3.1.82 \(\int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^5} \, dx\) [82]

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

[Out]

-1/4*(-a*d+b*c)^4*exp(-b*x-a)/d^5/(d*x+c)^4+4/3*b*(-a*d+b*c)^3*exp(-b*x-a)/d^5/(d*x+c)^3+1/12*b*(-a*d+b*c)^4*e
xp(-b*x-a)/d^6/(d*x+c)^3-3*b^2*(-a*d+b*c)^2*exp(-b*x-a)/d^5/(d*x+c)^2-2/3*b^2*(-a*d+b*c)^3*exp(-b*x-a)/d^6/(d*
x+c)^2-1/24*b^2*(-a*d+b*c)^4*exp(-b*x-a)/d^7/(d*x+c)^2+4*b^3*(-a*d+b*c)*exp(-b*x-a)/d^5/(d*x+c)+3*b^3*(-a*d+b*
c)^2*exp(-b*x-a)/d^6/(d*x+c)+2/3*b^3*(-a*d+b*c)^3*exp(-b*x-a)/d^7/(d*x+c)+1/24*b^3*(-a*d+b*c)^4*exp(-b*x-a)/d^
8/(d*x+c)+b^4*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^5+4*b^4*(-a*d+b*c)*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^6+3*b^4*(-a
*d+b*c)^2*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^7+2/3*b^4*(-a*d+b*c)^3*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^8+1/24*b^4*
(-a*d+b*c)^4*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^9

________________________________________________________________________________________

Rubi [A]
time = 0.48, antiderivative size = 557, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2230, 2208, 2209} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^5,x]

[Out]

-1/4*((b*c - a*d)^4*E^(-a - b*x))/(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 2208

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] &&  !TrueQ[$UseGamm
a]

Rule 2209

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

Rule 2230

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] &&  !TrueQ[$UseGamma]

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 = 2.28, size = 669, normalized size = 1.20 \begin {gather*} \frac {e^{-a} \left (\frac {d e^{-b x} \left (-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 \left (b^2 c^2-2 (-8+a) b c d+\left (72-16 a+a^2\right ) d^2\right ) (c+d x)^2+b^3 \left (b^4 c^4-4 (-4+a) b^3 c^3 d+6 \left (12-8 a+a^2\right ) b^2 c^2 d^2-4 \left (-24+36 a-12 a^2+a^3\right ) b c d^3+a \left (-96+72 a-16 a^2+a^3\right ) d^4\right ) (c+d x)^3\right )}{(c+d x)^4}+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 )-4 a b^7 c^3 d 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 )-48 a b^6 c^2 d^2 e^{\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )+6 a^2 b^6 c^2 d^2 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 )-144 a 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 )-4 a^3 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 )-96 a 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 )-16 a^3 b^4 d^4 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 )\right )}{24 d^9} \end {gather*}

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)

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 596, normalized size = 1.07

method	result	size
derivativedivides	\(-\frac {\frac {4 b^{5} \left (a d -c b \right )^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{8}}-\frac {6 b^{5} \left (a d -c b \right )^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {b^{5} {\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{d^{5}}+\frac {4 b^{5} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {b^{5} \left (a d -c b \right )^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{9}}}{b}\)	\(596\)
default	\(-\frac {\frac {4 b^{5} \left (a d -c b \right )^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{8}}-\frac {6 b^{5} \left (a d -c b \right )^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {b^{5} {\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{d^{5}}+\frac {4 b^{5} \left (a d -c b \right ) \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {b^{5} \left (a d -c b \right )^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{4 \left (-b x -a +\frac {a d -c b}{d}\right )^{4}}-\frac {{\mathrm e}^{-b x -a}}{12 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{24 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \expIntegral \left (1, b x +a -\frac {a d -c b}{d}\right )}{24}\right )}{d^{9}}}{b}\)	\(596\)
risch	\(\text {Expression too large to display}\)	\(2054\)

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,method=_RETURNVERBOSE)

[Out]

-1/b*(4*b^5*(a*d-b*c)^3/d^8*(-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))-6*b^5*(a*d-b*c)^2/d^7*(-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)+4*b^5*(a*d-b*c)/d^6*(-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))-b^5*(a*d-b*c)^4/d^9*(-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)))

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (524) = 1048\).
time = 0.38, size = 1084, normalized size = 1.95 \begin {gather*} \frac {{\left (b^{8} c^{8} - 4 \, {\left (a - 4\right )} b^{7} c^{7} d + 6 \, {\left (a^{2} - 8 \, a + 12\right )} b^{6} c^{6} d^{2} - 4 \, {\left (a^{3} - 12 \, a^{2} + 36 \, a - 24\right )} b^{5} c^{5} d^{3} + {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2} - 96 \, a + 24\right )} b^{4} c^{4} d^{4} + {\left (b^{8} c^{4} d^{4} - 4 \, {\left (a - 4\right )} b^{7} c^{3} d^{5} + 6 \, {\left (a^{2} - 8 \, a + 12\right )} b^{6} c^{2} d^{6} - 4 \, {\left (a^{3} - 12 \, a^{2} + 36 \, a - 24\right )} b^{5} c d^{7} + {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2} - 96 \, a + 24\right )} b^{4} d^{8}\right )} x^{4} + 4 \, {\left (b^{8} c^{5} d^{3} - 4 \, {\left (a - 4\right )} b^{7} c^{4} d^{4} + 6 \, {\left (a^{2} - 8 \, a + 12\right )} b^{6} c^{3} d^{5} - 4 \, {\left (a^{3} - 12 \, a^{2} + 36 \, a - 24\right )} b^{5} c^{2} d^{6} + {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2} - 96 \, a + 24\right )} b^{4} c d^{7}\right )} x^{3} + 6 \, {\left (b^{8} c^{6} d^{2} - 4 \, {\left (a - 4\right )} b^{7} c^{5} d^{3} + 6 \, {\left (a^{2} - 8 \, a + 12\right )} b^{6} c^{4} d^{4} - 4 \, {\left (a^{3} - 12 \, a^{2} + 36 \, a - 24\right )} b^{5} c^{3} d^{5} + {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2} - 96 \, a + 24\right )} b^{4} c^{2} d^{6}\right )} x^{2} + 4 \, {\left (b^{8} c^{7} d - 4 \, {\left (a - 4\right )} b^{7} c^{6} d^{2} + 6 \, {\left (a^{2} - 8 \, a + 12\right )} b^{6} c^{5} d^{3} - 4 \, {\left (a^{3} - 12 \, a^{2} + 36 \, a - 24\right )} b^{5} c^{4} d^{4} + {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2} - 96 \, a + 24\right )} b^{4} c^{3} d^{5}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{7} c^{7} d - {\left (4 \, a - 15\right )} b^{6} c^{6} d^{2} + 2 \, {\left (3 \, a^{2} - 22 \, a + 29\right )} b^{5} c^{5} d^{3} - 2 \, {\left (2 \, a^{3} - 21 \, a^{2} + 52 \, a - 25\right )} b^{4} c^{4} d^{4} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c^{3} d^{5} - 6 \, a^{4} d^{8} - {\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c^{2} d^{6} + 2 \, {\left (a^{4} - 4 \, a^{3}\right )} b c d^{7} + {\left (b^{7} c^{4} d^{4} - 4 \, {\left (a - 4\right )} b^{6} c^{3} d^{5} + 6 \, {\left (a^{2} - 8 \, a + 12\right )} b^{5} c^{2} d^{6} - 4 \, {\left (a^{3} - 12 \, a^{2} + 36 \, a - 24\right )} b^{4} c d^{7} + {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2} - 96 \, a\right )} b^{3} d^{8}\right )} x^{3} + {\left (3 \, b^{7} c^{5} d^{3} - {\left (12 \, a - 47\right )} b^{6} c^{4} d^{4} + 2 \, {\left (9 \, a^{2} - 70 \, a + 100\right )} b^{5} c^{3} d^{5} - 6 \, {\left (2 \, a^{3} - 23 \, a^{2} + 64 \, a - 36\right )} b^{4} c^{2} d^{6} + {\left (3 \, a^{4} - 44 \, a^{3} + 168 \, a^{2} - 144 \, a\right )} b^{3} c d^{7} - {\left (a^{4} - 16 \, a^{3} + 72 \, a^{2}\right )} b^{2} d^{8}\right )} x^{2} + {\left (3 \, b^{7} c^{6} d^{2} - 2 \, {\left (6 \, a - 23\right )} b^{6} c^{5} d^{3} + 2 \, {\left (9 \, a^{2} - 68 \, a + 93\right )} b^{5} c^{4} d^{4} - 4 \, {\left (3 \, a^{3} - 33 \, a^{2} + 86 \, a - 44\right )} b^{4} c^{3} d^{5} + {\left (3 \, a^{4} - 40 \, a^{3} + 132 \, a^{2} - 96 \, a\right )} b^{3} c^{2} d^{6} - 2 \, {\left (a^{4} - 12 \, a^{3} + 24 \, a^{2}\right )} b^{2} c d^{7} + 2 \, {\left (a^{4} - 16 \, a^{3}\right )} b d^{8}\right )} x\right )} e^{\left (-b x - a\right )}}{24 \, {\left (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}\right )}} \end {gather*}

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)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \left (\int \frac {a^{4}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c^{5} e^{b x} + 5 c^{4} d x e^{b x} + 10 c^{3} d^{2} x^{2} e^{b x} + 10 c^{2} d^{3} x^{3} e^{b x} + 5 c d^{4} x^{4} e^{b x} + d^{5} x^{5} e^{b x}}\, dx\right ) e^{- a} \end {gather*}

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]

(Integral(a**4/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 10*c**3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*exp(b*x)
+ 5*c*d**4*x**4*exp(b*x) + d**5*x**5*exp(b*x)), x) + Integral(b**4*x**4/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) +
10*c**3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*exp(b*x)), x) +
Integral(4*a*b**3*x**3/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 10*c**3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*e
xp(b*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*exp(b*x)), x) + Integral(6*a**2*b**2*x**2/(c**5*exp(b*x) + 5*c**4
*d*x*exp(b*x) + 10*c**3*d**2*x**2*exp(b*x) + 10*c**2*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*e
xp(b*x)), x) + Integral(4*a**3*b*x/(c**5*exp(b*x) + 5*c**4*d*x*exp(b*x) + 10*c**3*d**2*x**2*exp(b*x) + 10*c**2
*d**3*x**3*exp(b*x) + 5*c*d**4*x**4*exp(b*x) + d**5*x**5*exp(b*x)), x))*exp(-a)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 16988 vs. \(2 (524) = 1048\).
time = 2.85, size = 16988, normalized size = 30.50 \begin {gather*} \text {Too large to display} \end {gather*}

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*((d*x + c)^4*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^9*c^4*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x +
c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 4*(d*x + c)^3*(b - b*c/(d*x + c) + a*d/(d*x + c))^3*b^10*c^5*Ei(-((d*
x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 6*(d*x + c)^2*(b - b*c/(d*x + c
) + a*d/(d*x + c))^2*b^11*c^6*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d
)/d) + 4*(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^12*c^7*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x +
c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + b^13*c^8*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a
*d)/d)*e^((b*c - a*d)/d) - 4*(d*x + c)^4*a*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^8*c^3*d*Ei(-((d*x + c)*(b -
b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 20*(d*x + c)^3*a*(b - b*c/(d*x + c) + a*d/
(d*x + c))^3*b^9*c^4*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) -
36*(d*x + c)^2*a*(b - b*c/(d*x + c) + a*d/(d*x + c))^2*b^10*c^5*d*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x
+ c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 28*(d*x + c)*a*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^11*c^6*d*Ei(-(
(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 8*a*b^12*c^7*d*Ei(-((d*x + c
)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 6*(d*x + c)^4*a^2*(b - b*c/(d*x + c)
+ a*d/(d*x + c))^4*b^7*c^2*d^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a
*d)/d) + 40*(d*x + c)^3*a^2*(b - b*c/(d*x + c) + a*d/(d*x + c))^3*b^8*c^3*d^2*Ei(-((d*x + c)*(b - b*c/(d*x + c
) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 90*(d*x + c)^2*a^2*(b - b*c/(d*x + c) + a*d/(d*x + c))^
2*b^9*c^4*d^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 84*(d*x +
c)*a^2*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^10*c^5*d^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) +
b*c - a*d)/d)*e^((b*c - a*d)/d) + 28*a^2*b^11*c^6*d^2*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c
- a*d)/d)*e^((b*c - a*d)/d) - 4*(d*x + c)^4*a^3*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^6*c*d^3*Ei(-((d*x + c
)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 40*(d*x + c)^3*a^3*(b - b*c/(d*x + c
) + a*d/(d*x + c))^3*b^7*c^2*d^3*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c -
a*d)/d) - 120*(d*x + c)^2*a^3*(b - b*c/(d*x + c) + a*d/(d*x + c))^2*b^8*c^3*d^3*Ei(-((d*x + c)*(b - b*c/(d*x +
c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 140*(d*x + c)*a^3*(b - b*c/(d*x + c) + a*d/(d*x + c))
*b^9*c^4*d^3*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 56*a^3*b^1
0*c^5*d^3*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + (d*x + c)^4*a
^4*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^5*d^4*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*
d)/d)*e^((b*c - a*d)/d) + 20*(d*x + c)^3*a^4*(b - b*c/(d*x + c) + a*d/(d*x + c))^3*b^6*c*d^4*Ei(-((d*x + c)*(b
- b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 90*(d*x + c)^2*a^4*(b - b*c/(d*x + c) +
a*d/(d*x + c))^2*b^7*c^2*d^4*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)
/d) + 140*(d*x + c)*a^4*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^8*c^3*d^4*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*
d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 70*a^4*b^9*c^4*d^4*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d
*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 4*(d*x + c)^3*a^5*(b - b*c/(d*x + c) + a*d/(d*x + c))^3*b^5*d^5*E
i(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 36*(d*x + c)^2*a^5*(b -
b*c/(d*x + c) + a*d/(d*x + c))^2*b^6*c*d^5*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*
e^((b*c - a*d)/d) - 84*(d*x + c)*a^5*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^7*c^2*d^5*Ei(-((d*x + c)*(b - b*c/(
d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 56*a^5*b^8*c^3*d^5*Ei(-((d*x + c)*(b - b*c/(d*x
+ c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 6*(d*x + c)^2*a^6*(b - b*c/(d*x + c) + a*d/(d*x + c)
)^2*b^5*d^6*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 28*(d*x + c
)*a^6*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^6*c*d^6*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c -
a*d)/d)*e^((b*c - a*d)/d) + 28*a^6*b^7*c^2*d^6*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d
)/d)*e^((b*c - a*d)/d) - 4*(d*x + c)*a^7*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^5*d^7*Ei(-((d*x + c)*(b - b*c/(
d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) - 8*a^7*b^6*c*d^7*Ei(-((d*x + c)*(b - b*c/(d*x + c
) + a*d/(d*x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + a^8*b^5*d^8*Ei(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*
x + c)) + b*c - a*d)/d)*e^((b*c - a*d)/d) + 16*(d*x + c)^4*(b - b*c/(d*x + c) + a*d/(d*x + c))^4*b^8*c^3*d*Ei(
-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))...

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^5} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x)^5,x)

[Out]

int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x)^5, x)

________________________________________________________________________________________