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

Optimal. Leaf size=271 \[ -\frac {e^{-a-b x} (a+b x)^4 (b c-a d)}{b^2}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)}{b^2}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)}{b^2}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)}{b^2}-\frac {24 e^{-a-b x} (b c-a d)}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac {120 d e^{-a-b x} (a+b x)}{b^2}-\frac {120 d e^{-a-b x}}{b^2} \]

[Out]

-120*d*exp(-b*x-a)/b^2-24*(-a*d+b*c)*exp(-b*x-a)/b^2-120*d*exp(-b*x-a)*(b*x+a)/b^2-24*(-a*d+b*c)*exp(-b*x-a)*(
b*x+a)/b^2-60*d*exp(-b*x-a)*(b*x+a)^2/b^2-12*(-a*d+b*c)*exp(-b*x-a)*(b*x+a)^2/b^2-20*d*exp(-b*x-a)*(b*x+a)^3/b
^2-4*(-a*d+b*c)*exp(-b*x-a)*(b*x+a)^3/b^2-5*d*exp(-b*x-a)*(b*x+a)^4/b^2-(-a*d+b*c)*exp(-b*x-a)*(b*x+a)^4/b^2-d
*exp(-b*x-a)*(b*x+a)^5/b^2

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Rubi [A]  time = 0.34, antiderivative size = 271, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2196, 2176, 2194} \[ -\frac {e^{-a-b x} (a+b x)^4 (b c-a d)}{b^2}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)}{b^2}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)}{b^2}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)}{b^2}-\frac {24 e^{-a-b x} (b c-a d)}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac {120 d e^{-a-b x} (a+b x)}{b^2}-\frac {120 d e^{-a-b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-120*d*E^(-a - b*x))/b^2 - (24*(b*c - a*d)*E^(-a - b*x))/b^2 - (120*d*E^(-a - b*x)*(a + b*x))/b^2 - (24*(b*c
- a*d)*E^(-a - b*x)*(a + b*x))/b^2 - (60*d*E^(-a - b*x)*(a + b*x)^2)/b^2 - (12*(b*c - a*d)*E^(-a - b*x)*(a + b
*x)^2)/b^2 - (20*d*E^(-a - b*x)*(a + b*x)^3)/b^2 - (4*(b*c - a*d)*E^(-a - b*x)*(a + b*x)^3)/b^2 - (5*d*E^(-a -
b*x)*(a + b*x)^4)/b^2 - ((b*c - a*d)*E^(-a - b*x)*(a + b*x)^4)/b^2 - (d*E^(-a - b*x)*(a + b*x)^5)/b^2

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 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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c
}, x] && PolynomialQ[u, x] && LinearQ[v, x] &&  !$UseGamma === True

Rubi steps

\begin {align*} \int e^{-a-b x} (a+b x)^4 (c+d x) \, dx &=\int \left (\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b}+\frac {d e^{-a-b x} (a+b x)^5}{b}\right ) \, dx\\ &=\frac {d \int e^{-a-b x} (a+b x)^5 \, dx}{b}+\frac {(b c-a d) \int e^{-a-b x} (a+b x)^4 \, dx}{b}\\ &=-\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}+\frac {(5 d) \int e^{-a-b x} (a+b x)^4 \, dx}{b}+\frac {(4 (b c-a d)) \int e^{-a-b x} (a+b x)^3 \, dx}{b}\\ &=-\frac {4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}+\frac {(20 d) \int e^{-a-b x} (a+b x)^3 \, dx}{b}+\frac {(12 (b c-a d)) \int e^{-a-b x} (a+b x)^2 \, dx}{b}\\ &=-\frac {12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}+\frac {(60 d) \int e^{-a-b x} (a+b x)^2 \, dx}{b}+\frac {(24 (b c-a d)) \int e^{-a-b x} (a+b x) \, dx}{b}\\ &=-\frac {24 (b c-a d) e^{-a-b x} (a+b x)}{b^2}-\frac {60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac {12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}+\frac {(120 d) \int e^{-a-b x} (a+b x) \, dx}{b}+\frac {(24 (b c-a d)) \int e^{-a-b x} \, dx}{b}\\ &=-\frac {24 (b c-a d) e^{-a-b x}}{b^2}-\frac {120 d e^{-a-b x} (a+b x)}{b^2}-\frac {24 (b c-a d) e^{-a-b x} (a+b x)}{b^2}-\frac {60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac {12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}+\frac {(120 d) \int e^{-a-b x} \, dx}{b}\\ &=-\frac {120 d e^{-a-b x}}{b^2}-\frac {24 (b c-a d) e^{-a-b x}}{b^2}-\frac {120 d e^{-a-b x} (a+b x)}{b^2}-\frac {24 (b c-a d) e^{-a-b x} (a+b x)}{b^2}-\frac {60 d e^{-a-b x} (a+b x)^2}{b^2}-\frac {12 (b c-a d) e^{-a-b x} (a+b x)^2}{b^2}-\frac {20 d e^{-a-b x} (a+b x)^3}{b^2}-\frac {4 (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}-\frac {5 d e^{-a-b x} (a+b x)^4}{b^2}-\frac {(b c-a d) e^{-a-b x} (a+b x)^4}{b^2}-\frac {d e^{-a-b x} (a+b x)^5}{b^2}\\ \end {align*}

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Mathematica [A]  time = 0.27, size = 191, normalized size = 0.70 \[ \frac {e^{-a-b x} \left (-2 b^3 x^2 \left (3 \left (a^2+2 a+2\right ) c+\left (3 a^2+8 a+10\right ) d x\right )-2 b^2 x \left (2 \left (a^3+3 a^2+6 a+6\right ) c+\left (2 a^3+9 a^2+24 a+30\right ) d x\right )-b \left (\left (a^4+4 a^3+12 a^2+24 a+24\right ) c+\left (a^4+8 a^3+36 a^2+96 a+120\right ) d x\right )-\left (a^4+8 a^3+36 a^2+96 a+120\right ) d-b^4 x^3 (4 (a+1) c+(4 a+5) d x)+b^5 \left (-x^4\right ) (c+d x)\right )}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 197, normalized size = 0.73 \[ -\frac {{\left (b^{5} d x^{5} + {\left (b^{5} c + {\left (4 \, a + 5\right )} b^{4} d\right )} x^{4} + 2 \, {\left (2 \, {\left (a + 1\right )} b^{4} c + {\left (3 \, a^{2} + 8 \, a + 10\right )} b^{3} d\right )} x^{3} + {\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b c + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a + 2\right )} b^{3} c + {\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{2} d\right )} x^{2} + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} d + {\left (4 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{2} c + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b d\right )} x\right )} e^{\left (-b x - a\right )}}{b^{2}} \]

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

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giac [A]  time = 0.42, size = 331, normalized size = 1.22 \[ -\frac {{\left (b^{9} d x^{5} + b^{9} c x^{4} + 4 \, a b^{8} d x^{4} + 4 \, a b^{8} c x^{3} + 6 \, a^{2} b^{7} d x^{3} + 5 \, b^{8} d x^{4} + 6 \, a^{2} b^{7} c x^{2} + 4 \, a^{3} b^{6} d x^{2} + 4 \, b^{8} c x^{3} + 16 \, a b^{7} d x^{3} + 4 \, a^{3} b^{6} c x + a^{4} b^{5} d x + 12 \, a b^{7} c x^{2} + 18 \, a^{2} b^{6} d x^{2} + 20 \, b^{7} d x^{3} + a^{4} b^{5} c + 12 \, a^{2} b^{6} c x + 8 \, a^{3} b^{5} d x + 12 \, b^{7} c x^{2} + 48 \, a b^{6} d x^{2} + 4 \, a^{3} b^{5} c + a^{4} b^{4} d + 24 \, a b^{6} c x + 36 \, a^{2} b^{5} d x + 60 \, b^{6} d x^{2} + 12 \, a^{2} b^{5} c + 8 \, a^{3} b^{4} d + 24 \, b^{6} c x + 96 \, a b^{5} d x + 24 \, a b^{5} c + 36 \, a^{2} b^{4} d + 120 \, b^{5} d x + 24 \, b^{5} c + 96 \, a b^{4} d + 120 \, b^{4} d\right )} e^{\left (-b x - a\right )}}{b^{6}} \]

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^9*d*x^5 + b^9*c*x^4 + 4*a*b^8*d*x^4 + 4*a*b^8*c*x^3 + 6*a^2*b^7*d*x^3 + 5*b^8*d*x^4 + 6*a^2*b^7*c*x^2 + 4*
a^3*b^6*d*x^2 + 4*b^8*c*x^3 + 16*a*b^7*d*x^3 + 4*a^3*b^6*c*x + a^4*b^5*d*x + 12*a*b^7*c*x^2 + 18*a^2*b^6*d*x^2
+ 20*b^7*d*x^3 + a^4*b^5*c + 12*a^2*b^6*c*x + 8*a^3*b^5*d*x + 12*b^7*c*x^2 + 48*a*b^6*d*x^2 + 4*a^3*b^5*c + a
^4*b^4*d + 24*a*b^6*c*x + 36*a^2*b^5*d*x + 60*b^6*d*x^2 + 12*a^2*b^5*c + 8*a^3*b^4*d + 24*b^6*c*x + 96*a*b^5*d
*x + 24*a*b^5*c + 36*a^2*b^4*d + 120*b^5*d*x + 24*b^5*c + 96*a*b^4*d + 120*b^4*d)*e^(-b*x - a)/b^6

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maple [A]  time = 0.01, size = 297, normalized size = 1.10 \[ -\frac {\left (d \,b^{5} x^{5}+4 a \,b^{4} d \,x^{4}+b^{5} c \,x^{4}+6 a^{2} b^{3} d \,x^{3}+4 a \,b^{4} c \,x^{3}+5 b^{4} d \,x^{4}+4 a^{3} b^{2} d \,x^{2}+6 a^{2} b^{3} c \,x^{2}+16 a \,b^{3} d \,x^{3}+4 b^{4} c \,x^{3}+a^{4} b d x +4 a^{3} b^{2} c x +18 a^{2} b^{2} d \,x^{2}+12 a \,b^{3} c \,x^{2}+20 b^{3} d \,x^{3}+c \,a^{4} b +8 a^{3} b d x +12 a^{2} b^{2} c x +48 a \,b^{2} d \,x^{2}+12 b^{3} c \,x^{2}+d \,a^{4}+4 c \,a^{3} b +36 a^{2} b d x +24 a \,b^{2} c x +60 b^{2} d \,x^{2}+8 a^{3} d +12 a^{2} b c +96 a b d x +24 b^{2} c x +36 a^{2} d +24 a b c +120 b d x +96 a d +24 b c +120 d \right ) {\mathrm e}^{-b x -a}}{b^{2}} \]

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]

-(b^5*d*x^5+4*a*b^4*d*x^4+b^5*c*x^4+6*a^2*b^3*d*x^3+4*a*b^4*c*x^3+5*b^4*d*x^4+4*a^3*b^2*d*x^2+6*a^2*b^3*c*x^2+
16*a*b^3*d*x^3+4*b^4*c*x^3+a^4*b*d*x+4*a^3*b^2*c*x+18*a^2*b^2*d*x^2+12*a*b^3*c*x^2+20*b^3*d*x^3+a^4*b*c+8*a^3*
b*d*x+12*a^2*b^2*c*x+48*a*b^2*d*x^2+12*b^3*c*x^2+a^4*d+4*a^3*b*c+36*a^2*b*d*x+24*a*b^2*c*x+60*b^2*d*x^2+8*a^3*
d+12*a^2*b*c+96*a*b*d*x+24*b^2*c*x+36*a^2*d+24*a*b*c+120*b*d*x+96*a*d+24*b*c+120*d)*exp(-b*x-a)/b^2

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maxima [A]  time = 0.79, size = 344, normalized size = 1.27 \[ -\frac {4 \, {\left (b x + 1\right )} a^{3} c e^{\left (-b x - a\right )}}{b} - \frac {a^{4} c e^{\left (-b x - a\right )}}{b} - \frac {{\left (b x + 1\right )} a^{4} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c e^{\left (-b x - a\right )}}{b} - \frac {6 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c e^{\left (-b x - a\right )}}{b} - \frac {4 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} d e^{\left (-b x - a\right )}}{b^{2}} \]

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]

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

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mupad [B]  time = 0.18, size = 264, normalized size = 0.97 \[ -\frac {{\mathrm {e}}^{-a-b\,x}\,\left (120\,d+96\,a\,d+24\,b\,c+36\,a^2\,d+8\,a^3\,d+a^4\,d+24\,a\,b\,c+12\,a^2\,b\,c+4\,a^3\,b\,c+a^4\,b\,c\right )}{b^2}-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (60\,d+48\,a\,d+12\,b\,c+18\,a^2\,d+4\,a^3\,d+12\,a\,b\,c+6\,a^2\,b\,c\right )-x\,{\mathrm {e}}^{-a-b\,x}\,\left (24\,c+24\,a\,c+12\,a^2\,c+4\,a^3\,c+\frac {d\,a^4+8\,d\,a^3+36\,d\,a^2+96\,d\,a+120\,d}{b}\right )-b^3\,d\,x^5\,{\mathrm {e}}^{-a-b\,x}-b^2\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (5\,d+4\,a\,d+b\,c\right )-2\,b\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (10\,d+8\,a\,d+2\,b\,c+3\,a^2\,d+2\,a\,b\,c\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (exp(- a - b*x)*(120*d + 96*a*d + 24*b*c + 36*a^2*d + 8*a^3*d + a^4*d + 24*a*b*c + 12*a^2*b*c + 4*a^3*b*c +
a^4*b*c))/b^2 - x^2*exp(- a - b*x)*(60*d + 48*a*d + 12*b*c + 18*a^2*d + 4*a^3*d + 12*a*b*c + 6*a^2*b*c) - x*ex
p(- a - b*x)*(24*c + 24*a*c + 12*a^2*c + 4*a^3*c + (120*d + 96*a*d + 36*a^2*d + 8*a^3*d + a^4*d)/b) - b^3*d*x^
5*exp(- a - b*x) - b^2*x^4*exp(- a - b*x)*(5*d + 4*a*d + b*c) - 2*b*x^3*exp(- a - b*x)*(10*d + 8*a*d + 2*b*c +
3*a^2*d + 2*a*b*c)

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sympy [A]  time = 0.31, size = 447, normalized size = 1.65 \[ \begin {cases} \frac {\left (- a^{4} b c - a^{4} b d x - a^{4} d - 4 a^{3} b^{2} c x - 4 a^{3} b^{2} d x^{2} - 4 a^{3} b c - 8 a^{3} b d x - 8 a^{3} d - 6 a^{2} b^{3} c x^{2} - 6 a^{2} b^{3} d x^{3} - 12 a^{2} b^{2} c x - 18 a^{2} b^{2} d x^{2} - 12 a^{2} b c - 36 a^{2} b d x - 36 a^{2} d - 4 a b^{4} c x^{3} - 4 a b^{4} d x^{4} - 12 a b^{3} c x^{2} - 16 a b^{3} d x^{3} - 24 a b^{2} c x - 48 a b^{2} d x^{2} - 24 a b c - 96 a b d x - 96 a d - b^{5} c x^{4} - b^{5} d x^{5} - 4 b^{4} c x^{3} - 5 b^{4} d x^{4} - 12 b^{3} c x^{2} - 20 b^{3} d x^{3} - 24 b^{2} c x - 60 b^{2} d x^{2} - 24 b c - 120 b d x - 120 d\right ) e^{- a - b x}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\a^{4} c x + \frac {b^{4} d x^{6}}{6} + x^{5} \left (\frac {4 a b^{3} d}{5} + \frac {b^{4} c}{5}\right ) + x^{4} \left (\frac {3 a^{2} b^{2} d}{2} + a b^{3} c\right ) + x^{3} \left (\frac {4 a^{3} b d}{3} + 2 a^{2} b^{2} c\right ) + x^{2} \left (\frac {a^{4} d}{2} + 2 a^{3} b c\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-a**4*b*c - a**4*b*d*x - a**4*d - 4*a**3*b**2*c*x - 4*a**3*b**2*d*x**2 - 4*a**3*b*c - 8*a**3*b*d*x
- 8*a**3*d - 6*a**2*b**3*c*x**2 - 6*a**2*b**3*d*x**3 - 12*a**2*b**2*c*x - 18*a**2*b**2*d*x**2 - 12*a**2*b*c -
36*a**2*b*d*x - 36*a**2*d - 4*a*b**4*c*x**3 - 4*a*b**4*d*x**4 - 12*a*b**3*c*x**2 - 16*a*b**3*d*x**3 - 24*a*b*
*2*c*x - 48*a*b**2*d*x**2 - 24*a*b*c - 96*a*b*d*x - 96*a*d - b**5*c*x**4 - b**5*d*x**5 - 4*b**4*c*x**3 - 5*b**
4*d*x**4 - 12*b**3*c*x**2 - 20*b**3*d*x**3 - 24*b**2*c*x - 60*b**2*d*x**2 - 24*b*c - 120*b*d*x - 120*d)*exp(-a
- b*x)/b**2, Ne(b**2, 0)), (a**4*c*x + b**4*d*x**6/6 + x**5*(4*a*b**3*d/5 + b**4*c/5) + x**4*(3*a**2*b**2*d/2
+ a*b**3*c) + x**3*(4*a**3*b*d/3 + 2*a**2*b**2*c) + x**2*(a**4*d/2 + 2*a**3*b*c), True))

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