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

Optimal. Leaf size=495 \[ -\frac{2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^3}-\frac{e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^3}-\frac{10 d e^{-a-b x} (a+b x)^4 (b c-a d)}{b^3}-\frac{4 e^{-a-b x} (a+b x)^3 (b c-a d)^2}{b^3}-\frac{40 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac{12 e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac{120 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac{24 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac{240 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac{24 e^{-a-b x} (b c-a d)^2}{b^3}-\frac{240 d e^{-a-b x} (b c-a d)}{b^3}-\frac{d^2 e^{-a-b x} (a+b x)^6}{b^3}-\frac{6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac{30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac{120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac{360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac{720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac{720 d^2 e^{-a-b x}}{b^3} \]

[Out]

(-720*d^2*E^(-a - b*x))/b^3 - (240*d*(b*c - a*d)*E^(-a - b*x))/b^3 - (24*(b*c -
a*d)^2*E^(-a - b*x))/b^3 - (720*d^2*E^(-a - b*x)*(a + b*x))/b^3 - (240*d*(b*c -
a*d)*E^(-a - b*x)*(a + b*x))/b^3 - (24*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x))/b^3
- (360*d^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d*(b*c - a*d)*E^(-a - b*x)*(a +
b*x)^2)/b^3 - (12*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d^2*E^(-a
- b*x)*(a + b*x)^3)/b^3 - (40*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^3)/b^3 - (4*(
b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^3)/b^3 - (30*d^2*E^(-a - b*x)*(a + b*x)^4)/b
^3 - (10*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^4)/b^3 - ((b*c - a*d)^2*E^(-a - b*
x)*(a + b*x)^4)/b^3 - (6*d^2*E^(-a - b*x)*(a + b*x)^5)/b^3 - (2*d*(b*c - a*d)*E^
(-a - b*x)*(a + b*x)^5)/b^3 - (d^2*E^(-a - b*x)*(a + b*x)^6)/b^3

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Rubi [A]  time = 1.04744, antiderivative size = 495, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ -\frac{2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^3}-\frac{e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^3}-\frac{10 d e^{-a-b x} (a+b x)^4 (b c-a d)}{b^3}-\frac{4 e^{-a-b x} (a+b x)^3 (b c-a d)^2}{b^3}-\frac{40 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac{12 e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac{120 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac{24 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac{240 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac{24 e^{-a-b x} (b c-a d)^2}{b^3}-\frac{240 d e^{-a-b x} (b c-a d)}{b^3}-\frac{d^2 e^{-a-b x} (a+b x)^6}{b^3}-\frac{6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac{30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac{120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac{360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac{720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac{720 d^2 e^{-a-b x}}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(-720*d^2*E^(-a - b*x))/b^3 - (240*d*(b*c - a*d)*E^(-a - b*x))/b^3 - (24*(b*c -
a*d)^2*E^(-a - b*x))/b^3 - (720*d^2*E^(-a - b*x)*(a + b*x))/b^3 - (240*d*(b*c -
a*d)*E^(-a - b*x)*(a + b*x))/b^3 - (24*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x))/b^3
- (360*d^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d*(b*c - a*d)*E^(-a - b*x)*(a +
b*x)^2)/b^3 - (12*(b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (120*d^2*E^(-a
- b*x)*(a + b*x)^3)/b^3 - (40*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^3)/b^3 - (4*(
b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^3)/b^3 - (30*d^2*E^(-a - b*x)*(a + b*x)^4)/b
^3 - (10*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^4)/b^3 - ((b*c - a*d)^2*E^(-a - b*
x)*(a + b*x)^4)/b^3 - (6*d^2*E^(-a - b*x)*(a + b*x)^5)/b^3 - (2*d*(b*c - a*d)*E^
(-a - b*x)*(a + b*x)^5)/b^3 - (d^2*E^(-a - b*x)*(a + b*x)^6)/b^3

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Rubi in Sympy [A]  time = 83.4931, size = 454, normalized size = 0.92 \[ - \frac{d^{2} \left (a + b x\right )^{6} e^{- a - b x}}{b^{3}} - \frac{6 d^{2} \left (a + b x\right )^{5} e^{- a - b x}}{b^{3}} - \frac{30 d^{2} \left (a + b x\right )^{4} e^{- a - b x}}{b^{3}} - \frac{120 d^{2} \left (a + b x\right )^{3} e^{- a - b x}}{b^{3}} - \frac{360 d^{2} \left (a + b x\right )^{2} e^{- a - b x}}{b^{3}} - \frac{720 d^{2} \left (a + b x\right ) e^{- a - b x}}{b^{3}} - \frac{720 d^{2} e^{- a - b x}}{b^{3}} + \frac{2 d \left (a + b x\right )^{5} \left (a d - b c\right ) e^{- a - b x}}{b^{3}} + \frac{10 d \left (a + b x\right )^{4} \left (a d - b c\right ) e^{- a - b x}}{b^{3}} + \frac{40 d \left (a + b x\right )^{3} \left (a d - b c\right ) e^{- a - b x}}{b^{3}} + \frac{120 d \left (a + b x\right )^{2} \left (a d - b c\right ) e^{- a - b x}}{b^{3}} + \frac{240 d \left (a + b x\right ) \left (a d - b c\right ) e^{- a - b x}}{b^{3}} + \frac{240 d \left (a d - b c\right ) e^{- a - b x}}{b^{3}} - \frac{\left (a + b x\right )^{4} \left (a d - b c\right )^{2} e^{- a - b x}}{b^{3}} - \frac{4 \left (a + b x\right )^{3} \left (a d - b c\right )^{2} e^{- a - b x}}{b^{3}} - \frac{12 \left (a + b x\right )^{2} \left (a d - b c\right )^{2} e^{- a - b x}}{b^{3}} - \frac{24 \left (a + b x\right ) \left (a d - b c\right )^{2} e^{- a - b x}}{b^{3}} - \frac{24 \left (a d - b c\right )^{2} e^{- a - b x}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(-b*x-a)*(b*x+a)**4*(d*x+c)**2,x)

[Out]

-d**2*(a + b*x)**6*exp(-a - b*x)/b**3 - 6*d**2*(a + b*x)**5*exp(-a - b*x)/b**3 -
30*d**2*(a + b*x)**4*exp(-a - b*x)/b**3 - 120*d**2*(a + b*x)**3*exp(-a - b*x)/b
**3 - 360*d**2*(a + b*x)**2*exp(-a - b*x)/b**3 - 720*d**2*(a + b*x)*exp(-a - b*x
)/b**3 - 720*d**2*exp(-a - b*x)/b**3 + 2*d*(a + b*x)**5*(a*d - b*c)*exp(-a - b*x
)/b**3 + 10*d*(a + b*x)**4*(a*d - b*c)*exp(-a - b*x)/b**3 + 40*d*(a + b*x)**3*(a
*d - b*c)*exp(-a - b*x)/b**3 + 120*d*(a + b*x)**2*(a*d - b*c)*exp(-a - b*x)/b**3
+ 240*d*(a + b*x)*(a*d - b*c)*exp(-a - b*x)/b**3 + 240*d*(a*d - b*c)*exp(-a - b
*x)/b**3 - (a + b*x)**4*(a*d - b*c)**2*exp(-a - b*x)/b**3 - 4*(a + b*x)**3*(a*d
- b*c)**2*exp(-a - b*x)/b**3 - 12*(a + b*x)**2*(a*d - b*c)**2*exp(-a - b*x)/b**3
- 24*(a + b*x)*(a*d - b*c)**2*exp(-a - b*x)/b**3 - 24*(a*d - b*c)**2*exp(-a - b
*x)/b**3

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Mathematica [A]  time = 0.172015, size = 320, normalized size = 0.65 \[ \frac{e^{-a-b x} \left (-2 b^4 x^2 \left (3 \left (a^2+2 a+2\right ) c^2+2 \left (3 a^2+8 a+10\right ) c d x+\left (3 a^2+10 a+15\right ) d^2 x^2\right )-4 b^3 x \left (\left (a^3+3 a^2+6 a+6\right ) c^2+\left (2 a^3+9 a^2+24 a+30\right ) c d x+\left (a^3+6 a^2+20 a+30\right ) d^2 x^2\right )-b^2 \left (\left (a^4+4 a^3+12 a^2+24 a+24\right ) c^2+2 \left (a^4+8 a^3+36 a^2+96 a+120\right ) c d x+\left (a^4+12 a^3+72 a^2+240 a+360\right ) d^2 x^2\right )-2 b d \left (\left (a^4+8 a^3+36 a^2+96 a+120\right ) c+\left (a^4+12 a^3+72 a^2+240 a+360\right ) d x\right )-2 \left (a^4+12 a^3+72 a^2+240 a+360\right ) d^2-2 b^5 x^3 (c+d x) (2 (a+1) c+(2 a+3) d x)+b^6 \left (-x^4\right ) (c+d x)^2\right )}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(E^(-a - b*x)*(-2*(360 + 240*a + 72*a^2 + 12*a^3 + a^4)*d^2 - b^6*x^4*(c + d*x)^
2 - 2*b^5*x^3*(c + d*x)*(2*(1 + a)*c + (3 + 2*a)*d*x) - 2*b*d*((120 + 96*a + 36*
a^2 + 8*a^3 + a^4)*c + (360 + 240*a + 72*a^2 + 12*a^3 + a^4)*d*x) - 2*b^4*x^2*(3
*(2 + 2*a + a^2)*c^2 + 2*(10 + 8*a + 3*a^2)*c*d*x + (15 + 10*a + 3*a^2)*d^2*x^2)
- 4*b^3*x*((6 + 6*a + 3*a^2 + a^3)*c^2 + (30 + 24*a + 9*a^2 + 2*a^3)*c*d*x + (3
0 + 20*a + 6*a^2 + a^3)*d^2*x^2) - b^2*((24 + 24*a + 12*a^2 + 4*a^3 + a^4)*c^2 +
2*(120 + 96*a + 36*a^2 + 8*a^3 + a^4)*c*d*x + (360 + 240*a + 72*a^2 + 12*a^3 +
a^4)*d^2*x^2)))/b^3

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Maple [A]  time = 0.008, size = 640, normalized size = 1.3 \[ -{\frac{ \left ({d}^{2}{b}^{6}{x}^{6}+4\,a{b}^{5}{d}^{2}{x}^{5}+2\,{b}^{6}cd{x}^{5}+6\,{a}^{2}{b}^{4}{d}^{2}{x}^{4}+8\,a{b}^{5}cd{x}^{4}+{b}^{6}{c}^{2}{x}^{4}+6\,{b}^{5}{d}^{2}{x}^{5}+4\,{a}^{3}{b}^{3}{d}^{2}{x}^{3}+12\,{a}^{2}{b}^{4}cd{x}^{3}+4\,a{b}^{5}{c}^{2}{x}^{3}+20\,a{b}^{4}{d}^{2}{x}^{4}+10\,{b}^{5}cd{x}^{4}+{a}^{4}{b}^{2}{d}^{2}{x}^{2}+8\,{a}^{3}{b}^{3}cd{x}^{2}+6\,{a}^{2}{b}^{4}{c}^{2}{x}^{2}+24\,{a}^{2}{b}^{3}{d}^{2}{x}^{3}+32\,a{b}^{4}cd{x}^{3}+4\,{b}^{5}{c}^{2}{x}^{3}+30\,{b}^{4}{d}^{2}{x}^{4}+2\,{a}^{4}{b}^{2}cdx+4\,{a}^{3}{b}^{3}{c}^{2}x+12\,{a}^{3}{b}^{2}{d}^{2}{x}^{2}+36\,{a}^{2}{b}^{3}cd{x}^{2}+12\,a{b}^{4}{c}^{2}{x}^{2}+80\,a{b}^{3}{d}^{2}{x}^{3}+40\,{b}^{4}cd{x}^{3}+{c}^{2}{a}^{4}{b}^{2}+2\,{a}^{4}b{d}^{2}x+16\,{a}^{3}{b}^{2}cdx+12\,{a}^{2}{b}^{3}{c}^{2}x+72\,{a}^{2}{b}^{2}{d}^{2}{x}^{2}+96\,a{b}^{3}cd{x}^{2}+12\,{b}^{4}{c}^{2}{x}^{2}+120\,{b}^{3}{d}^{2}{x}^{3}+2\,cd{a}^{4}b+4\,{c}^{2}{a}^{3}{b}^{2}+24\,{a}^{3}b{d}^{2}x+72\,{a}^{2}{b}^{2}cdx+24\,a{b}^{3}{c}^{2}x+240\,a{b}^{2}{d}^{2}{x}^{2}+120\,{b}^{3}cd{x}^{2}+2\,{d}^{2}{a}^{4}+16\,cd{a}^{3}b+12\,{c}^{2}{a}^{2}{b}^{2}+144\,{a}^{2}b{d}^{2}x+192\,a{b}^{2}cdx+24\,{b}^{3}{c}^{2}x+360\,{b}^{2}{d}^{2}{x}^{2}+24\,{a}^{3}{d}^{2}+72\,{a}^{2}bcd+24\,a{b}^{2}{c}^{2}+480\,ab{d}^{2}x+240\,x{b}^{2}dc+144\,{a}^{2}{d}^{2}+192\,abcd+24\,{b}^{2}{c}^{2}+720\,b{d}^{2}x+480\,a{d}^{2}+240\,bcd+720\,{d}^{2} \right ){{\rm e}^{-bx-a}}}{{b}^{3}}} \]

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]

-(b^6*d^2*x^6+4*a*b^5*d^2*x^5+2*b^6*c*d*x^5+6*a^2*b^4*d^2*x^4+8*a*b^5*c*d*x^4+b^
6*c^2*x^4+6*b^5*d^2*x^5+4*a^3*b^3*d^2*x^3+12*a^2*b^4*c*d*x^3+4*a*b^5*c^2*x^3+20*
a*b^4*d^2*x^4+10*b^5*c*d*x^4+a^4*b^2*d^2*x^2+8*a^3*b^3*c*d*x^2+6*a^2*b^4*c^2*x^2
+24*a^2*b^3*d^2*x^3+32*a*b^4*c*d*x^3+4*b^5*c^2*x^3+30*b^4*d^2*x^4+2*a^4*b^2*c*d*
x+4*a^3*b^3*c^2*x+12*a^3*b^2*d^2*x^2+36*a^2*b^3*c*d*x^2+12*a*b^4*c^2*x^2+80*a*b^
3*d^2*x^3+40*b^4*c*d*x^3+a^4*b^2*c^2+2*a^4*b*d^2*x+16*a^3*b^2*c*d*x+12*a^2*b^3*c
^2*x+72*a^2*b^2*d^2*x^2+96*a*b^3*c*d*x^2+12*b^4*c^2*x^2+120*b^3*d^2*x^3+2*a^4*b*
c*d+4*a^3*b^2*c^2+24*a^3*b*d^2*x+72*a^2*b^2*c*d*x+24*a*b^3*c^2*x+240*a*b^2*d^2*x
^2+120*b^3*c*d*x^2+2*a^4*d^2+16*a^3*b*c*d+12*a^2*b^2*c^2+144*a^2*b*d^2*x+192*a*b
^2*c*d*x+24*b^3*c^2*x+360*b^2*d^2*x^2+24*a^3*d^2+72*a^2*b*c*d+24*a*b^2*c^2+480*a
*b*d^2*x+240*b^2*c*d*x+144*a^2*d^2+192*a*b*c*d+24*b^2*c^2+720*b*d^2*x+480*a*d^2+
240*b*c*d+720*d^2)*exp(-b*x-a)/b^3

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Maxima [A]  time = 0.792814, size = 809, normalized size = 1.63 \[ -\frac{4 \,{\left (b x + 1\right )} a^{3} c^{2} e^{\left (-b x - a\right )}}{b} - \frac{a^{4} c^{2} e^{\left (-b x - a\right )}}{b} - \frac{2 \,{\left (b x + 1\right )} a^{4} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac{6 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c^{2} e^{\left (-b x - a\right )}}{b} - \frac{8 \,{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac{{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{4} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac{4 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c^{2} e^{\left (-b x - a\right )}}{b} - \frac{12 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac{4 \,{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac{{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c^{2} e^{\left (-b x - a\right )}}{b} - \frac{8 \,{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a c d e^{\left (-b x - a\right )}}{b^{2}} - \frac{6 \,{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac{2 \,{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac{4 \,{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} a d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac{{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\right )} d^{2} e^{\left (-b x - a\right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^2*e^(-b*x - a),x, algorithm="maxima")

[Out]

-4*(b*x + 1)*a^3*c^2*e^(-b*x - a)/b - a^4*c^2*e^(-b*x - a)/b - 2*(b*x + 1)*a^4*c
*d*e^(-b*x - a)/b^2 - 6*(b^2*x^2 + 2*b*x + 2)*a^2*c^2*e^(-b*x - a)/b - 8*(b^2*x^
2 + 2*b*x + 2)*a^3*c*d*e^(-b*x - a)/b^2 - (b^2*x^2 + 2*b*x + 2)*a^4*d^2*e^(-b*x
- a)/b^3 - 4*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*a*c^2*e^(-b*x - a)/b - 12*(b^3*x^
3 + 3*b^2*x^2 + 6*b*x + 6)*a^2*c*d*e^(-b*x - a)/b^2 - 4*(b^3*x^3 + 3*b^2*x^2 + 6
*b*x + 6)*a^3*d^2*e^(-b*x - a)/b^3 - (b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x
+ 24)*c^2*e^(-b*x - a)/b - 8*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*a*
c*d*e^(-b*x - a)/b^2 - 6*(b^4*x^4 + 4*b^3*x^3 + 12*b^2*x^2 + 24*b*x + 24)*a^2*d^
2*e^(-b*x - a)/b^3 - 2*(b^5*x^5 + 5*b^4*x^4 + 20*b^3*x^3 + 60*b^2*x^2 + 120*b*x
+ 120)*c*d*e^(-b*x - a)/b^2 - 4*(b^5*x^5 + 5*b^4*x^4 + 20*b^3*x^3 + 60*b^2*x^2 +
120*b*x + 120)*a*d^2*e^(-b*x - a)/b^3 - (b^6*x^6 + 6*b^5*x^5 + 30*b^4*x^4 + 120
*b^3*x^3 + 360*b^2*x^2 + 720*b*x + 720)*d^2*e^(-b*x - a)/b^3

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Fricas [A]  time = 0.250619, size = 478, normalized size = 0.97 \[ -\frac{{\left (b^{6} d^{2} x^{6} + 2 \,{\left (b^{6} c d +{\left (2 \, a + 3\right )} b^{5} d^{2}\right )} x^{5} +{\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b^{2} c^{2} +{\left (b^{6} c^{2} + 2 \,{\left (4 \, a + 5\right )} b^{5} c d + 2 \,{\left (3 \, a^{2} + 10 \, a + 15\right )} b^{4} d^{2}\right )} x^{4} + 2 \,{\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b c d + 4 \,{\left ({\left (a + 1\right )} b^{5} c^{2} +{\left (3 \, a^{2} + 8 \, a + 10\right )} b^{4} c d +{\left (a^{3} + 6 \, a^{2} + 20 \, a + 30\right )} b^{3} d^{2}\right )} x^{3} + 2 \,{\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} d^{2} +{\left (6 \,{\left (a^{2} + 2 \, a + 2\right )} b^{4} c^{2} + 4 \,{\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{3} c d +{\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b^{2} d^{2}\right )} x^{2} + 2 \,{\left (2 \,{\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{3} c^{2} +{\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b^{2} c d +{\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^2*e^(-b*x - a),x, algorithm="fricas")

[Out]

-(b^6*d^2*x^6 + 2*(b^6*c*d + (2*a + 3)*b^5*d^2)*x^5 + (a^4 + 4*a^3 + 12*a^2 + 24
*a + 24)*b^2*c^2 + (b^6*c^2 + 2*(4*a + 5)*b^5*c*d + 2*(3*a^2 + 10*a + 15)*b^4*d^
2)*x^4 + 2*(a^4 + 8*a^3 + 36*a^2 + 96*a + 120)*b*c*d + 4*((a + 1)*b^5*c^2 + (3*a
^2 + 8*a + 10)*b^4*c*d + (a^3 + 6*a^2 + 20*a + 30)*b^3*d^2)*x^3 + 2*(a^4 + 12*a^
3 + 72*a^2 + 240*a + 360)*d^2 + (6*(a^2 + 2*a + 2)*b^4*c^2 + 4*(2*a^3 + 9*a^2 +
24*a + 30)*b^3*c*d + (a^4 + 12*a^3 + 72*a^2 + 240*a + 360)*b^2*d^2)*x^2 + 2*(2*(
a^3 + 3*a^2 + 6*a + 6)*b^3*c^2 + (a^4 + 8*a^3 + 36*a^2 + 96*a + 120)*b^2*c*d + (
a^4 + 12*a^3 + 72*a^2 + 240*a + 360)*b*d^2)*x)*e^(-b*x - a)/b^3

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Sympy [A]  time = 1.2155, size = 899, normalized size = 1.82 \[ \begin{cases} \frac{\left (- a^{4} b^{2} c^{2} - 2 a^{4} b^{2} c d x - a^{4} b^{2} d^{2} x^{2} - 2 a^{4} b c d - 2 a^{4} b d^{2} x - 2 a^{4} d^{2} - 4 a^{3} b^{3} c^{2} x - 8 a^{3} b^{3} c d x^{2} - 4 a^{3} b^{3} d^{2} x^{3} - 4 a^{3} b^{2} c^{2} - 16 a^{3} b^{2} c d x - 12 a^{3} b^{2} d^{2} x^{2} - 16 a^{3} b c d - 24 a^{3} b d^{2} x - 24 a^{3} d^{2} - 6 a^{2} b^{4} c^{2} x^{2} - 12 a^{2} b^{4} c d x^{3} - 6 a^{2} b^{4} d^{2} x^{4} - 12 a^{2} b^{3} c^{2} x - 36 a^{2} b^{3} c d x^{2} - 24 a^{2} b^{3} d^{2} x^{3} - 12 a^{2} b^{2} c^{2} - 72 a^{2} b^{2} c d x - 72 a^{2} b^{2} d^{2} x^{2} - 72 a^{2} b c d - 144 a^{2} b d^{2} x - 144 a^{2} d^{2} - 4 a b^{5} c^{2} x^{3} - 8 a b^{5} c d x^{4} - 4 a b^{5} d^{2} x^{5} - 12 a b^{4} c^{2} x^{2} - 32 a b^{4} c d x^{3} - 20 a b^{4} d^{2} x^{4} - 24 a b^{3} c^{2} x - 96 a b^{3} c d x^{2} - 80 a b^{3} d^{2} x^{3} - 24 a b^{2} c^{2} - 192 a b^{2} c d x - 240 a b^{2} d^{2} x^{2} - 192 a b c d - 480 a b d^{2} x - 480 a d^{2} - b^{6} c^{2} x^{4} - 2 b^{6} c d x^{5} - b^{6} d^{2} x^{6} - 4 b^{5} c^{2} x^{3} - 10 b^{5} c d x^{4} - 6 b^{5} d^{2} x^{5} - 12 b^{4} c^{2} x^{2} - 40 b^{4} c d x^{3} - 30 b^{4} d^{2} x^{4} - 24 b^{3} c^{2} x - 120 b^{3} c d x^{2} - 120 b^{3} d^{2} x^{3} - 24 b^{2} c^{2} - 240 b^{2} c d x - 360 b^{2} d^{2} x^{2} - 240 b c d - 720 b d^{2} x - 720 d^{2}\right ) e^{- a - b x}}{b^{3}} & \text{for}\: b^{3} \neq 0 \\a^{4} c^{2} x + \frac{b^{4} d^{2} x^{7}}{7} + x^{6} \left (\frac{2 a b^{3} d^{2}}{3} + \frac{b^{4} c d}{3}\right ) + x^{5} \left (\frac{6 a^{2} b^{2} d^{2}}{5} + \frac{8 a b^{3} c d}{5} + \frac{b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac{a^{4} d^{2}}{3} + \frac{8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) & \text{otherwise} \end{cases} \]

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]

Piecewise(((-a**4*b**2*c**2 - 2*a**4*b**2*c*d*x - a**4*b**2*d**2*x**2 - 2*a**4*b
*c*d - 2*a**4*b*d**2*x - 2*a**4*d**2 - 4*a**3*b**3*c**2*x - 8*a**3*b**3*c*d*x**2
- 4*a**3*b**3*d**2*x**3 - 4*a**3*b**2*c**2 - 16*a**3*b**2*c*d*x - 12*a**3*b**2*
d**2*x**2 - 16*a**3*b*c*d - 24*a**3*b*d**2*x - 24*a**3*d**2 - 6*a**2*b**4*c**2*x
**2 - 12*a**2*b**4*c*d*x**3 - 6*a**2*b**4*d**2*x**4 - 12*a**2*b**3*c**2*x - 36*a
**2*b**3*c*d*x**2 - 24*a**2*b**3*d**2*x**3 - 12*a**2*b**2*c**2 - 72*a**2*b**2*c*
d*x - 72*a**2*b**2*d**2*x**2 - 72*a**2*b*c*d - 144*a**2*b*d**2*x - 144*a**2*d**2
- 4*a*b**5*c**2*x**3 - 8*a*b**5*c*d*x**4 - 4*a*b**5*d**2*x**5 - 12*a*b**4*c**2*
x**2 - 32*a*b**4*c*d*x**3 - 20*a*b**4*d**2*x**4 - 24*a*b**3*c**2*x - 96*a*b**3*c
*d*x**2 - 80*a*b**3*d**2*x**3 - 24*a*b**2*c**2 - 192*a*b**2*c*d*x - 240*a*b**2*d
**2*x**2 - 192*a*b*c*d - 480*a*b*d**2*x - 480*a*d**2 - b**6*c**2*x**4 - 2*b**6*c
*d*x**5 - b**6*d**2*x**6 - 4*b**5*c**2*x**3 - 10*b**5*c*d*x**4 - 6*b**5*d**2*x**
5 - 12*b**4*c**2*x**2 - 40*b**4*c*d*x**3 - 30*b**4*d**2*x**4 - 24*b**3*c**2*x -
120*b**3*c*d*x**2 - 120*b**3*d**2*x**3 - 24*b**2*c**2 - 240*b**2*c*d*x - 360*b**
2*d**2*x**2 - 240*b*c*d - 720*b*d**2*x - 720*d**2)*exp(-a - b*x)/b**3, Ne(b**3,
0)), (a**4*c**2*x + b**4*d**2*x**7/7 + x**6*(2*a*b**3*d**2/3 + b**4*c*d/3) + x**
5*(6*a**2*b**2*d**2/5 + 8*a*b**3*c*d/5 + b**4*c**2/5) + x**4*(a**3*b*d**2 + 3*a*
*2*b**2*c*d + a*b**3*c**2) + x**3*(a**4*d**2/3 + 8*a**3*b*c*d/3 + 2*a**2*b**2*c*
*2) + x**2*(a**4*c*d + 2*a**3*b*c**2), True))

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GIAC/XCAS [A]  time = 0.255389, size = 910, normalized size = 1.84 \[ -\frac{{\left (b^{10} d^{2} x^{6} + 2 \, b^{10} c d x^{5} + 4 \, a b^{9} d^{2} x^{5} + b^{10} c^{2} x^{4} + 8 \, a b^{9} c d x^{4} + 6 \, a^{2} b^{8} d^{2} x^{4} + 6 \, b^{9} d^{2} x^{5} + 4 \, a b^{9} c^{2} x^{3} + 12 \, a^{2} b^{8} c d x^{3} + 4 \, a^{3} b^{7} d^{2} x^{3} + 10 \, b^{9} c d x^{4} + 20 \, a b^{8} d^{2} x^{4} + 6 \, a^{2} b^{8} c^{2} x^{2} + 8 \, a^{3} b^{7} c d x^{2} + a^{4} b^{6} d^{2} x^{2} + 4 \, b^{9} c^{2} x^{3} + 32 \, a b^{8} c d x^{3} + 24 \, a^{2} b^{7} d^{2} x^{3} + 30 \, b^{8} d^{2} x^{4} + 4 \, a^{3} b^{7} c^{2} x + 2 \, a^{4} b^{6} c d x + 12 \, a b^{8} c^{2} x^{2} + 36 \, a^{2} b^{7} c d x^{2} + 12 \, a^{3} b^{6} d^{2} x^{2} + 40 \, b^{8} c d x^{3} + 80 \, a b^{7} d^{2} x^{3} + a^{4} b^{6} c^{2} + 12 \, a^{2} b^{7} c^{2} x + 16 \, a^{3} b^{6} c d x + 2 \, a^{4} b^{5} d^{2} x + 12 \, b^{8} c^{2} x^{2} + 96 \, a b^{7} c d x^{2} + 72 \, a^{2} b^{6} d^{2} x^{2} + 120 \, b^{7} d^{2} x^{3} + 4 \, a^{3} b^{6} c^{2} + 2 \, a^{4} b^{5} c d + 24 \, a b^{7} c^{2} x + 72 \, a^{2} b^{6} c d x + 24 \, a^{3} b^{5} d^{2} x + 120 \, b^{7} c d x^{2} + 240 \, a b^{6} d^{2} x^{2} + 12 \, a^{2} b^{6} c^{2} + 16 \, a^{3} b^{5} c d + 2 \, a^{4} b^{4} d^{2} + 24 \, b^{7} c^{2} x + 192 \, a b^{6} c d x + 144 \, a^{2} b^{5} d^{2} x + 360 \, b^{6} d^{2} x^{2} + 24 \, a b^{6} c^{2} + 72 \, a^{2} b^{5} c d + 24 \, a^{3} b^{4} d^{2} + 240 \, b^{6} c d x + 480 \, a b^{5} d^{2} x + 24 \, b^{6} c^{2} + 192 \, a b^{5} c d + 144 \, a^{2} b^{4} d^{2} + 720 \, b^{5} d^{2} x + 240 \, b^{5} c d + 480 \, a b^{4} d^{2} + 720 \, b^{4} d^{2}\right )} e^{\left (-b x - a\right )}}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^4*(d*x + c)^2*e^(-b*x - a),x, algorithm="giac")

[Out]

-(b^10*d^2*x^6 + 2*b^10*c*d*x^5 + 4*a*b^9*d^2*x^5 + b^10*c^2*x^4 + 8*a*b^9*c*d*x
^4 + 6*a^2*b^8*d^2*x^4 + 6*b^9*d^2*x^5 + 4*a*b^9*c^2*x^3 + 12*a^2*b^8*c*d*x^3 +
4*a^3*b^7*d^2*x^3 + 10*b^9*c*d*x^4 + 20*a*b^8*d^2*x^4 + 6*a^2*b^8*c^2*x^2 + 8*a^
3*b^7*c*d*x^2 + a^4*b^6*d^2*x^2 + 4*b^9*c^2*x^3 + 32*a*b^8*c*d*x^3 + 24*a^2*b^7*
d^2*x^3 + 30*b^8*d^2*x^4 + 4*a^3*b^7*c^2*x + 2*a^4*b^6*c*d*x + 12*a*b^8*c^2*x^2
+ 36*a^2*b^7*c*d*x^2 + 12*a^3*b^6*d^2*x^2 + 40*b^8*c*d*x^3 + 80*a*b^7*d^2*x^3 +
a^4*b^6*c^2 + 12*a^2*b^7*c^2*x + 16*a^3*b^6*c*d*x + 2*a^4*b^5*d^2*x + 12*b^8*c^2
*x^2 + 96*a*b^7*c*d*x^2 + 72*a^2*b^6*d^2*x^2 + 120*b^7*d^2*x^3 + 4*a^3*b^6*c^2 +
2*a^4*b^5*c*d + 24*a*b^7*c^2*x + 72*a^2*b^6*c*d*x + 24*a^3*b^5*d^2*x + 120*b^7*
c*d*x^2 + 240*a*b^6*d^2*x^2 + 12*a^2*b^6*c^2 + 16*a^3*b^5*c*d + 2*a^4*b^4*d^2 +
24*b^7*c^2*x + 192*a*b^6*c*d*x + 144*a^2*b^5*d^2*x + 360*b^6*d^2*x^2 + 24*a*b^6*
c^2 + 72*a^2*b^5*c*d + 24*a^3*b^4*d^2 + 240*b^6*c*d*x + 480*a*b^5*d^2*x + 24*b^6
*c^2 + 192*a*b^5*c*d + 144*a^2*b^4*d^2 + 720*b^5*d^2*x + 240*b^5*c*d + 480*a*b^4
*d^2 + 720*b^4*d^2)*e^(-b*x - a)/b^7