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

Optimal. Leaf size=396 \[ -\frac{b^3 e^{\frac{b c}{d}-a} (b c-a d)^4 \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{6 d^8}-\frac{2 b^3 e^{\frac{b c}{d}-a} (b c-a d)^3 \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^7}-\frac{6 b^3 e^{\frac{b c}{d}-a} (b c-a d)^2 \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^6}-\frac{4 b^3 e^{\frac{b c}{d}-a} (b c-a d) \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^5}-\frac{b^3 e^{-a-b x}}{d^4}-\frac{b^2 e^{-a-b x} (b c-a d)^4}{6 d^7 (c+d x)}-\frac{2 b^2 e^{-a-b x} (b c-a d)^3}{d^6 (c+d x)}-\frac{6 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)}+\frac{b e^{-a-b x} (b c-a d)^4}{6 d^6 (c+d x)^2}-\frac{e^{-a-b x} (b c-a d)^4}{3 d^5 (c+d x)^3}+\frac{2 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)^2} \]

[Out]

-((b^3*E^(-a - b*x))/d^4) - ((b*c - a*d)^4*E^(-a - b*x))/(3*d^5*(c + d*x)^3) + (
2*b*(b*c - a*d)^3*E^(-a - b*x))/(d^5*(c + d*x)^2) + (b*(b*c - a*d)^4*E^(-a - b*x
))/(6*d^6*(c + d*x)^2) - (6*b^2*(b*c - a*d)^2*E^(-a - b*x))/(d^5*(c + d*x)) - (2
*b^2*(b*c - a*d)^3*E^(-a - b*x))/(d^6*(c + d*x)) - (b^2*(b*c - a*d)^4*E^(-a - b*
x))/(6*d^7*(c + d*x)) - (4*b^3*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(
c + d*x))/d)])/d^5 - (6*b^3*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c
+ d*x))/d)])/d^6 - (2*b^3*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c
+ d*x))/d)])/d^7 - (b^3*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d
*x))/d)])/(6*d^8)

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Rubi [A]  time = 0.859049, antiderivative size = 396, 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 \[ -\frac{b^3 e^{\frac{b c}{d}-a} (b c-a d)^4 \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{6 d^8}-\frac{2 b^3 e^{\frac{b c}{d}-a} (b c-a d)^3 \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^7}-\frac{6 b^3 e^{\frac{b c}{d}-a} (b c-a d)^2 \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^6}-\frac{4 b^3 e^{\frac{b c}{d}-a} (b c-a d) \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )}{d^5}-\frac{b^3 e^{-a-b x}}{d^4}-\frac{b^2 e^{-a-b x} (b c-a d)^4}{6 d^7 (c+d x)}-\frac{2 b^2 e^{-a-b x} (b c-a d)^3}{d^6 (c+d x)}-\frac{6 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)}+\frac{b e^{-a-b x} (b c-a d)^4}{6 d^6 (c+d x)^2}-\frac{e^{-a-b x} (b c-a d)^4}{3 d^5 (c+d x)^3}+\frac{2 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)^2} \]

Antiderivative was successfully verified.

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

[Out]

-((b^3*E^(-a - b*x))/d^4) - ((b*c - a*d)^4*E^(-a - b*x))/(3*d^5*(c + d*x)^3) + (
2*b*(b*c - a*d)^3*E^(-a - b*x))/(d^5*(c + d*x)^2) + (b*(b*c - a*d)^4*E^(-a - b*x
))/(6*d^6*(c + d*x)^2) - (6*b^2*(b*c - a*d)^2*E^(-a - b*x))/(d^5*(c + d*x)) - (2
*b^2*(b*c - a*d)^3*E^(-a - b*x))/(d^6*(c + d*x)) - (b^2*(b*c - a*d)^4*E^(-a - b*
x))/(6*d^7*(c + d*x)) - (4*b^3*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(
c + d*x))/d)])/d^5 - (6*b^3*(b*c - a*d)^2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c
+ d*x))/d)])/d^6 - (2*b^3*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c
+ d*x))/d)])/d^7 - (b^3*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d
*x))/d)])/(6*d^8)

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Rubi in Sympy [A]  time = 87.5981, size = 347, normalized size = 0.88 \[ - \frac{b^{3} e^{- a - b x}}{d^{4}} + \frac{4 b^{3} \left (a d - b c\right ) e^{- a} e^{\frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{d^{5}} - \frac{6 b^{3} \left (a d - b c\right )^{2} e^{- a + \frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{d^{6}} + \frac{2 b^{3} \left (a d - b c\right )^{3} e^{- a} e^{\frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{d^{7}} - \frac{b^{3} \left (a d - b c\right )^{4} e^{- a + \frac{b c}{d}} \operatorname{Ei}{\left (\frac{b \left (- c - d x\right )}{d} \right )}}{6 d^{8}} - \frac{6 b^{2} \left (a d - b c\right )^{2} e^{- a - b x}}{d^{5} \left (c + d x\right )} + \frac{2 b^{2} \left (a d - b c\right )^{3} e^{- a - b x}}{d^{6} \left (c + d x\right )} - \frac{b^{2} \left (a d - b c\right )^{4} e^{- a - b x}}{6 d^{7} \left (c + d x\right )} - \frac{2 b \left (a d - b c\right )^{3} e^{- a - b x}}{d^{5} \left (c + d x\right )^{2}} + \frac{b \left (a d - b c\right )^{4} e^{- a - b x}}{6 d^{6} \left (c + d x\right )^{2}} - \frac{\left (a d - b c\right )^{4} e^{- a - b x}}{3 d^{5} \left (c + d x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(-b*x-a)*(b*x+a)**4/(d*x+c)**4,x)

[Out]

-b**3*exp(-a - b*x)/d**4 + 4*b**3*(a*d - b*c)*exp(-a)*exp(b*c/d)*Ei(b*(-c - d*x)
/d)/d**5 - 6*b**3*(a*d - b*c)**2*exp(-a + b*c/d)*Ei(b*(-c - d*x)/d)/d**6 + 2*b**
3*(a*d - b*c)**3*exp(-a)*exp(b*c/d)*Ei(b*(-c - d*x)/d)/d**7 - b**3*(a*d - b*c)**
4*exp(-a + b*c/d)*Ei(b*(-c - d*x)/d)/(6*d**8) - 6*b**2*(a*d - b*c)**2*exp(-a - b
*x)/(d**5*(c + d*x)) + 2*b**2*(a*d - b*c)**3*exp(-a - b*x)/(d**6*(c + d*x)) - b*
*2*(a*d - b*c)**4*exp(-a - b*x)/(6*d**7*(c + d*x)) - 2*b*(a*d - b*c)**3*exp(-a -
b*x)/(d**5*(c + d*x)**2) + b*(a*d - b*c)**4*exp(-a - b*x)/(6*d**6*(c + d*x)**2)
- (a*d - b*c)**4*exp(-a - b*x)/(3*d**5*(c + d*x)**3)

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Mathematica [A]  time = 0.872146, size = 389, normalized size = 0.98 \[ \frac{e^{-a} \left (b^3 e^{\frac{b c}{d}} \left (-\left (6 \left (a^2-6 a+6\right ) b^2 c^2 d^2-4 \left (a^3-9 a^2+18 a-6\right ) b c d^3+a \left (a^3-12 a^2+36 a-24\right ) d^4-4 (a-3) b^3 c^3 d+b^4 c^4\right )\right ) \text{ExpIntegralEi}\left (-\frac{b (c+d x)}{d}\right )-\frac{d e^{-b x} \left (2 a^4 d^6-a^3 b d^5 ((a-4) c+(a-12) d x)+2 b^4 c^2 d^2 \left (\left (3 a^2-16 a+13\right ) c^2+2 \left (3 a^2-17 a+15\right ) c d x+3 \left (a^2-6 a+6\right ) d^2 x^2\right )+a^2 b^2 d^4 \left (\left (a^2-8 a+12\right ) c^2+2 \left (a^2-10 a+18\right ) c d x+(a-6)^2 d^2 x^2\right )+2 b^3 d^3 \left (\left (-2 a^3+15 a^2-22 a+3\right ) c^3+\left (-4 a^3+33 a^2-54 a+9\right ) c^2 d x+\left (-2 a^3+18 a^2-36 a+9\right ) c d^2 x^2+3 d^3 x^3\right )-b^5 c^3 d (c+d x) ((4 a-11) c+4 (a-3) d x)+b^6 c^4 (c+d x)^2\right )}{(c+d x)^3}\right )}{6 d^8} \]

Antiderivative was successfully verified.

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

[Out]

(-((d*(2*a^4*d^6 + b^6*c^4*(c + d*x)^2 - a^3*b*d^5*((-4 + a)*c + (-12 + a)*d*x)
- b^5*c^3*d*(c + d*x)*((-11 + 4*a)*c + 4*(-3 + a)*d*x) + a^2*b^2*d^4*((12 - 8*a
+ a^2)*c^2 + 2*(18 - 10*a + a^2)*c*d*x + (-6 + a)^2*d^2*x^2) + 2*b^4*c^2*d^2*((1
3 - 16*a + 3*a^2)*c^2 + 2*(15 - 17*a + 3*a^2)*c*d*x + 3*(6 - 6*a + a^2)*d^2*x^2)
+ 2*b^3*d^3*((3 - 22*a + 15*a^2 - 2*a^3)*c^3 + (9 - 54*a + 33*a^2 - 4*a^3)*c^2*
d*x + (9 - 36*a + 18*a^2 - 2*a^3)*c*d^2*x^2 + 3*d^3*x^3)))/(E^(b*x)*(c + d*x)^3)
) - b^3*(b^4*c^4 - 4*(-3 + a)*b^3*c^3*d + 6*(6 - 6*a + a^2)*b^2*c^2*d^2 - 4*(-6
+ 18*a - 9*a^2 + a^3)*b*c*d^3 + a*(-24 + 36*a - 12*a^2 + a^3)*d^4)*E^((b*c)/d)*E
xpIntegralEi[-((b*(c + d*x))/d)])/(6*d^8*E^a)

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Maple [A]  time = 0.019, size = 511, normalized size = 1.3 \[ -{\frac{1}{b} \left ({\frac{{b}^{4}{{\rm e}^{-bx-a}}}{{d}^{4}}}+4\,{\frac{ \left ( ad-cb \right ){b}^{4}}{{d}^{5}}{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) }+6\,{\frac{ \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ){b}^{4}}{{d}^{6}} \left ( -{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) \right ) }-4\,{\frac{ \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ){b}^{4}}{{d}^{7}} \left ( -1/2\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-2}}-1/2\,{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-1/2\,{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) \right ) }+{\frac{ \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{d}^{3}+6\,{a}^{2}{b}^{2}{c}^{2}{d}^{2}-4\,a{b}^{3}{c}^{3}d+{b}^{4}{c}^{4} \right ){b}^{4}}{{d}^{8}} \left ( -{\frac{{{\rm e}^{-bx-a}}}{3} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-3}}-{\frac{{{\rm e}^{-bx-a}}}{6} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-2}}-{\frac{{{\rm e}^{-bx-a}}}{6} \left ( -bx-a+{\frac{ad-cb}{d}} \right ) ^{-1}}-{\frac{1}{6}{{\rm e}^{-{\frac{ad-cb}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-cb}{d}} \right ) } \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/b*(b^4/d^4*exp(-b*x-a)+4/d^5*(a*d-b*c)*b^4*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-
b*c)/d)+6/d^6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*b^4*(-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))-4/d^7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2
*c^2*d-b^3*c^3)*b^4*(-1/2*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/2*exp(-b*x-a)/(-b
*x-a+(a*d-b*c)/d)-1/2*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^4/d^8*(-1/3*exp(-b*x-a)/(-b*x
-a+(a*d-b*c)/d)^3-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/(-b*x-a
+(a*d-b*c)/d)-1/6*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. \[ -\frac{a^{4} e^{\left (-a + \frac{b c}{d}\right )} exp_integral_e\left (4, \frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{3} d} - \frac{{\left (b^{3} d^{2} x^{4} + 4 \, a b^{2} d^{2} x^{3} + 2 \,{\left (3 \, a^{2} b d^{2} + 2 \, b^{2} c d - 2 \, a b d^{2}\right )} x^{2} + 4 \,{\left (a^{3} d^{2} - b^{2} c^{2} - 3 \, a^{2} d^{2} - 2 \, b c d + 2 \,{\left (2 \, b c d + d^{2}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{6} x^{4} e^{a} + 4 \, c d^{5} x^{3} e^{a} + 6 \, c^{2} d^{4} x^{2} e^{a} + 4 \, c^{3} d^{3} x e^{a} + c^{4} d^{2} e^{a}} - \int -\frac{4 \,{\left (a^{3} c d^{2} - b^{2} c^{3} - 3 \, a^{2} c d^{2} - 2 \, b c^{2} d + 2 \,{\left (2 \, b c^{2} d + c d^{2}\right )} a +{\left (b^{3} c^{3} - 3 \, a^{3} d^{3} + 7 \, b^{2} c^{2} d + 6 \, b c d^{2} + 3 \,{\left (2 \, b c d^{2} + 3 \, d^{3}\right )} a^{2} - 2 \,{\left (2 \, b^{2} c^{2} d + 8 \, b c d^{2} + 3 \, d^{3}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{7} x^{5} e^{a} + 5 \, c d^{6} x^{4} e^{a} + 10 \, c^{2} d^{5} x^{3} e^{a} + 10 \, c^{3} d^{4} x^{2} e^{a} + 5 \, c^{4} d^{3} x e^{a} + c^{5} d^{2} e^{a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*e^(-b*x - a)/(d*x + c)^4,x, algorithm="maxima")

[Out]

-a^4*e^(-a + b*c/d)*exp_integral_e(4, (d*x + c)*b/d)/((d*x + c)^3*d) - (b^3*d^2*
x^4 + 4*a*b^2*d^2*x^3 + 2*(3*a^2*b*d^2 + 2*b^2*c*d - 2*a*b*d^2)*x^2 + 4*(a^3*d^2
- b^2*c^2 - 3*a^2*d^2 - 2*b*c*d + 2*(2*b*c*d + d^2)*a)*x)*e^(-b*x)/(d^6*x^4*e^a
+ 4*c*d^5*x^3*e^a + 6*c^2*d^4*x^2*e^a + 4*c^3*d^3*x*e^a + c^4*d^2*e^a) - integr
ate(-4*(a^3*c*d^2 - b^2*c^3 - 3*a^2*c*d^2 - 2*b*c^2*d + 2*(2*b*c^2*d + c*d^2)*a
+ (b^3*c^3 - 3*a^3*d^3 + 7*b^2*c^2*d + 6*b*c*d^2 + 3*(2*b*c*d^2 + 3*d^3)*a^2 - 2
*(2*b^2*c^2*d + 8*b*c*d^2 + 3*d^3)*a)*x)*e^(-b*x)/(d^7*x^5*e^a + 5*c*d^6*x^4*e^a
+ 10*c^2*d^5*x^3*e^a + 10*c^3*d^4*x^2*e^a + 5*c^4*d^3*x*e^a + c^5*d^2*e^a), x)

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Fricas [A]  time = 0.271633, size = 1071, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*e^(-b*x - a)/(d*x + c)^4,x, algorithm="fricas")

[Out]

-1/6*((b^7*c^7 - 4*(a - 3)*b^6*c^6*d + 6*(a^2 - 6*a + 6)*b^5*c^5*d^2 - 4*(a^3 -
9*a^2 + 18*a - 6)*b^4*c^4*d^3 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*c^3*d^4 + (b^
7*c^4*d^3 - 4*(a - 3)*b^6*c^3*d^4 + 6*(a^2 - 6*a + 6)*b^5*c^2*d^5 - 4*(a^3 - 9*a
^2 + 18*a - 6)*b^4*c*d^6 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*d^7)*x^3 + 3*(b^7*
c^5*d^2 - 4*(a - 3)*b^6*c^4*d^3 + 6*(a^2 - 6*a + 6)*b^5*c^3*d^4 - 4*(a^3 - 9*a^2
+ 18*a - 6)*b^4*c^2*d^5 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*c*d^6)*x^2 + 3*(b^
7*c^6*d - 4*(a - 3)*b^6*c^5*d^2 + 6*(a^2 - 6*a + 6)*b^5*c^4*d^3 - 4*(a^3 - 9*a^2
+ 18*a - 6)*b^4*c^3*d^4 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*c^2*d^5)*x)*Ei(-(b
*d*x + b*c)/d)*e^((b*c - a*d)/d) + (b^6*c^6*d - (4*a - 11)*b^5*c^5*d^2 + 6*b^3*d
^7*x^3 + 2*(3*a^2 - 16*a + 13)*b^4*c^4*d^3 - 2*(2*a^3 - 15*a^2 + 22*a - 3)*b^3*c
^3*d^4 + 2*a^4*d^7 + (a^4 - 8*a^3 + 12*a^2)*b^2*c^2*d^5 - (a^4 - 4*a^3)*b*c*d^6
+ (b^6*c^4*d^3 - 4*(a - 3)*b^5*c^3*d^4 + 6*(a^2 - 6*a + 6)*b^4*c^2*d^5 - 2*(2*a^
3 - 18*a^2 + 36*a - 9)*b^3*c*d^6 + (a^4 - 12*a^3 + 36*a^2)*b^2*d^7)*x^2 + (2*b^6
*c^5*d^2 - (8*a - 23)*b^5*c^4*d^3 + 4*(3*a^2 - 17*a + 15)*b^4*c^3*d^4 - 2*(4*a^3
- 33*a^2 + 54*a - 9)*b^3*c^2*d^5 + 2*(a^4 - 10*a^3 + 18*a^2)*b^2*c*d^6 - (a^4 -
12*a^3)*b*d^7)*x)*e^(-b*x - a))/(d^11*x^3 + 3*c*d^10*x^2 + 3*c^2*d^9*x + c^3*d^
8)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.254977, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*e^(-b*x - a)/(d*x + c)^4,x, algorithm="giac")

[Out]

Done