Integrand size = 18, antiderivative size = 111 \[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=-\frac {e^{-a-b x}}{3 d (c+d x)^3}+\frac {b e^{-a-b x}}{6 d^2 (c+d x)^2}-\frac {b^2 e^{-a-b x}}{6 d^3 (c+d x)}-\frac {b^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{6 d^4} \] Output:
-1/3*exp(-b*x-a)/d/(d*x+c)^3+1/6*b*exp(-b*x-a)/d^2/(d*x+c)^2-1/6*b^2*exp(- b*x-a)/d^3/(d*x+c)-1/6*b^3*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^4
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.79 \[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=\frac {e^{-a-b x} \left (-2 d^3+b d^2 (c+d x)-b^2 d (c+d x)^2-b^3 e^{b \left (\frac {c}{d}+x\right )} (c+d x)^3 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{6 d^4 (c+d x)^3} \] Input:
Integrate[E^(-a - b*x)/(c + d*x)^4,x]
Output:
(E^(-a - b*x)*(-2*d^3 + b*d^2*(c + d*x) - b^2*d*(c + d*x)^2 - b^3*E^(b*(c/ d + x))*(c + d*x)^3*ExpIntegralEi[-((b*(c + d*x))/d)]))/(6*d^4*(c + d*x)^3 )
Time = 0.57 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2608, 2608, 2608, 2609}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle -\frac {b \int \frac {e^{-a-b x}}{(c+d x)^3}dx}{3 d}-\frac {e^{-a-b x}}{3 d (c+d x)^3}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle -\frac {b \left (-\frac {b \int \frac {e^{-a-b x}}{(c+d x)^2}dx}{2 d}-\frac {e^{-a-b x}}{2 d (c+d x)^2}\right )}{3 d}-\frac {e^{-a-b x}}{3 d (c+d x)^3}\) |
\(\Big \downarrow \) 2608 |
\(\displaystyle -\frac {b \left (-\frac {b \left (-\frac {b \int \frac {e^{-a-b x}}{c+d x}dx}{d}-\frac {e^{-a-b x}}{d (c+d x)}\right )}{2 d}-\frac {e^{-a-b x}}{2 d (c+d x)^2}\right )}{3 d}-\frac {e^{-a-b x}}{3 d (c+d x)^3}\) |
\(\Big \downarrow \) 2609 |
\(\displaystyle -\frac {b \left (-\frac {b \left (-\frac {b e^{\frac {b c}{d}-a} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^2}-\frac {e^{-a-b x}}{d (c+d x)}\right )}{2 d}-\frac {e^{-a-b x}}{2 d (c+d x)^2}\right )}{3 d}-\frac {e^{-a-b x}}{3 d (c+d x)^3}\) |
Input:
Int[E^(-a - b*x)/(c + d*x)^4,x]
Output:
-1/3*E^(-a - b*x)/(d*(c + d*x)^3) - (b*(-1/2*E^(-a - b*x)/(d*(c + d*x)^2) - (b*(-(E^(-a - b*x)/(d*(c + d*x))) - (b*E^(-a + (b*c)/d)*ExpIntegralEi[-( (b*(c + d*x))/d)])/d^2))/(2*d)))/(3*d)
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] - Simp[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] && In tegerQ[2*m] && !TrueQ[$UseGamma]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Si mp[(F^(g*(e - c*(f/d)))/d)*ExpIntegralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; F reeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Time = 0.70 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.23
method | result | size |
risch | \(\frac {b^{3} {\mathrm e}^{-b x -a}}{3 d^{4} \left (-b x -\frac {c b}{d}\right )^{3}}+\frac {b^{3} {\mathrm e}^{-b x -a}}{6 d^{4} \left (-b x -\frac {c b}{d}\right )^{2}}+\frac {b^{3} {\mathrm e}^{-b x -a}}{6 d^{4} \left (-b x -\frac {c b}{d}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6 d^{4}}\) | \(137\) |
derivativedivides | \(-\frac {b^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{4}}\) | \(145\) |
default | \(-\frac {b^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{4}}\) | \(145\) |
Input:
int(exp(-b*x-a)/(d*x+c)^4,x,method=_RETURNVERBOSE)
Output:
1/3*b^3/d^4*exp(-b*x-a)/(-b*x-c*b/d)^3+1/6*b^3/d^4*exp(-b*x-a)/(-b*x-c*b/d )^2+1/6*b^3/d^4*exp(-b*x-a)/(-b*x-c*b/d)+1/6*b^3/d^4*exp(-(a*d-b*c)/d)*Ei( 1,b*x+a-(a*d-b*c)/d)
Time = 0.09 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.49 \[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=-\frac {{\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{2} d^{3} x^{2} + b^{2} c^{2} d - b c d^{2} + 2 \, d^{3} + {\left (2 \, b^{2} c d^{2} - b d^{3}\right )} x\right )} e^{\left (-b x - a\right )}}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \] Input:
integrate(exp(-b*x-a)/(d*x+c)^4,x, algorithm="fricas")
Output:
-1/6*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) + (b^2*d^3*x^2 + b^2*c^2*d - b*c*d^2 + 2*d^3 + (2*b^2*c*d^2 - b*d^3)*x)*e^(-b*x - a))/(d^7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^ 5*x + c^3*d^4)
\[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=e^{- a} \int \frac {1}{c^{4} e^{b x} + 4 c^{3} d x e^{b x} + 6 c^{2} d^{2} x^{2} e^{b x} + 4 c d^{3} x^{3} e^{b x} + d^{4} x^{4} e^{b x}}\, dx \] Input:
integrate(exp(-b*x-a)/(d*x+c)**4,x)
Output:
exp(-a)*Integral(1/(c**4*exp(b*x) + 4*c**3*d*x*exp(b*x) + 6*c**2*d**2*x**2 *exp(b*x) + 4*c*d**3*x**3*exp(b*x) + d**4*x**4*exp(b*x)), x)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.32 \[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=-\frac {e^{\left (-a + \frac {b c}{d}\right )} E_{4}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{3} d} \] Input:
integrate(exp(-b*x-a)/(d*x+c)^4,x, algorithm="maxima")
Output:
-e^(-a + b*c/d)*exp_integral_e(4, (d*x + c)*b/d)/((d*x + c)^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (99) = 198\).
Time = 0.11 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.50 \[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=-\frac {b^{3} d^{3} x^{3} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 3 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 3 \, b^{3} c^{2} d x {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{2} d^{3} x^{2} e^{\left (-b x - a\right )} + b^{3} c^{3} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 2 \, b^{2} c d^{2} x e^{\left (-b x - a\right )} + b^{2} c^{2} d e^{\left (-b x - a\right )} - b d^{3} x e^{\left (-b x - a\right )} - b c d^{2} e^{\left (-b x - a\right )} + 2 \, d^{3} e^{\left (-b x - a\right )}}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \] Input:
integrate(exp(-b*x-a)/(d*x+c)^4,x, algorithm="giac")
Output:
-1/6*(b^3*d^3*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*b^3*c*d^2*x^2*Ei (-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*b^3*c^2*d*x*Ei(-(b*d*x + b*c)/d)*e^( -a + b*c/d) + b^2*d^3*x^2*e^(-b*x - a) + b^3*c^3*Ei(-(b*d*x + b*c)/d)*e^(- a + b*c/d) + 2*b^2*c*d^2*x*e^(-b*x - a) + b^2*c^2*d*e^(-b*x - a) - b*d^3*x *e^(-b*x - a) - b*c*d^2*e^(-b*x - a) + 2*d^3*e^(-b*x - a))/(d^7*x^3 + 3*c* d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
Timed out. \[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}}{{\left (c+d\,x\right )}^4} \,d x \] Input:
int(exp(- a - b*x)/(c + d*x)^4,x)
Output:
int(exp(- a - b*x)/(c + d*x)^4, x)
\[ \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx=\frac {\int \frac {1}{e^{b x} c^{4}+4 e^{b x} c^{3} d x +6 e^{b x} c^{2} d^{2} x^{2}+4 e^{b x} c \,d^{3} x^{3}+e^{b x} d^{4} x^{4}}d x}{e^{a}} \] Input:
int(exp(-b*x-a)/(d*x+c)^4,x)
Output:
int(1/(e**(b*x)*c**4 + 4*e**(b*x)*c**3*d*x + 6*e**(b*x)*c**2*d**2*x**2 + 4 *e**(b*x)*c*d**3*x**3 + e**(b*x)*d**4*x**4),x)/e**a