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

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

[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)*ExpIntegralEi[-((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)

________________________________________________________________________________________

Rubi [A]  time = 0.684236, 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, Rules used = {2199, 2177, 2178} \[ \frac{b^4 (b c-a d)^4 e^{\frac{b c}{d}-a} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{24 d^9}+\frac{2 b^4 (b c-a d)^3 e^{\frac{b c}{d}-a} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{3 d^8}+\frac{3 b^4 (b c-a d)^2 e^{\frac{b c}{d}-a} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^7}+\frac{4 b^4 (b c-a d) e^{\frac{b c}{d}-a} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{b^4 e^{\frac{b c}{d}-a} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^3 e^{-a-b x} (b c-a d)^4}{24 d^8 (c+d x)}+\frac{2 b^3 e^{-a-b x} (b c-a d)^3}{3 d^7 (c+d x)}+\frac{3 b^3 e^{-a-b x} (b c-a d)^2}{d^6 (c+d x)}+\frac{4 b^3 e^{-a-b x} (b c-a d)}{d^5 (c+d x)}-\frac{b^2 e^{-a-b x} (b c-a d)^4}{24 d^7 (c+d x)^2}-\frac{2 b^2 e^{-a-b x} (b c-a d)^3}{3 d^6 (c+d x)^2}-\frac{3 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)^2}+\frac{b e^{-a-b x} (b c-a d)^4}{12 d^6 (c+d x)^3}+\frac{4 b e^{-a-b x} (b c-a d)^3}{3 d^5 (c+d x)^3}-\frac{e^{-a-b x} (b c-a d)^4}{4 d^5 (c+d x)^4} \]

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)*ExpIntegralEi[-((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)

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx &=\int \left (\frac{(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)^5}-\frac{4 b (b c-a d)^3 e^{-a-b x}}{d^4 (c+d x)^4}+\frac{6 b^2 (b c-a d)^2 e^{-a-b x}}{d^4 (c+d x)^3}-\frac{4 b^3 (b c-a d) e^{-a-b x}}{d^4 (c+d x)^2}+\frac{b^4 e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx\\ &=\frac{b^4 \int \frac{e^{-a-b x}}{c+d x} \, dx}{d^4}-\frac{\left (4 b^3 (b c-a d)\right ) \int \frac{e^{-a-b x}}{(c+d x)^2} \, dx}{d^4}+\frac{\left (6 b^2 (b c-a d)^2\right ) \int \frac{e^{-a-b x}}{(c+d x)^3} \, dx}{d^4}-\frac{\left (4 b (b c-a d)^3\right ) \int \frac{e^{-a-b x}}{(c+d x)^4} \, dx}{d^4}+\frac{(b c-a d)^4 \int \frac{e^{-a-b x}}{(c+d x)^5} \, dx}{d^4}\\ &=-\frac{(b c-a d)^4 e^{-a-b x}}{4 d^5 (c+d x)^4}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{3 d^5 (c+d x)^3}-\frac{3 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)^2}+\frac{4 b^3 (b c-a d) e^{-a-b x}}{d^5 (c+d x)}+\frac{b^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{\left (4 b^4 (b c-a d)\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{d^5}-\frac{\left (3 b^3 (b c-a d)^2\right ) \int \frac{e^{-a-b x}}{(c+d x)^2} \, dx}{d^5}+\frac{\left (4 b^2 (b c-a d)^3\right ) \int \frac{e^{-a-b x}}{(c+d x)^3} \, dx}{3 d^5}-\frac{\left (b (b c-a d)^4\right ) \int \frac{e^{-a-b x}}{(c+d x)^4} \, dx}{4 d^5}\\ &=-\frac{(b c-a d)^4 e^{-a-b x}}{4 d^5 (c+d x)^4}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac{b (b c-a d)^4 e^{-a-b x}}{12 d^6 (c+d x)^3}-\frac{3 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)^2}-\frac{2 b^2 (b c-a d)^3 e^{-a-b x}}{3 d^6 (c+d x)^2}+\frac{4 b^3 (b c-a d) e^{-a-b x}}{d^5 (c+d x)}+\frac{3 b^3 (b c-a d)^2 e^{-a-b x}}{d^6 (c+d x)}+\frac{b^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{4 b^4 (b c-a d) e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{\left (3 b^4 (b c-a d)^2\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{d^6}-\frac{\left (2 b^3 (b c-a d)^3\right ) \int \frac{e^{-a-b x}}{(c+d x)^2} \, dx}{3 d^6}+\frac{\left (b^2 (b c-a d)^4\right ) \int \frac{e^{-a-b x}}{(c+d x)^3} \, dx}{12 d^6}\\ &=-\frac{(b c-a d)^4 e^{-a-b x}}{4 d^5 (c+d x)^4}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac{b (b c-a d)^4 e^{-a-b x}}{12 d^6 (c+d x)^3}-\frac{3 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)^2}-\frac{2 b^2 (b c-a d)^3 e^{-a-b x}}{3 d^6 (c+d x)^2}-\frac{b^2 (b c-a d)^4 e^{-a-b x}}{24 d^7 (c+d x)^2}+\frac{4 b^3 (b c-a d) e^{-a-b x}}{d^5 (c+d x)}+\frac{3 b^3 (b c-a d)^2 e^{-a-b x}}{d^6 (c+d x)}+\frac{2 b^3 (b c-a d)^3 e^{-a-b x}}{3 d^7 (c+d x)}+\frac{b^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{4 b^4 (b c-a d) e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{3 b^4 (b c-a d)^2 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^7}+\frac{\left (2 b^4 (b c-a d)^3\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{3 d^7}-\frac{\left (b^3 (b c-a d)^4\right ) \int \frac{e^{-a-b x}}{(c+d x)^2} \, dx}{24 d^7}\\ &=-\frac{(b c-a d)^4 e^{-a-b x}}{4 d^5 (c+d x)^4}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac{b (b c-a d)^4 e^{-a-b x}}{12 d^6 (c+d x)^3}-\frac{3 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)^2}-\frac{2 b^2 (b c-a d)^3 e^{-a-b x}}{3 d^6 (c+d x)^2}-\frac{b^2 (b c-a d)^4 e^{-a-b x}}{24 d^7 (c+d x)^2}+\frac{4 b^3 (b c-a d) e^{-a-b x}}{d^5 (c+d x)}+\frac{3 b^3 (b c-a d)^2 e^{-a-b x}}{d^6 (c+d x)}+\frac{2 b^3 (b c-a d)^3 e^{-a-b x}}{3 d^7 (c+d x)}+\frac{b^3 (b c-a d)^4 e^{-a-b x}}{24 d^8 (c+d x)}+\frac{b^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{4 b^4 (b c-a d) e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{3 b^4 (b c-a d)^2 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^7}+\frac{2 b^4 (b c-a d)^3 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{3 d^8}+\frac{\left (b^4 (b c-a d)^4\right ) \int \frac{e^{-a-b x}}{c+d x} \, dx}{24 d^8}\\ &=-\frac{(b c-a d)^4 e^{-a-b x}}{4 d^5 (c+d x)^4}+\frac{4 b (b c-a d)^3 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac{b (b c-a d)^4 e^{-a-b x}}{12 d^6 (c+d x)^3}-\frac{3 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)^2}-\frac{2 b^2 (b c-a d)^3 e^{-a-b x}}{3 d^6 (c+d x)^2}-\frac{b^2 (b c-a d)^4 e^{-a-b x}}{24 d^7 (c+d x)^2}+\frac{4 b^3 (b c-a d) e^{-a-b x}}{d^5 (c+d x)}+\frac{3 b^3 (b c-a d)^2 e^{-a-b x}}{d^6 (c+d x)}+\frac{2 b^3 (b c-a d)^3 e^{-a-b x}}{3 d^7 (c+d x)}+\frac{b^3 (b c-a d)^4 e^{-a-b x}}{24 d^8 (c+d x)}+\frac{b^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{4 b^4 (b c-a d) e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{3 b^4 (b c-a d)^2 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^7}+\frac{2 b^4 (b c-a d)^3 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{3 d^8}+\frac{b^4 (b c-a d)^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{24 d^9}\\ \end{align*}

Mathematica [A]  time = 0.725518, size = 669, normalized size = 1.2 \[ \frac{e^{-a} \left (6 a^2 b^6 c^2 d^2 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+\frac{d e^{-b x} \left (-b^2 d (c+d x)^2 \left (\left (a^2-16 a+72\right ) d^2-2 (a-8) b c d+b^2 c^2\right ) (b c-a d)^2+b^3 (c+d x)^3 \left (6 \left (a^2-8 a+12\right ) b^2 c^2 d^2-4 \left (a^3-12 a^2+36 a-24\right ) b c d^3+a \left (a^3-16 a^2+72 a-96\right ) d^4-4 (a-4) b^3 c^3 d+b^4 c^4\right )+2 b d^2 (c+d x) (b c-(a-16) d) (b c-a d)^3-6 d^3 (b c-a d)^4\right )}{(c+d x)^4}-4 a^3 b^5 c d^3 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+48 a^2 b^5 c d^3 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+a^4 b^4 d^4 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )-16 a^3 b^4 d^4 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+72 a^2 b^4 d^4 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )-48 a b^6 c^2 d^2 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )-4 a b^7 c^3 d e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )-144 a b^5 c d^3 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )-96 a b^4 d^4 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+72 b^6 c^2 d^2 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+b^8 c^4 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+16 b^7 c^3 d e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+96 b^5 c d^3 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+24 b^4 d^4 e^{\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )\right )}{24 d^9} \]

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)*ExpInt
egralEi[-((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)*ExpIntegralEi[-((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)] - 96*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)*ExpIntegralEi[-((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 = 0.013, size = 596, normalized size = 1.1 \begin{align*} -{\frac{1}{b} \left ( -6\,{\frac{ \left ( ad-bc \right ) ^{2}{b}^{5}}{{d}^{7}} \left ( -1/2\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-2}}-1/2\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-1/2\,{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) \right ) }+{\frac{{b}^{5}}{{d}^{5}}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) }-{\frac{ \left ( ad-bc \right ) ^{4}{b}^{5}}{{d}^{9}} \left ( -{\frac{{{\rm e}^{-bx-a}}}{4} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-4}}-{\frac{{{\rm e}^{-bx-a}}}{12} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-3}}-{\frac{{{\rm e}^{-bx-a}}}{24} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-2}}-{\frac{{{\rm e}^{-bx-a}}}{24} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{\frac{1}{24}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) } \right ) }+4\,{\frac{ \left ( ad-bc \right ) ^{3}{b}^{5}}{{d}^{8}} \left ( -1/3\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-3}}-1/6\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-2}}-1/6\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-1/6\,{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) \right ) }+4\,{\frac{ \left ( ad-bc \right ){b}^{5}}{{d}^{6}} \left ( -{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) \right ) } \right ) } \end{align*}

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*(-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))+b^5/d^5*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))+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))+4*(a*d-b*c)/d^6*b^5*(-exp(-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)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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 )} E_{5}\left (\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} \end{align*}

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, 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)*exp_integra
l_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 [B]  time = 1.64307, size = 2442, normalized size = 4.38 \begin{align*} \text{result too large to display} \end{align*}

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, 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 + 1
2)*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 - 12*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 + 24*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. \begin{align*} \text{Timed out} \end{align*}

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 [B]  time = 1.51575, size = 6118, normalized size = 10.98 \begin{align*} \text{result too large to display} \end{align*}

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, algorithm="giac")

[Out]

1/24*(b^8*c^4*d^4*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^7*c^3*d^5*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b
*c/d) + 6*a^2*b^6*c^2*d^6*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^5*c*d^7*x^4*Ei(-(b*d*x + b*c)/d)*e
^(-a + b*c/d) + a^4*b^4*d^8*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 4*b^8*c^5*d^3*x^3*Ei(-(b*d*x + b*c)/d)*e
^(-a + b*c/d) - 16*a*b^7*c^4*d^4*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 24*a^2*b^6*c^3*d^5*x^3*Ei(-(b*d*x +
b*c)/d)*e^(-a + b*c/d) - 16*a^3*b^5*c^2*d^6*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 4*a^4*b^4*c*d^7*x^3*Ei(
-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 16*b^7*c^3*d^5*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 48*a*b^6*c^2*d^6*x
^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 48*a^2*b^5*c*d^7*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 16*a^3*b^4
*d^8*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 6*b^8*c^6*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 24*a*b^
7*c^5*d^3*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 36*a^2*b^6*c^4*d^4*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d)
- 24*a^3*b^5*c^3*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 6*a^4*b^4*c^2*d^6*x^2*Ei(-(b*d*x + b*c)/d)*e^(
-a + b*c/d) + 64*b^7*c^4*d^4*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 192*a*b^6*c^3*d^5*x^3*Ei(-(b*d*x + b*c)
/d)*e^(-a + b*c/d) + 192*a^2*b^5*c^2*d^6*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 64*a^3*b^4*c*d^7*x^3*Ei(-(b
*d*x + b*c)/d)*e^(-a + b*c/d) + 72*b^6*c^2*d^6*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 144*a*b^5*c*d^7*x^4*E
i(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 72*a^2*b^4*d^8*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^7*c^4*d^4*x^3*
e^(-b*x - a) - 4*a*b^6*c^3*d^5*x^3*e^(-b*x - a) + 6*a^2*b^5*c^2*d^6*x^3*e^(-b*x - a) - 4*a^3*b^4*c*d^7*x^3*e^(
-b*x - a) + a^4*b^3*d^8*x^3*e^(-b*x - a) + 4*b^8*c^7*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 16*a*b^7*c^6*d^
2*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 24*a^2*b^6*c^5*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 16*a^3*b^
5*c^4*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 4*a^4*b^4*c^3*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 96
*b^7*c^5*d^3*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 288*a*b^6*c^4*d^4*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/
d) + 288*a^2*b^5*c^3*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 96*a^3*b^4*c^2*d^6*x^2*Ei(-(b*d*x + b*c)/d)
*e^(-a + b*c/d) + 288*b^6*c^3*d^5*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 576*a*b^5*c^2*d^6*x^3*Ei(-(b*d*x +
b*c)/d)*e^(-a + b*c/d) + 288*a^2*b^4*c*d^7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 96*b^5*c*d^7*x^4*Ei(-(b*
d*x + b*c)/d)*e^(-a + b*c/d) - 96*a*b^4*d^8*x^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*b^7*c^5*d^3*x^2*e^(-b*
x - a) - 12*a*b^6*c^4*d^4*x^2*e^(-b*x - a) + 18*a^2*b^5*c^3*d^5*x^2*e^(-b*x - a) - 12*a^3*b^4*c^2*d^6*x^2*e^(-
b*x - a) + 3*a^4*b^3*c*d^7*x^2*e^(-b*x - a) + 16*b^6*c^3*d^5*x^3*e^(-b*x - a) - 48*a*b^5*c^2*d^6*x^3*e^(-b*x -
a) + 48*a^2*b^4*c*d^7*x^3*e^(-b*x - a) - 16*a^3*b^3*d^8*x^3*e^(-b*x - a) + b^8*c^8*Ei(-(b*d*x + b*c)/d)*e^(-a
+ b*c/d) - 4*a*b^7*c^7*d*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 6*a^2*b^6*c^6*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a +
b*c/d) - 4*a^3*b^5*c^5*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + a^4*b^4*c^4*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a +
b*c/d) + 64*b^7*c^6*d^2*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 192*a*b^6*c^5*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-
a + b*c/d) + 192*a^2*b^5*c^4*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 64*a^3*b^4*c^3*d^5*x*Ei(-(b*d*x + b*c
)/d)*e^(-a + b*c/d) + 432*b^6*c^4*d^4*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 864*a*b^5*c^3*d^5*x^2*Ei(-(b*d
*x + b*c)/d)*e^(-a + b*c/d) + 432*a^2*b^4*c^2*d^6*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 384*b^5*c^2*d^6*x^
3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 384*a*b^4*c*d^7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 24*b^4*d^8*x
^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*b^7*c^6*d^2*x*e^(-b*x - a) - 12*a*b^6*c^5*d^3*x*e^(-b*x - a) + 18*a
^2*b^5*c^4*d^4*x*e^(-b*x - a) - 12*a^3*b^4*c^3*d^5*x*e^(-b*x - a) + 3*a^4*b^3*c^2*d^6*x*e^(-b*x - a) + 47*b^6*
c^4*d^4*x^2*e^(-b*x - a) - 140*a*b^5*c^3*d^5*x^2*e^(-b*x - a) + 138*a^2*b^4*c^2*d^6*x^2*e^(-b*x - a) - 44*a^3*
b^3*c*d^7*x^2*e^(-b*x - a) - a^4*b^2*d^8*x^2*e^(-b*x - a) + 72*b^5*c^2*d^6*x^3*e^(-b*x - a) - 144*a*b^4*c*d^7*
x^3*e^(-b*x - a) + 72*a^2*b^3*d^8*x^3*e^(-b*x - a) + 16*b^7*c^7*d*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 48*a*b
^6*c^6*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 48*a^2*b^5*c^5*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 16*a
^3*b^4*c^4*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 288*b^6*c^5*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 5
76*a*b^5*c^4*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 288*a^2*b^4*c^3*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*
c/d) + 576*b^5*c^3*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 576*a*b^4*c^2*d^6*x^2*Ei(-(b*d*x + b*c)/d)*e^
(-a + b*c/d) + 96*b^4*c*d^7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^7*c^7*d*e^(-b*x - a) - 4*a*b^6*c^6*d^2
*e^(-b*x - a) + 6*a^2*b^5*c^5*d^3*e^(-b*x - a) - 4*a^3*b^4*c^4*d^4*e^(-b*x - a) + a^4*b^3*c^3*d^5*e^(-b*x - a)
+ 46*b^6*c^5*d^3*x*e^(-b*x - a) - 136*a*b^5*c^4*d^4*x*e^(-b*x - a) + 132*a^2*b^4*c^3*d^5*x*e^(-b*x - a) - 40*
a^3*b^3*c^2*d^6*x*e^(-b*x - a) - 2*a^4*b^2*c*d^7*x*e^(-b*x - a) + 200*b^5*c^3*d^5*x^2*e^(-b*x - a) - 384*a*b^4
*c^2*d^6*x^2*e^(-b*x - a) + 168*a^2*b^3*c*d^7*x^2*e^(-b*x - a) + 16*a^3*b^2*d^8*x^2*e^(-b*x - a) + 96*b^4*c*d^
7*x^3*e^(-b*x - a) - 96*a*b^3*d^8*x^3*e^(-b*x - a) + 72*b^6*c^6*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 144*
a*b^5*c^5*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 72*a^2*b^4*c^4*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3
84*b^5*c^4*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 384*a*b^4*c^3*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d)
+ 144*b^4*c^2*d^6*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 15*b^6*c^6*d^2*e^(-b*x - a) - 44*a*b^5*c^5*d^3*e^
(-b*x - a) + 42*a^2*b^4*c^4*d^4*e^(-b*x - a) - 12*a^3*b^3*c^3*d^5*e^(-b*x - a) - a^4*b^2*c^2*d^6*e^(-b*x - a)
+ 186*b^5*c^4*d^4*x*e^(-b*x - a) - 344*a*b^4*c^3*d^5*x*e^(-b*x - a) + 132*a^2*b^3*c^2*d^6*x*e^(-b*x - a) + 24*
a^3*b^2*c*d^7*x*e^(-b*x - a) + 2*a^4*b*d^8*x*e^(-b*x - a) + 216*b^4*c^2*d^6*x^2*e^(-b*x - a) - 144*a*b^3*c*d^7
*x^2*e^(-b*x - a) - 72*a^2*b^2*d^8*x^2*e^(-b*x - a) + 96*b^5*c^5*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 96*
a*b^4*c^4*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 96*b^4*c^3*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 58*
b^5*c^5*d^3*e^(-b*x - a) - 104*a*b^4*c^4*d^4*e^(-b*x - a) + 36*a^2*b^3*c^3*d^5*e^(-b*x - a) + 8*a^3*b^2*c^2*d^
6*e^(-b*x - a) + 2*a^4*b*c*d^7*e^(-b*x - a) + 176*b^4*c^3*d^5*x*e^(-b*x - a) - 96*a*b^3*c^2*d^6*x*e^(-b*x - a)
- 48*a^2*b^2*c*d^7*x*e^(-b*x - a) - 32*a^3*b*d^8*x*e^(-b*x - a) + 24*b^4*c^4*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a +
b*c/d) + 50*b^4*c^4*d^4*e^(-b*x - a) - 24*a*b^3*c^3*d^5*e^(-b*x - a) - 12*a^2*b^2*c^2*d^6*e^(-b*x - a) - 8*a^
3*b*c*d^7*e^(-b*x - a) - 6*a^4*d^8*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)