3.78 \(\int \frac{e^{-a-b x} (a+b x)^4}{c+d x} \, dx\)

Optimal. Leaf size=277 \[ \frac{e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{e^{-a-b x} (b c-a d)^3}{d^4}-\frac{e^{-a-b x} (b c-a d)^2}{d^3}-\frac{e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}+\frac{2 e^{-a-b x} (b c-a d)}{d^2}+\frac{e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac{2 e^{-a-b x} (a+b x) (b c-a d)}{d^2}-\frac{6 e^{-a-b x}}{d}-\frac{e^{-a-b x} (a+b x)^3}{d}-\frac{3 e^{-a-b x} (a+b x)^2}{d}-\frac{6 e^{-a-b x} (a+b x)}{d} \]

[Out]

(-6*E^(-a - b*x))/d + (2*(b*c - a*d)*E^(-a - b*x))/d^2 - ((b*c - a*d)^2*E^(-a - b*x))/d^3 + ((b*c - a*d)^3*E^(
-a - b*x))/d^4 - (6*E^(-a - b*x)*(a + b*x))/d + (2*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/d^2 - ((b*c - a*d)^2*E^
(-a - b*x)*(a + b*x))/d^3 - (3*E^(-a - b*x)*(a + b*x)^2)/d + ((b*c - a*d)*E^(-a - b*x)*(a + b*x)^2)/d^2 - (E^(
-a - b*x)*(a + b*x)^3)/d + ((b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5

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Rubi [A]  time = 0.338364, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2199, 2194, 2176, 2178} \[ \frac{e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{e^{-a-b x} (b c-a d)^3}{d^4}-\frac{e^{-a-b x} (b c-a d)^2}{d^3}-\frac{e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}+\frac{2 e^{-a-b x} (b c-a d)}{d^2}+\frac{e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac{2 e^{-a-b x} (a+b x) (b c-a d)}{d^2}-\frac{6 e^{-a-b x}}{d}-\frac{e^{-a-b x} (a+b x)^3}{d}-\frac{3 e^{-a-b x} (a+b x)^2}{d}-\frac{6 e^{-a-b x} (a+b x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x),x]

[Out]

(-6*E^(-a - b*x))/d + (2*(b*c - a*d)*E^(-a - b*x))/d^2 - ((b*c - a*d)^2*E^(-a - b*x))/d^3 + ((b*c - a*d)^3*E^(
-a - b*x))/d^4 - (6*E^(-a - b*x)*(a + b*x))/d + (2*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/d^2 - ((b*c - a*d)^2*E^
(-a - b*x)*(a + b*x))/d^3 - (3*E^(-a - b*x)*(a + b*x)^2)/d + ((b*c - a*d)*E^(-a - b*x)*(a + b*x)^2)/d^2 - (E^(
-a - b*x)*(a + b*x)^3)/d + ((b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5

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 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

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

Rubi steps

\begin{align*} \int \frac{e^{-a-b x} (a+b x)^4}{c+d x} \, dx &=\int \left (-\frac{b (b c-a d)^3 e^{-a-b x}}{d^4}+\frac{b (b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac{b (b c-a d) e^{-a-b x} (a+b x)^2}{d^2}+\frac{b e^{-a-b x} (a+b x)^3}{d}+\frac{(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx\\ &=\frac{b \int e^{-a-b x} (a+b x)^3 \, dx}{d}-\frac{(b (b c-a d)) \int e^{-a-b x} (a+b x)^2 \, dx}{d^2}+\frac{\left (b (b c-a d)^2\right ) \int e^{-a-b x} (a+b x) \, dx}{d^3}-\frac{\left (b (b c-a d)^3\right ) \int e^{-a-b x} \, dx}{d^4}+\frac{(b c-a d)^4 \int \frac{e^{-a-b x}}{c+d x} \, dx}{d^4}\\ &=\frac{(b c-a d)^3 e^{-a-b x}}{d^4}-\frac{(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}+\frac{(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac{e^{-a-b x} (a+b x)^3}{d}+\frac{(b c-a d)^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{(3 b) \int e^{-a-b x} (a+b x)^2 \, dx}{d}-\frac{(2 b (b c-a d)) \int e^{-a-b x} (a+b x) \, dx}{d^2}+\frac{\left (b (b c-a d)^2\right ) \int e^{-a-b x} \, dx}{d^3}\\ &=-\frac{(b c-a d)^2 e^{-a-b x}}{d^3}+\frac{(b c-a d)^3 e^{-a-b x}}{d^4}+\frac{2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac{(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac{3 e^{-a-b x} (a+b x)^2}{d}+\frac{(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac{e^{-a-b x} (a+b x)^3}{d}+\frac{(b c-a d)^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{(6 b) \int e^{-a-b x} (a+b x) \, dx}{d}-\frac{(2 b (b c-a d)) \int e^{-a-b x} \, dx}{d^2}\\ &=\frac{2 (b c-a d) e^{-a-b x}}{d^2}-\frac{(b c-a d)^2 e^{-a-b x}}{d^3}+\frac{(b c-a d)^3 e^{-a-b x}}{d^4}-\frac{6 e^{-a-b x} (a+b x)}{d}+\frac{2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac{(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac{3 e^{-a-b x} (a+b x)^2}{d}+\frac{(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac{e^{-a-b x} (a+b x)^3}{d}+\frac{(b c-a d)^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{(6 b) \int e^{-a-b x} \, dx}{d}\\ &=-\frac{6 e^{-a-b x}}{d}+\frac{2 (b c-a d) e^{-a-b x}}{d^2}-\frac{(b c-a d)^2 e^{-a-b x}}{d^3}+\frac{(b c-a d)^3 e^{-a-b x}}{d^4}-\frac{6 e^{-a-b x} (a+b x)}{d}+\frac{2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac{(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac{3 e^{-a-b x} (a+b x)^2}{d}+\frac{(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac{e^{-a-b x} (a+b x)^3}{d}+\frac{(b c-a d)^4 e^{-a+\frac{b c}{d}} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}\\ \end{align*}

Mathematica [A]  time = 0.298496, size = 175, normalized size = 0.63 \[ \frac{e^{-a-b x} \left ((b c-a d)^4 e^{b \left (\frac{c}{d}+x\right )} \text{Ei}\left (-\frac{b (c+d x)}{d}\right )-d \left (2 b d^2 \left (\left (3 a^2+4 a+3\right ) d x-\left (3 a^2+2 a+1\right ) c\right )+2 \left (2 a^3+3 a^2+4 a+3\right ) d^3+b^2 d \left ((4 a+1) c^2-2 (2 a+1) c d x+(4 a+3) d^2 x^2\right )+b^3 \left (c^2 d x-c^3-c d^2 x^2+d^3 x^3\right )\right )\right )}{d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^(-a - b*x)*(-(d*(2*(3 + 4*a + 3*a^2 + 2*a^3)*d^3 + 2*b*d^2*(-((1 + 2*a + 3*a^2)*c) + (3 + 4*a + 3*a^2)*d*x)
+ b^2*d*((1 + 4*a)*c^2 - 2*(1 + 2*a)*c*d*x + (3 + 4*a)*d^2*x^2) + b^3*(-c^3 + c^2*d*x - c*d^2*x^2 + d^3*x^3))
) + (b*c - a*d)^4*E^(b*(c/d + x))*ExpIntegralEi[-((b*(c + d*x))/d)]))/d^5

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Maple [A]  time = 0.012, size = 489, normalized size = 1.8 \begin{align*} -{\frac{1}{b} \left ( -{\frac{b \left ( \left ( -bx-a \right ) ^{3}{{\rm e}^{-bx-a}}-3\, \left ( -bx-a \right ) ^{2}{{\rm e}^{-bx-a}}+6\, \left ( -bx-a \right ){{\rm e}^{-bx-a}}-6\,{{\rm e}^{-bx-a}} \right ) }{d}}+{\frac{ab \left ( \left ( -bx-a \right ) ^{2}{{\rm e}^{-bx-a}}-2\, \left ( -bx-a \right ){{\rm e}^{-bx-a}}+2\,{{\rm e}^{-bx-a}} \right ) }{d}}-{\frac{{b}^{2}c \left ( \left ( -bx-a \right ) ^{2}{{\rm e}^{-bx-a}}-2\, \left ( -bx-a \right ){{\rm e}^{-bx-a}}+2\,{{\rm e}^{-bx-a}} \right ) }{{d}^{2}}}-{\frac{{a}^{2}b \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{d}}+2\,{\frac{a{b}^{2}c \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{{d}^{2}}}-{\frac{{b}^{3}{c}^{2} \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{{d}^{3}}}+{\frac{{a}^{3}b{{\rm e}^{-bx-a}}}{d}}-3\,{\frac{{b}^{2}{a}^{2}c{{\rm e}^{-bx-a}}}{{d}^{2}}}+3\,{\frac{a{b}^{3}{c}^{2}{{\rm e}^{-bx-a}}}{{d}^{3}}}-{\frac{{b}^{4}{c}^{3}{{\rm e}^{-bx-a}}}{{d}^{4}}}+{\frac{ \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ) b}{{d}^{5}}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/b*(-b/d*((-b*x-a)^3*exp(-b*x-a)-3*(-b*x-a)^2*exp(-b*x-a)+6*(-b*x-a)*exp(-b*x-a)-6*exp(-b*x-a))+b/d*a*((-b*x
-a)^2*exp(-b*x-a)-2*(-b*x-a)*exp(-b*x-a)+2*exp(-b*x-a))-b^2/d^2*c*((-b*x-a)^2*exp(-b*x-a)-2*(-b*x-a)*exp(-b*x-
a)+2*exp(-b*x-a))-b/d*a^2*((-b*x-a)*exp(-b*x-a)-exp(-b*x-a))+2*b^2/d^2*a*c*((-b*x-a)*exp(-b*x-a)-exp(-b*x-a))-
b^3/d^3*c^2*((-b*x-a)*exp(-b*x-a)-exp(-b*x-a))+b/d*a^3*exp(-b*x-a)-3*b^2/d^2*a^2*c*exp(-b*x-a)+3*b^3/d^3*a*c^2
*exp(-b*x-a)-b^4/d^4*c^3*exp(-b*x-a)+(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*b/d^5*exp
(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{4} e^{\left (-a + \frac{b c}{d}\right )} E_{1}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{d} - \frac{{\left (b^{3} d^{2} x^{4} +{\left (4 \, a b^{2} d^{2} + 3 \, b^{2} d^{2}\right )} x^{3} +{\left (6 \, a^{2} b d^{2} + b^{2} c d + 8 \, a b d^{2} + 6 \, b d^{2}\right )} x^{2} +{\left (4 \, a^{3} d^{2} - b^{2} c^{2} + 6 \, a^{2} d^{2} + 4 \, b c d + 4 \,{\left (b c d + 2 \, d^{2}\right )} a + 6 \, d^{2}\right )} x\right )} e^{\left (-b x\right )}}{d^{3} x e^{a} + c d^{2} e^{a}} + \int \frac{{\left (4 \, a^{3} c d^{2} - b^{2} c^{3} + 6 \, a^{2} c d^{2} + 4 \, b c^{2} d + 6 \, c d^{2} + 4 \,{\left (b c^{2} d + 2 \, c d^{2}\right )} a +{\left (b^{3} c^{3} + 6 \, a^{2} b c d^{2} - 2 \, b^{2} c^{2} d + 6 \, b c d^{2} - 4 \,{\left (b^{2} c^{2} d - 2 \, b c d^{2}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{4} x^{2} e^{a} + 2 \, c d^{3} x e^{a} + c^{2} d^{2} e^{a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^4*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d - (b^3*d^2*x^4 + (4*a*b^2*d^2 + 3*b^2*d^2)*x^3 + (6*a^2
*b*d^2 + b^2*c*d + 8*a*b*d^2 + 6*b*d^2)*x^2 + (4*a^3*d^2 - b^2*c^2 + 6*a^2*d^2 + 4*b*c*d + 4*(b*c*d + 2*d^2)*a
+ 6*d^2)*x)*e^(-b*x)/(d^3*x*e^a + c*d^2*e^a) + integrate((4*a^3*c*d^2 - b^2*c^3 + 6*a^2*c*d^2 + 4*b*c^2*d + 6
*c*d^2 + 4*(b*c^2*d + 2*c*d^2)*a + (b^3*c^3 + 6*a^2*b*c*d^2 - 2*b^2*c^2*d + 6*b*c*d^2 - 4*(b^2*c^2*d - 2*b*c*d
^2)*a)*x)*e^(-b*x)/(d^4*x^2*e^a + 2*c*d^3*x*e^a + c^2*d^2*e^a), x)

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Fricas [A]  time = 1.52102, size = 479, normalized size = 1.73 \begin{align*} \frac{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (\frac{b c - a d}{d}\right )} -{\left (b^{3} d^{4} x^{3} - b^{3} c^{3} d +{\left (4 \, a + 1\right )} b^{2} c^{2} d^{2} - 2 \,{\left (3 \, a^{2} + 2 \, a + 1\right )} b c d^{3} + 2 \,{\left (2 \, a^{3} + 3 \, a^{2} + 4 \, a + 3\right )} d^{4} -{\left (b^{3} c d^{3} -{\left (4 \, a + 3\right )} b^{2} d^{4}\right )} x^{2} +{\left (b^{3} c^{2} d^{2} - 2 \,{\left (2 \, a + 1\right )} b^{2} c d^{3} + 2 \,{\left (3 \, a^{2} + 4 \, a + 3\right )} b d^{4}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d
) - (b^3*d^4*x^3 - b^3*c^3*d + (4*a + 1)*b^2*c^2*d^2 - 2*(3*a^2 + 2*a + 1)*b*c*d^3 + 2*(2*a^3 + 3*a^2 + 4*a +
3)*d^4 - (b^3*c*d^3 - (4*a + 3)*b^2*d^4)*x^2 + (b^3*c^2*d^2 - 2*(2*a + 1)*b^2*c*d^3 + 2*(3*a^2 + 4*a + 3)*b*d^
4)*x)*e^(-b*x - a))/d^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \left (\int \frac{a^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac{b^{4} x^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac{4 a b^{3} x^{3}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac{6 a^{2} b^{2} x^{2}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac{4 a^{3} b x}{c e^{b x} + d x e^{b x}}\, dx\right ) e^{- a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(Integral(a**4/(c*exp(b*x) + d*x*exp(b*x)), x) + Integral(b**4*x**4/(c*exp(b*x) + d*x*exp(b*x)), x) + Integral
(4*a*b**3*x**3/(c*exp(b*x) + d*x*exp(b*x)), x) + Integral(6*a**2*b**2*x**2/(c*exp(b*x) + d*x*exp(b*x)), x) + I
ntegral(4*a**3*b*x/(c*exp(b*x) + d*x*exp(b*x)), x))*exp(-a)

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Giac [A]  time = 1.18869, size = 240, normalized size = 0.87 \begin{align*} \frac{b^{4} c^{4}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - 4 \, a b^{3} c^{3} d{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + 6 \, a^{2} b^{2} c^{2} d^{2}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} - 4 \, a^{3} b c d^{3}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )} + a^{4} d^{4}{\rm Ei}\left (-\frac{b d x + b c}{d}\right ) e^{\left (-a + \frac{b c}{d}\right )}}{d^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^4*c^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^3*c^3*d*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 6*a^2*b^2*c
^2*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b*c*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + a^4*d^4*Ei(-(
b*d*x + b*c)/d)*e^(-a + b*c/d))/d^5