Integrand size = 21, antiderivative size = 198 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=-\frac {a^3 e^{-a-b x}}{3 x^3}-\frac {3 a^2 b e^{-a-b x}}{2 x^2}+\frac {a^3 b e^{-a-b x}}{6 x^2}-\frac {3 a b^2 e^{-a-b x}}{x}+\frac {3 a^2 b^2 e^{-a-b x}}{2 x}-\frac {a^3 b^2 e^{-a-b x}}{6 x}+b^3 e^{-a} \operatorname {ExpIntegralEi}(-b x)-3 a b^3 e^{-a} \operatorname {ExpIntegralEi}(-b x)+\frac {3}{2} a^2 b^3 e^{-a} \operatorname {ExpIntegralEi}(-b x)-\frac {1}{6} a^3 b^3 e^{-a} \operatorname {ExpIntegralEi}(-b x) \]
-1/3*a^3*exp(-b*x-a)/x^3-3/2*a^2*b*exp(-b*x-a)/x^2+1/6*a^3*b*exp(-b*x-a)/x ^2-3*a*b^2*exp(-b*x-a)/x+3/2*a^2*b^2*exp(-b*x-a)/x-1/6*a^3*b^2*exp(-b*x-a) /x+b^3*Ei(-b*x)/exp(a)-3*a*b^3*Ei(-b*x)/exp(a)+3/2*a^2*b^3*Ei(-b*x)/exp(a) -1/6*a^3*b^3*Ei(-b*x)/exp(a)
Time = 0.20 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=\frac {1}{6} e^{-a} \left (-\frac {a e^{-b x} \left (18 b^2 x^2-9 a b x (-1+b x)+a^2 \left (2-b x+b^2 x^2\right )\right )}{x^3}-\left (-6+18 a-9 a^2+a^3\right ) b^3 \operatorname {ExpIntegralEi}(-b x)\right ) \]
(-((a*(18*b^2*x^2 - 9*a*b*x*(-1 + b*x) + a^2*(2 - b*x + b^2*x^2)))/(E^(b*x )*x^3)) - (-6 + 18*a - 9*a^2 + a^3)*b^3*ExpIntegralEi[-(b*x)])/(6*E^a)
Time = 0.47 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2629, 2009}
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.
\(\displaystyle \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \int \left (\frac {a^3 e^{-a-b x}}{x^4}+\frac {3 a^2 b e^{-a-b x}}{x^3}+\frac {b^3 e^{-a-b x}}{x}+\frac {3 a b^2 e^{-a-b x}}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{6} e^{-a} a^3 b^3 \operatorname {ExpIntegralEi}(-b x)-\frac {a^3 b^2 e^{-a-b x}}{6 x}-\frac {a^3 e^{-a-b x}}{3 x^3}+\frac {a^3 b e^{-a-b x}}{6 x^2}+\frac {3}{2} e^{-a} a^2 b^3 \operatorname {ExpIntegralEi}(-b x)+\frac {3 a^2 b^2 e^{-a-b x}}{2 x}-\frac {3 a^2 b e^{-a-b x}}{2 x^2}-3 e^{-a} a b^3 \operatorname {ExpIntegralEi}(-b x)+e^{-a} b^3 \operatorname {ExpIntegralEi}(-b x)-\frac {3 a b^2 e^{-a-b x}}{x}\) |
-1/3*(a^3*E^(-a - b*x))/x^3 - (3*a^2*b*E^(-a - b*x))/(2*x^2) + (a^3*b*E^(- a - b*x))/(6*x^2) - (3*a*b^2*E^(-a - b*x))/x + (3*a^2*b^2*E^(-a - b*x))/(2 *x) - (a^3*b^2*E^(-a - b*x))/(6*x) + (b^3*ExpIntegralEi[-(b*x)])/E^a - (3* a*b^3*ExpIntegralEi[-(b*x)])/E^a + (3*a^2*b^3*ExpIntegralEi[-(b*x)])/(2*E^ a) - (a^3*b^3*ExpIntegralEi[-(b*x)])/(6*E^a)
3.1.63.3.1 Defintions of rubi rules used
Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.21 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(b^{3} \left (-3 a \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\right )-a^{3} \left (\frac {{\mathrm e}^{-b x -a}}{3 b^{3} x^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{6 b x}-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{6}\right )-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )+3 a^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{2}\right )\right )\) | \(167\) |
default | \(b^{3} \left (-3 a \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )\right )-a^{3} \left (\frac {{\mathrm e}^{-b x -a}}{3 b^{3} x^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{6 b x}-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{6}\right )-{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )+3 a^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{2}\right )\right )\) | \(167\) |
risch | \(-\frac {3 a \,b^{2} {\mathrm e}^{-b x -a}}{x}+3 b^{3} a \,{\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )+\frac {b^{3} a^{3} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{6}-\frac {a^{3} b^{2} {\mathrm e}^{-b x -a}}{6 x}+\frac {a^{3} b \,{\mathrm e}^{-b x -a}}{6 x^{2}}-\frac {a^{3} {\mathrm e}^{-b x -a}}{3 x^{3}}-b^{3} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )-\frac {3 a^{2} b \,{\mathrm e}^{-b x -a}}{2 x^{2}}+\frac {3 a^{2} b^{2} {\mathrm e}^{-b x -a}}{2 x}-\frac {3 b^{3} a^{2} {\mathrm e}^{-a} \operatorname {Ei}_{1}\left (b x \right )}{2}\) | \(176\) |
meijerg | \(b^{3} {\mathrm e}^{-a} \left (\ln \left (x \right )+\ln \left (b \right )-\ln \left (b x \right )-\operatorname {Ei}_{1}\left (b x \right )\right )+3 b^{3} {\mathrm e}^{-a} a \left (-\frac {1}{b x}+1-\ln \left (x \right )-\ln \left (b \right )+\frac {-2 b x +2}{2 b x}-\frac {{\mathrm e}^{-b x}}{b x}+\ln \left (b x \right )+\operatorname {Ei}_{1}\left (b x \right )\right )+3 b^{3} {\mathrm e}^{-a} a^{2} \left (-\frac {1}{2 b^{2} x^{2}}+\frac {1}{b x}-\frac {3}{4}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (b \right )}{2}+\frac {9 b^{2} x^{2}-12 b x +6}{12 b^{2} x^{2}}-\frac {\left (-3 b x +3\right ) {\mathrm e}^{-b x}}{6 b^{2} x^{2}}-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Ei}_{1}\left (b x \right )}{2}\right )+{\mathrm e}^{-a} a^{3} b^{3} \left (-\frac {1}{3 b^{3} x^{3}}+\frac {1}{2 b^{2} x^{2}}-\frac {1}{2 b x}+\frac {11}{36}-\frac {\ln \left (x \right )}{6}-\frac {\ln \left (b \right )}{6}+\frac {-22 b^{3} x^{3}+36 b^{2} x^{2}-36 b x +24}{72 b^{3} x^{3}}-\frac {\left (4 b^{2} x^{2}-4 b x +8\right ) {\mathrm e}^{-b x}}{24 b^{3} x^{3}}+\frac {\ln \left (b x \right )}{6}+\frac {\operatorname {Ei}_{1}\left (b x \right )}{6}\right )\) | \(298\) |
b^3*(-3*a*(exp(-b*x-a)/b/x-exp(-a)*Ei(1,b*x))-a^3*(1/3*exp(-b*x-a)/b^3/x^3 -1/6*exp(-b*x-a)/b^2/x^2+1/6*exp(-b*x-a)/b/x-1/6*exp(-a)*Ei(1,b*x))-exp(-a )*Ei(1,b*x)+3*a^2*(-1/2*exp(-b*x-a)/b^2/x^2+1/2*exp(-b*x-a)/b/x-1/2*exp(-a )*Ei(1,b*x)))
Time = 0.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.42 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=-\frac {{\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\left ({\left (a^{3} - 9 \, a^{2} + 18 \, a\right )} b^{2} x^{2} + 2 \, a^{3} - {\left (a^{3} - 9 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{6 \, x^{3}} \]
-1/6*((a^3 - 9*a^2 + 18*a - 6)*b^3*x^3*Ei(-b*x)*e^(-a) + ((a^3 - 9*a^2 + 1 8*a)*b^2*x^2 + 2*a^3 - (a^3 - 9*a^2)*b*x)*e^(-b*x - a))/x^3
Time = 1.17 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.27 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=\left (- \frac {a^{3} \operatorname {E}_{4}\left (b x\right )}{x^{3}} - \frac {3 a^{2} b \operatorname {E}_{3}\left (b x\right )}{x^{2}} - \frac {3 a b^{2} \operatorname {E}_{2}\left (b x\right )}{x} + b^{3} \operatorname {Ei}{\left (- b x \right )}\right ) e^{- a} \]
(-a**3*expint(4, b*x)/x**3 - 3*a**2*b*expint(3, b*x)/x**2 - 3*a*b**2*expin t(2, b*x)/x + b**3*Ei(-b*x))*exp(-a)
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.32 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=-a^{3} b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - 3 \, a^{2} b^{3} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a b^{3} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + b^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} \]
-a^3*b^3*e^(-a)*gamma(-3, b*x) - 3*a^2*b^3*e^(-a)*gamma(-2, b*x) - 3*a*b^3 *e^(-a)*gamma(-1, b*x) + b^3*Ei(-b*x)*e^(-a)
Time = 0.29 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=-\frac {a^{3} b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 18 \, a b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b^{2} x^{2} e^{\left (-b x - a\right )} - 6 \, b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} b x e^{\left (-b x - a\right )} + 18 \, a b^{2} x^{2} e^{\left (-b x - a\right )} + 9 \, a^{2} b x e^{\left (-b x - a\right )} + 2 \, a^{3} e^{\left (-b x - a\right )}}{6 \, x^{3}} \]
-1/6*(a^3*b^3*x^3*Ei(-b*x)*e^(-a) - 9*a^2*b^3*x^3*Ei(-b*x)*e^(-a) + 18*a*b ^3*x^3*Ei(-b*x)*e^(-a) + a^3*b^2*x^2*e^(-b*x - a) - 6*b^3*x^3*Ei(-b*x)*e^( -a) - 9*a^2*b^2*x^2*e^(-b*x - a) - a^3*b*x*e^(-b*x - a) + 18*a*b^2*x^2*e^( -b*x - a) + 9*a^2*b*x*e^(-b*x - a) + 2*a^3*e^(-b*x - a))/x^3
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.72 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx=3\,a\,b^3\,{\mathrm {e}}^{-a}\,\left (\mathrm {expint}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b\,x}}{b\,x}\right )-b^3\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right )+\frac {a^3\,b^3\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right )}{6}+3\,a^2\,b^3\,{\mathrm {e}}^{-a}\,\left ({\mathrm {e}}^{-b\,x}\,\left (\frac {1}{2\,b\,x}-\frac {1}{2\,b^2\,x^2}\right )-\frac {\mathrm {expint}\left (b\,x\right )}{2}\right )-a^3\,b^3\,{\mathrm {e}}^{-a-b\,x}\,\left (\frac {1}{6\,b\,x}-\frac {1}{6\,b^2\,x^2}+\frac {1}{3\,b^3\,x^3}\right ) \]