3.63 \(\int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx\)

Optimal. Leaf size=198 \[ -\frac {1}{6} e^{-a} a^3 b^3 \text {Ei}(-b x)-\frac {a^3 b^2 e^{-a-b x}}{6 x}-\frac {a^3 e^{-a-b x}}{3 x^3}+\frac {a^3 b e^{-a-b x}}{6 x^2}+\frac {3}{2} e^{-a} a^2 b^3 \text {Ei}(-b x)+\frac {3 a^2 b^2 e^{-a-b x}}{2 x}-\frac {3 a^2 b e^{-a-b x}}{2 x^2}-3 e^{-a} a b^3 \text {Ei}(-b x)+e^{-a} b^3 \text {Ei}(-b x)-\frac {3 a b^2 e^{-a-b x}}{x} \]

[Out]

-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)

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Rubi [A]  time = 0.29, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2199, 2177, 2178} \[ -\frac {1}{6} e^{-a} a^3 b^3 \text {Ei}(-b x)+\frac {3}{2} e^{-a} a^2 b^3 \text {Ei}(-b x)-\frac {a^3 b^2 e^{-a-b x}}{6 x}+\frac {3 a^2 b^2 e^{-a-b x}}{2 x}+\frac {a^3 b e^{-a-b x}}{6 x^2}-\frac {a^3 e^{-a-b x}}{3 x^3}-\frac {3 a^2 b e^{-a-b x}}{2 x^2}-3 e^{-a} a b^3 \text {Ei}(-b x)+e^{-a} b^3 \text {Ei}(-b x)-\frac {3 a b^2 e^{-a-b x}}{x} \]

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^3)/x^4,x]

[Out]

-(a^3*E^(-a - b*x))/(3*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)

Rule 2177

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] &&  !$UseGamma ===
True

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 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

Rubi steps

\begin {align*} \int \frac {e^{-a-b x} (a+b x)^3}{x^4} \, dx &=\int \left (\frac {a^3 e^{-a-b x}}{x^4}+\frac {3 a^2 b e^{-a-b x}}{x^3}+\frac {3 a b^2 e^{-a-b x}}{x^2}+\frac {b^3 e^{-a-b x}}{x}\right ) \, dx\\ &=a^3 \int \frac {e^{-a-b x}}{x^4} \, dx+\left (3 a^2 b\right ) \int \frac {e^{-a-b x}}{x^3} \, dx+\left (3 a b^2\right ) \int \frac {e^{-a-b x}}{x^2} \, dx+b^3 \int \frac {e^{-a-b x}}{x} \, dx\\ &=-\frac {a^3 e^{-a-b x}}{3 x^3}-\frac {3 a^2 b e^{-a-b x}}{2 x^2}-\frac {3 a b^2 e^{-a-b x}}{x}+b^3 e^{-a} \text {Ei}(-b x)-\frac {1}{3} \left (a^3 b\right ) \int \frac {e^{-a-b x}}{x^3} \, dx-\frac {1}{2} \left (3 a^2 b^2\right ) \int \frac {e^{-a-b x}}{x^2} \, dx-\left (3 a b^3\right ) \int \frac {e^{-a-b x}}{x} \, 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}+b^3 e^{-a} \text {Ei}(-b x)-3 a b^3 e^{-a} \text {Ei}(-b x)+\frac {1}{6} \left (a^3 b^2\right ) \int \frac {e^{-a-b x}}{x^2} \, dx+\frac {1}{2} \left (3 a^2 b^3\right ) \int \frac {e^{-a-b x}}{x} \, 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} \text {Ei}(-b x)-3 a b^3 e^{-a} \text {Ei}(-b x)+\frac {3}{2} a^2 b^3 e^{-a} \text {Ei}(-b x)-\frac {1}{6} \left (a^3 b^3\right ) \int \frac {e^{-a-b x}}{x} \, 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} \text {Ei}(-b x)-3 a b^3 e^{-a} \text {Ei}(-b x)+\frac {3}{2} a^2 b^3 e^{-a} \text {Ei}(-b x)-\frac {1}{6} a^3 b^3 e^{-a} \text {Ei}(-b x)\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 81, normalized size = 0.41 \[ \frac {1}{6} e^{-a} \left (-\frac {a e^{-b x} \left (a^2 \left (b^2 x^2-b x+2\right )-9 a b x (b x-1)+18 b^2 x^2\right )}{x^3}-\left (\left (a^3-9 a^2+18 a-6\right ) b^3 \text {Ei}(-b x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(-a - b*x)*(a + b*x)^3)/x^4,x]

[Out]

(-((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)

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fricas [A]  time = 0.40, size = 83, normalized size = 0.42 \[ -\frac {{\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + {\left ({\left (a^{3} - 9 \, a^{2} + 18 \, a\right )} b^{2} x^{2} + 2 \, a^{3} - {\left (a^{3} - 9 \, a^{2}\right )} b x\right )} e^{\left (-b x - a\right )}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x^4,x, algorithm="fricas")

[Out]

-1/6*((a^3 - 9*a^2 + 18*a - 6)*b^3*x^3*Ei(-b*x)*e^(-a) + ((a^3 - 9*a^2 + 18*a)*b^2*x^2 + 2*a^3 - (a^3 - 9*a^2)
*b*x)*e^(-b*x - a))/x^3

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giac [A]  time = 0.41, size = 183, normalized size = 0.92 \[ -\frac {a^{3} b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + 18 \, a b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} + a^{3} b^{2} x^{2} e^{\left (-b x - a\right )} - 6 \, b^{3} x^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 9 \, a^{2} b^{2} x^{2} e^{\left (-b x - a\right )} - a^{3} b x e^{\left (-b x - a\right )} + 18 \, a b^{2} x^{2} e^{\left (-b x - a\right )} + 9 \, a^{2} b x e^{\left (-b x - a\right )} + 2 \, a^{3} e^{\left (-b x - a\right )}}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x^4,x, algorithm="giac")

[Out]

-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

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maple [A]  time = 0.02, size = 167, normalized size = 0.84 \[ \left (-\left (-\frac {\Ei \left (1, b x \right ) {\mathrm e}^{-a}}{6}+\frac {{\mathrm e}^{-b x -a}}{6 b x}-\frac {{\mathrm e}^{-b x -a}}{6 b^{2} x^{2}}+\frac {{\mathrm e}^{-b x -a}}{3 b^{3} x^{3}}\right ) a^{3}+3 \left (-\frac {\Ei \left (1, b x \right ) {\mathrm e}^{-a}}{2}+\frac {{\mathrm e}^{-b x -a}}{2 b x}-\frac {{\mathrm e}^{-b x -a}}{2 b^{2} x^{2}}\right ) a^{2}-\Ei \left (1, b x \right ) {\mathrm e}^{-a}-3 \left (-\Ei \left (1, b x \right ) {\mathrm e}^{-a}+\frac {{\mathrm e}^{-b x -a}}{b x}\right ) a \right ) b^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b*x-a)*(b*x+a)^3/x^4,x)

[Out]

b^3*(-3*a*(1/b/x*exp(-b*x-a)-exp(-a)*Ei(1,b*x))-a^3*(1/3*exp(-b*x-a)/b^3/x^3-1/6/b^2/x^2*exp(-b*x-a)+1/6/b/x*e
xp(-b*x-a)-1/6*exp(-a)*Ei(1,b*x))-exp(-a)*Ei(1,b*x)+3*a^2*(-1/2/b^2/x^2*exp(-b*x-a)+1/2/b/x*exp(-b*x-a)-1/2*ex
p(-a)*Ei(1,b*x)))

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maxima [A]  time = 1.17, size = 63, normalized size = 0.32 \[ -a^{3} b^{3} e^{\left (-a\right )} \Gamma \left (-3, b x\right ) - 3 \, a^{2} b^{3} e^{\left (-a\right )} \Gamma \left (-2, b x\right ) - 3 \, a b^{3} e^{\left (-a\right )} \Gamma \left (-1, b x\right ) + b^{3} {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)^3/x^4,x, algorithm="maxima")

[Out]

-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)

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mupad [B]  time = 3.57, size = 142, normalized size = 0.72 \[ 3\,a\,b^3\,{\mathrm {e}}^{-a}\,\left (\mathrm {expint}\left (b\,x\right )-\frac {{\mathrm {e}}^{-b\,x}}{b\,x}\right )-b^3\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right )+\frac {a^3\,b^3\,{\mathrm {e}}^{-a}\,\mathrm {expint}\left (b\,x\right )}{6}+3\,a^2\,b^3\,{\mathrm {e}}^{-a}\,\left ({\mathrm {e}}^{-b\,x}\,\left (\frac {1}{2\,b\,x}-\frac {1}{2\,b^2\,x^2}\right )-\frac {\mathrm {expint}\left (b\,x\right )}{2}\right )-a^3\,b^3\,{\mathrm {e}}^{-a-b\,x}\,\left (\frac {1}{6\,b\,x}-\frac {1}{6\,b^2\,x^2}+\frac {1}{3\,b^3\,x^3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- a - b*x)*(a + b*x)^3)/x^4,x)

[Out]

3*a*b^3*exp(-a)*(expint(b*x) - exp(-b*x)/(b*x)) - b^3*exp(-a)*expint(b*x) + (a^3*b^3*exp(-a)*expint(b*x))/6 +
3*a^2*b^3*exp(-a)*(exp(-b*x)*(1/(2*b*x) - 1/(2*b^2*x^2)) - expint(b*x)/2) - a^3*b^3*exp(- a - b*x)*(1/(6*b*x)
- 1/(6*b^2*x^2) + 1/(3*b^3*x^3))

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sympy [A]  time = 5.33, size = 53, normalized size = 0.27 \[ \left (- \frac {a^{3} \operatorname {E}_{4}\left (b x\right )}{x^{3}} - \frac {3 a^{2} b \operatorname {E}_{3}\left (b x\right )}{x^{2}} - \frac {3 a b^{2} \operatorname {E}_{2}\left (b x\right )}{x} + b^{3} \operatorname {Ei}{\left (- b x \right )}\right ) e^{- a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*(b*x+a)**3/x**4,x)

[Out]

(-a**3*expint(4, b*x)/x**3 - 3*a**2*b*expint(3, b*x)/x**2 - 3*a*b**2*expint(2, b*x)/x + b**3*Ei(-b*x))*exp(-a)

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