + b*x)^4, x)

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

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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