3.60 \(\int \frac{e^{-a-b x} (a+b x)^3}{x} \, dx\)

Optimal. Leaf size=102 \[ e^{-a} a^3 \text{Ei}(-b x)-3 a^2 e^{-a-b x}-b^2 x^2 e^{-a-b x}-3 a e^{-a-b x}-3 a b x e^{-a-b x}-2 e^{-a-b x}-2 b x e^{-a-b x} \]

[Out]

-2*E^(-a - b*x) - 3*a*E^(-a - b*x) - 3*a^2*E^(-a - b*x) - 2*b*E^(-a - b*x)*x - 3*a*b*E^(-a - b*x)*x - b^2*E^(-
a - b*x)*x^2 + (a^3*ExpIntegralEi[-(b*x)])/E^a

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Rubi [A]  time = 0.155647, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2199, 2194, 2178, 2176} \[ e^{-a} a^3 \text{Ei}(-b x)-3 a^2 e^{-a-b x}-b^2 x^2 e^{-a-b x}-3 a e^{-a-b x}-3 a b x e^{-a-b x}-2 e^{-a-b x}-2 b x e^{-a-b x} \]

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^3)/x,x]

[Out]

-2*E^(-a - b*x) - 3*a*E^(-a - b*x) - 3*a^2*E^(-a - b*x) - 2*b*E^(-a - b*x)*x - 3*a*b*E^(-a - b*x)*x - b^2*E^(-
a - b*x)*x^2 + (a^3*ExpIntegralEi[-(b*x)])/E^a

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 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), 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] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{e^{-a-b x} (a+b x)^3}{x} \, dx &=\int \left (3 a^2 b e^{-a-b x}+\frac{a^3 e^{-a-b x}}{x}+3 a b^2 e^{-a-b x} x+b^3 e^{-a-b x} x^2\right ) \, dx\\ &=a^3 \int \frac{e^{-a-b x}}{x} \, dx+\left (3 a^2 b\right ) \int e^{-a-b x} \, dx+\left (3 a b^2\right ) \int e^{-a-b x} x \, dx+b^3 \int e^{-a-b x} x^2 \, dx\\ &=-3 a^2 e^{-a-b x}-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \text{Ei}(-b x)+(3 a b) \int e^{-a-b x} \, dx+\left (2 b^2\right ) \int e^{-a-b x} x \, dx\\ &=-3 a e^{-a-b x}-3 a^2 e^{-a-b x}-2 b e^{-a-b x} x-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \text{Ei}(-b x)+(2 b) \int e^{-a-b x} \, dx\\ &=-2 e^{-a-b x}-3 a e^{-a-b x}-3 a^2 e^{-a-b x}-2 b e^{-a-b x} x-3 a b e^{-a-b x} x-b^2 e^{-a-b x} x^2+a^3 e^{-a} \text{Ei}(-b x)\\ \end{align*}

Mathematica [A]  time = 0.0502071, size = 52, normalized size = 0.51 \[ e^{-a-b x} \left (a^3 e^{b x} \text{Ei}(-b x)-3 a^2-3 a (b x+1)-b^2 x^2-2 b x-2\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(-a - b*x)*(a + b*x)^3)/x,x]

[Out]

E^(-a - b*x)*(-2 - 3*a^2 - 2*b*x - b^2*x^2 - 3*a*(1 + b*x) + a^3*E^(b*x)*ExpIntegralEi[-(b*x)])

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Maple [A]  time = 0.008, size = 113, normalized size = 1.1 \begin{align*} - \left ( -bx-a \right ) ^{2}{{\rm e}^{-bx-a}}+2\, \left ( -bx-a \right ){{\rm e}^{-bx-a}}-2\,{{\rm e}^{-bx-a}}+a \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) -{a}^{2}{{\rm e}^{-bx-a}}-{a}^{3}{{\rm e}^{-a}}{\it Ei} \left ( 1,bx \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b*x-a)*(b*x+a)^3/x,x)

[Out]

-(-b*x-a)^2*exp(-b*x-a)+2*(-b*x-a)*exp(-b*x-a)-2*exp(-b*x-a)+a*((-b*x-a)*exp(-b*x-a)-exp(-b*x-a))-a^2*exp(-b*x
-a)-a^3*exp(-a)*Ei(1,b*x)

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Maxima [A]  time = 1.15238, size = 93, normalized size = 0.91 \begin{align*} a^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \,{\left (b x + 1\right )} a e^{\left (-b x - a\right )} - 3 \, a^{2} e^{\left (-b x - a\right )} -{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x,x, algorithm="maxima")

[Out]

a^3*Ei(-b*x)*e^(-a) - 3*(b*x + 1)*a*e^(-b*x - a) - 3*a^2*e^(-b*x - a) - (b^2*x^2 + 2*b*x + 2)*e^(-b*x - a)

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Fricas [A]  time = 1.46542, size = 108, normalized size = 1.06 \begin{align*} a^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} -{\left (b^{2} x^{2} +{\left (3 \, a + 2\right )} b x + 3 \, a^{2} + 3 \, a + 2\right )} e^{\left (-b x - a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x,x, algorithm="fricas")

[Out]

a^3*Ei(-b*x)*e^(-a) - (b^2*x^2 + (3*a + 2)*b*x + 3*a^2 + 3*a + 2)*e^(-b*x - a)

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Sympy [A]  time = 15.072, size = 70, normalized size = 0.69 \begin{align*} \left (a^{3} \operatorname{Ei}{\left (- b x \right )} - 3 a^{2} e^{- b x} - 3 a \left (b x e^{- b x} + e^{- b x}\right ) - b^{2} x^{2} e^{- b x} - 2 b x e^{- b x} - 2 e^{- b x}\right ) e^{- a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)**3/x,x)

[Out]

(a**3*Ei(-b*x) - 3*a**2*exp(-b*x) - 3*a*(b*x*exp(-b*x) + exp(-b*x)) - b**2*x**2*exp(-b*x) - 2*b*x*exp(-b*x) -
2*exp(-b*x))*exp(-a)

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Giac [A]  time = 1.31972, size = 128, normalized size = 1.25 \begin{align*} -b^{2} x^{2} e^{\left (-b x - a\right )} + a^{3}{\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, a b x e^{\left (-b x - a\right )} - 3 \, a^{2} e^{\left (-b x - a\right )} - 2 \, b x e^{\left (-b x - a\right )} - 3 \, a e^{\left (-b x - a\right )} - 2 \, e^{\left (-b x - a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x,x, algorithm="giac")

[Out]

-b^2*x^2*e^(-b*x - a) + a^3*Ei(-b*x)*e^(-a) - 3*a*b*x*e^(-b*x - a) - 3*a^2*e^(-b*x - a) - 2*b*x*e^(-b*x - a) -
3*a*e^(-b*x - a) - 2*e^(-b*x - a)