3.79 \(\int \frac{e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx\)

Optimal. Leaf size=258 \[ \frac{4 b^2 e^{-a-b x} (c+d x) (b c-a d)}{d^4}-\frac{b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac{2 b^2 e^{-a-b x} (c+d x)}{d^3}-\frac{b e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}-\frac{4 b e^{\frac{b c}{d}-a} (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}-\frac{e^{-a-b x} (b c-a d)^4}{d^5 (c+d x)}-\frac{6 b e^{-a-b x} (b c-a d)^2}{d^4}+\frac{4 b e^{-a-b x} (b c-a d)}{d^3}-\frac{2 b e^{-a-b x}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.378844, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2199, 2194, 2177, 2178, 2176} \[ \frac{4 b^2 e^{-a-b x} (c+d x) (b c-a d)}{d^4}-\frac{b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac{2 b^2 e^{-a-b x} (c+d x)}{d^3}-\frac{b e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}-\frac{4 b e^{\frac{b c}{d}-a} (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}-\frac{e^{-a-b x} (b c-a d)^4}{d^5 (c+d x)}-\frac{6 b e^{-a-b x} (b c-a d)^2}{d^4}+\frac{4 b e^{-a-b x} (b c-a d)}{d^3}-\frac{2 b e^{-a-b x}}{d^2} \]

Antiderivative was successfully verified.

[In]

Int[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.437347, size = 163, normalized size = 0.63 \[ \frac{e^{-a} \left (-\frac{d e^{-b x} \left (b d (c+d x) \left (2 \left (3 a^2+2 a+1\right ) d^2-2 (4 a+1) b c d+3 b^2 c^2\right )-2 b^2 d^2 x (c+d x) (b c-(2 a+1) d)+(b c-a d)^4+b^3 d^3 x^2 (c+d x)\right )}{c+d x}-b e^{\frac{b c}{d}} (b c-(a-4) d) (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )\right )}{d^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^2,x]

[Out]

(-((d*((b*c - a*d)^4 + b*d*(3*b^2*c^2 - 2*(1 + 4*a)*b*c*d + 2*(1 + 2*a + 3*a^2)*d^2)*(c + d*x) - 2*b^2*d^2*(b*
c - (1 + 2*a)*d)*x*(c + d*x) + b^3*d^3*x^2*(c + d*x)))/(E^(b*x)*(c + d*x))) - b*(b*c - (-4 + a)*d)*(b*c - a*d)
^3*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d^6*E^a)

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Maple [A]  time = 0.015, size = 406, normalized size = 1.6 \begin{align*} -{\frac{1}{b} \left ({\frac{{b}^{2} \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}}}-2\,{\frac{a{b}^{2} \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{{d}^{2}}}+2\,{\frac{{b}^{3}c \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{{d}^{3}}}+3\,{\frac{{b}^{2}{a}^{2}{{\rm e}^{-bx-a}}}{{d}^{2}}}-6\,{\frac{a{b}^{3}c{{\rm e}^{-bx-a}}}{{d}^{3}}}+3\,{\frac{{c}^{2}{b}^{4}{{\rm e}^{-bx-a}}}{{d}^{4}}}+4\,{\frac{ \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ){b}^{2}}{{d}^{5}}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) }+{\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}^{2}}{{d}^{6}} \left ( -{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) \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)^2,x)

[Out]

-1/b*(b^2/d^2*((-b*x-a)^2*exp(-b*x-a)-2*(-b*x-a)*exp(-b*x-a)+2*exp(-b*x-a))-2*b^2/d^2*a*((-b*x-a)*exp(-b*x-a)-
exp(-b*x-a))+2*b^3/d^3*c*((-b*x-a)*exp(-b*x-a)-exp(-b*x-a))+3*b^2/d^2*a^2*exp(-b*x-a)-6*b^3/d^3*a*c*exp(-b*x-a
)+3*b^4/d^4*c^2*exp(-b*x-a)+4/d^5*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*b^2*exp(-(a*d-b*c)/d)*Ei(1,b*x
+a-(a*d-b*c)/d)+(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^2/d^6*(-exp(-b*x-a)/(-b*x-a+
(a*d-b*c)/d)-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_{2}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )} d} - \frac{{\left (b^{3} d^{2} x^{4} + 2 \,{\left (2 \, a b^{2} d^{2} + b^{2} d^{2}\right )} x^{3} + 2 \,{\left (3 \, a^{2} b d^{2} + b^{2} c d + 2 \, a b d^{2} + b d^{2}\right )} x^{2} + 2 \,{\left (2 \, a^{3} d^{2} - b^{2} c^{2} + 4 \, a b c d + 2 \, b c d\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}} - \int -\frac{2 \,{\left (2 \, a^{3} c d^{2} - b^{2} c^{3} + 4 \, a b c^{2} d + 2 \, b c^{2} d +{\left (b^{3} c^{3} - 4 \, a b^{2} c^{2} d + 6 \, a^{2} b c d^{2} - 2 \, a^{3} d^{3} + b^{2} c^{2} d\right )} x\right )} e^{\left (-b x\right )}}{d^{5} x^{3} e^{a} + 3 \, c d^{4} x^{2} e^{a} + 3 \, c^{2} d^{3} x e^{a} + c^{3} 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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.53417, size = 729, normalized size = 2.83 \begin{align*} -\frac{{\left (b^{5} c^{5} - 4 \,{\left (a - 1\right )} b^{4} c^{4} d + 6 \,{\left (a^{2} - 2 \, a\right )} b^{3} c^{3} d^{2} - 4 \,{\left (a^{3} - 3 \, a^{2}\right )} b^{2} c^{2} d^{3} +{\left (a^{4} - 4 \, a^{3}\right )} b c d^{4} +{\left (b^{5} c^{4} d - 4 \,{\left (a - 1\right )} b^{4} c^{3} d^{2} + 6 \,{\left (a^{2} - 2 \, a\right )} b^{3} c^{2} d^{3} - 4 \,{\left (a^{3} - 3 \, a^{2}\right )} b^{2} c d^{4} +{\left (a^{4} - 4 \, a^{3}\right )} b d^{5}\right )} x\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^{5} x^{3} + b^{4} c^{4} d -{\left (4 \, a - 3\right )} b^{3} c^{3} d^{2} + a^{4} d^{5} + 2 \,{\left (3 \, a^{2} - 4 \, a - 1\right )} b^{2} c^{2} d^{3} - 2 \,{\left (2 \, a^{3} - 3 \, a^{2} - 2 \, a - 1\right )} b c d^{4} -{\left (b^{3} c d^{4} - 2 \,{\left (2 \, a + 1\right )} b^{2} d^{5}\right )} x^{2} +{\left (b^{3} c^{2} d^{3} - 4 \, a b^{2} c d^{4} + 2 \,{\left (3 \, a^{2} + 2 \, a + 1\right )} b d^{5}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{7} x + c d^{6}} \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)^2,x, algorithm="fricas")

[Out]

-((b^5*c^5 - 4*(a - 1)*b^4*c^4*d + 6*(a^2 - 2*a)*b^3*c^3*d^2 - 4*(a^3 - 3*a^2)*b^2*c^2*d^3 + (a^4 - 4*a^3)*b*c
*d^4 + (b^5*c^4*d - 4*(a - 1)*b^4*c^3*d^2 + 6*(a^2 - 2*a)*b^3*c^2*d^3 - 4*(a^3 - 3*a^2)*b^2*c*d^4 + (a^4 - 4*a
^3)*b*d^5)*x)*Ei(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) + (b^3*d^5*x^3 + b^4*c^4*d - (4*a - 3)*b^3*c^3*d^2 + a^4*
d^5 + 2*(3*a^2 - 4*a - 1)*b^2*c^2*d^3 - 2*(2*a^3 - 3*a^2 - 2*a - 1)*b*c*d^4 - (b^3*c*d^4 - 2*(2*a + 1)*b^2*d^5
)*x^2 + (b^3*c^2*d^3 - 4*a*b^2*c*d^4 + 2*(3*a^2 + 2*a + 1)*b*d^5)*x)*e^(-b*x - a))/(d^7*x + c*d^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**2,x)

[Out]

Timed out

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Giac [B]  time = 1.74218, size = 1037, normalized size = 4.02 \begin{align*} \text{result too large to display} \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)^2,x, algorithm="giac")

[Out]

-(b^5*c^4*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^4*c^3*d^2*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 6*
a^2*b^3*c^2*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^2*c*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d)
+ a^4*b*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^5*c^5*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^4*c^4*
d*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 6*a^2*b^3*c^3*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^2*c^2*
d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + a^4*b*c*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 4*b^4*c^3*d^2*x*Ei
(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 12*a*b^3*c^2*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*a^2*b^2*c*d^4*
x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 4*b^4*c^4*d*Ei(-(b
*d*x + b*c)/d)*e^(-a + b*c/d) - 12*a*b^3*c^3*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*a^2*b^2*c^2*d^3*Ei(-
(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b*c*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^4*c^4*d*e^(-b*x - a) -
4*a*b^3*c^3*d^2*e^(-b*x - a) + 6*a^2*b^2*c^2*d^3*e^(-b*x - a) - 4*a^3*b*c*d^4*e^(-b*x - a) + a^4*d^5*e^(-b*x
- a))/(d^7*x + c*d^6)