Integrand size = 21, antiderivative size = 130 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=-b^2 e^{-a-b x}-\frac {a^3 e^{-a-b x}}{2 x^2}-\frac {3 a^2 b e^{-a-b x}}{x}+\frac {a^3 b e^{-a-b x}}{2 x}+3 a b^2 e^{-a} \operatorname {ExpIntegralEi}(-b x)-3 a^2 b^2 e^{-a} \operatorname {ExpIntegralEi}(-b x)+\frac {1}{2} a^3 b^2 e^{-a} \operatorname {ExpIntegralEi}(-b x) \] Output:
-b^2*exp(-b*x-a)-1/2*a^3*exp(-b*x-a)/x^2-3*a^2*b*exp(-b*x-a)/x+1/2*a^3*b*e xp(-b*x-a)/x+3*a*b^2*Ei(-b*x)/exp(a)-3*a^2*b^2*Ei(-b*x)/exp(a)+1/2*a^3*b^2 *Ei(-b*x)/exp(a)
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.52 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=\frac {e^{-a-b x} \left (-6 a^2 b x-2 b^2 x^2+a^3 (-1+b x)+a \left (6-6 a+a^2\right ) b^2 e^{b x} x^2 \operatorname {ExpIntegralEi}(-b x)\right )}{2 x^2} \] Input:
Integrate[(E^(-a - b*x)*(a + b*x)^3)/x^3,x]
Output:
(E^(-a - b*x)*(-6*a^2*b*x - 2*b^2*x^2 + a^3*(-1 + b*x) + a*(6 - 6*a + a^2) *b^2*E^(b*x)*x^2*ExpIntegralEi[-(b*x)]))/(2*x^2)
Time = 0.64 (sec) , antiderivative size = 130, 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.
\(\displaystyle \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \int \left (\frac {a^3 e^{-a-b x}}{x^3}+\frac {3 a^2 b e^{-a-b x}}{x^2}+b^3 e^{-a-b x}+\frac {3 a b^2 e^{-a-b x}}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} e^{-a} a^3 b^2 \operatorname {ExpIntegralEi}(-b x)-\frac {a^3 e^{-a-b x}}{2 x^2}+\frac {a^3 b e^{-a-b x}}{2 x}-3 e^{-a} a^2 b^2 \operatorname {ExpIntegralEi}(-b x)-\frac {3 a^2 b e^{-a-b x}}{x}+3 e^{-a} a b^2 \operatorname {ExpIntegralEi}(-b x)-b^2 e^{-a-b x}\) |
Input:
Int[(E^(-a - b*x)*(a + b*x)^3)/x^3,x]
Output:
-(b^2*E^(-a - b*x)) - (a^3*E^(-a - b*x))/(2*x^2) - (3*a^2*b*E^(-a - b*x))/ x + (a^3*b*E^(-a - b*x))/(2*x) + (3*a*b^2*ExpIntegralEi[-(b*x)])/E^a - (3* a^2*b^2*ExpIntegralEi[-(b*x)])/E^a + (a^3*b^2*ExpIntegralEi[-(b*x)])/(2*E^ a)
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 = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-b^{2} \left ({\mathrm e}^{-b x -a}-a^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )}{2}\right )+3 a^{2} \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )\right )+3 a \,{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )\right )\) | \(112\) |
default | \(-b^{2} \left ({\mathrm e}^{-b x -a}-a^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )}{2}\right )+3 a^{2} \left (\frac {{\mathrm e}^{-b x -a}}{b x}-{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )\right )+3 a \,{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )\right )\) | \(112\) |
risch | \(-b^{2} {\mathrm e}^{-b x -a}-\frac {a^{3} {\mathrm e}^{-b x -a}}{2 x^{2}}+\frac {a^{3} b \,{\mathrm e}^{-b x -a}}{2 x}-\frac {b^{2} a^{3} {\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )}{2}-\frac {3 a^{2} b \,{\mathrm e}^{-b x -a}}{x}+3 b^{2} a^{2} {\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )-3 b^{2} a \,{\mathrm e}^{-a} \operatorname {expIntegral}_{1}\left (b x \right )\) | \(118\) |
meijerg | \({\mathrm e}^{-a} b^{2} \left (1-{\mathrm e}^{-b x}\right )+3 b^{2} {\mathrm e}^{-a} a \left (\ln \left (x \right )+\ln \left (b \right )-\ln \left (b x \right )-\operatorname {expIntegral}_{1}\left (b x \right )\right )+3 b^{2} {\mathrm e}^{-a} a^{2} \left (-\frac {1}{b x}+1-\ln \left (x \right )-\ln \left (b \right )+\frac {-2 b x +2}{2 b x}-\frac {{\mathrm e}^{-b x}}{b x}+\ln \left (b x \right )+\operatorname {expIntegral}_{1}\left (b x \right )\right )+{\mathrm e}^{-a} a^{3} b^{2} \left (-\frac {1}{2 b^{2} x^{2}}+\frac {1}{b x}-\frac {3}{4}+\frac {\ln \left (x \right )}{2}+\frac {\ln \left (b \right )}{2}+\frac {9 b^{2} x^{2}-12 b x +6}{12 b^{2} x^{2}}-\frac {\left (-3 b x +3\right ) {\mathrm e}^{-b x}}{6 b^{2} x^{2}}-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {expIntegral}_{1}\left (b x \right )}{2}\right )\) | \(203\) |
Input:
int(exp(-b*x-a)*(b*x+a)^3/x^3,x,method=_RETURNVERBOSE)
Output:
-b^2*(exp(-b*x-a)-a^3*(-1/2*exp(-b*x-a)/b^2/x^2+1/2*exp(-b*x-a)/b/x-1/2*ex p(-a)*Ei(1,b*x))+3*a^2*(exp(-b*x-a)/b/x-exp(-a)*Ei(1,b*x))+3*a*exp(-a)*Ei( 1,b*x))
Time = 0.07 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.54 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=\frac {{\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - {\left (2 \, b^{2} x^{2} + a^{3} - {\left (a^{3} - 6 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{2 \, x^{2}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^3/x^3,x, algorithm="fricas")
Output:
1/2*((a^3 - 6*a^2 + 6*a)*b^2*x^2*Ei(-b*x)*e^(-a) - (2*b^2*x^2 + a^3 - (a^3 - 6*a^2)*b*x)*e^(-b*x - a))/x^2
Time = 1.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.43 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=\left (- \frac {a^{3} \operatorname {E}_{3}\left (b x\right )}{x^{2}} - \frac {3 a^{2} b \operatorname {E}_{2}\left (b x\right )}{x} + 3 a b^{2} \operatorname {Ei}{\left (- b x \right )} + b^{3} \left (\begin {cases} x & \text {for}\: b = 0 \\- \frac {e^{- b x}}{b} & \text {otherwise} \end {cases}\right )\right ) e^{- a} \] Input:
integrate(exp(-b*x-a)*(b*x+a)**3/x**3,x)
Output:
(-a**3*expint(3, b*x)/x**2 - 3*a**2*b*expint(2, b*x)/x + 3*a*b**2*Ei(-b*x) + b**3*Piecewise((x, Eq(b, 0)), (-exp(-b*x)/b, True)))*exp(-a)
Time = 0.13 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.49 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=-a^{3} b^{2} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a^{2} b^{2} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + 3 \, a b^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - b^{2} e^{\left (-b x - a\right )} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^3/x^3,x, algorithm="maxima")
Output:
-a^3*b^2*e^(-a)*gamma(-2, b*x) - 3*a^2*b^2*e^(-a)*gamma(-1, b*x) + 3*a*b^2 *Ei(-b*x)*e^(-a) - b^2*e^(-b*x - a)
Time = 0.11 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=\frac {a^{3} b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 6 \, a^{2} b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 6 \, a b^{2} x^{2} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b x e^{\left (-b x - a\right )} - 6 \, a^{2} b x e^{\left (-b x - a\right )} - 2 \, b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} e^{\left (-b x - a\right )}}{2 \, x^{2}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^3/x^3,x, algorithm="giac")
Output:
1/2*(a^3*b^2*x^2*Ei(-b*x)*e^(-a) - 6*a^2*b^2*x^2*Ei(-b*x)*e^(-a) + 6*a*b^2 *x^2*Ei(-b*x)*e^(-a) + a^3*b*x*e^(-b*x - a) - 6*a^2*b*x*e^(-b*x - a) - 2*b ^2*x^2*e^(-b*x - a) - a^3*e^(-b*x - a))/x^2
Time = 22.64 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.77 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=3\,a^2\,b^2\,{\mathrm {e}}^{-a}\,\left (\mathrm {expint}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b\,x}}{b\,x}\right )-3\,a\,b^2\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right )-b^2\,{\mathrm {e}}^{-a-b\,x}+a^3\,b^2\,{\mathrm {e}}^{-a}\,\left ({\mathrm {e}}^{-b\,x}\,\left (\frac {1}{2\,b\,x}-\frac {1}{2\,b^2\,x^2}\right )-\frac {\mathrm {expint}\left (b\,x\right )}{2}\right ) \] Input:
int((exp(- a - b*x)*(a + b*x)^3)/x^3,x)
Output:
3*a^2*b^2*exp(-a)*(expint(b*x) - exp(-b*x)/(b*x)) - 3*a*b^2*exp(-a)*expint (b*x) - b^2*exp(- a - b*x) + a^3*b^2*exp(-a)*(exp(-b*x)*(1/(2*b*x) - 1/(2* b^2*x^2)) - expint(b*x)/2)
Time = 0.20 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.78 \[ \int \frac {e^{-a-b x} (a+b x)^3}{x^3} \, dx=\frac {e^{b x} \mathit {ei} \left (-b x \right ) a^{3} b^{2} x^{2}-6 e^{b x} \mathit {ei} \left (-b x \right ) a^{2} b^{2} x^{2}+6 e^{b x} \mathit {ei} \left (-b x \right ) a \,b^{2} x^{2}+a^{3} b x -a^{3}-6 a^{2} b x -2 b^{2} x^{2}}{2 e^{b x +a} x^{2}} \] Input:
int(exp(-b*x-a)*(b*x+a)^3/x^3,x)
Output:
(e**(b*x)*ei( - b*x)*a**3*b**2*x**2 - 6*e**(b*x)*ei( - b*x)*a**2*b**2*x**2 + 6*e**(b*x)*ei( - b*x)*a*b**2*x**2 + a**3*b*x - a**3 - 6*a**2*b*x - 2*b* *2*x**2)/(2*e**(a + b*x)*x**2)