\(\int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx\) [75]

Optimal result

Integrand size = 25, antiderivative size = 495 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {720 d^2 e^{-a-b x}}{b^3}-\frac {240 d (b c-a d) e^{-a-b x}}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2227, 2207, 2225} \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {2 d e^{-a-b x} (a+b x)^5 (b c-a d)}{b^3}-\frac {e^{-a-b x} (a+b x)^4 (b c-a d)^2}{b^3}-\frac {10 d e^{-a-b x} (a+b x)^4 (b c-a d)}{b^3}-\frac {4 e^{-a-b x} (a+b x)^3 (b c-a d)^2}{b^3}-\frac {40 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac {12 e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac {120 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac {24 e^{-a-b x} (b c-a d)^2}{b^3}-\frac {240 d e^{-a-b x} (b c-a d)}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {720 d^2 e^{-a-b x}}{b^3} \]

[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 - (72
0*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

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 2227

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] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^2}+\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^2}+\frac {d^2 e^{-a-b x} (a+b x)^6}{b^2}\right ) \, dx \\ & = \frac {d^2 \int e^{-a-b x} (a+b x)^6 \, dx}{b^2}+\frac {(2 d (b c-a d)) \int e^{-a-b x} (a+b x)^5 \, dx}{b^2}+\frac {(b c-a d)^2 \int e^{-a-b x} (a+b x)^4 \, dx}{b^2} \\ & = -\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (6 d^2\right ) \int e^{-a-b x} (a+b x)^5 \, dx}{b^2}+\frac {(10 d (b c-a d)) \int e^{-a-b x} (a+b x)^4 \, dx}{b^2}+\frac {\left (4 (b c-a d)^2\right ) \int e^{-a-b x} (a+b x)^3 \, dx}{b^2} \\ & = -\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (30 d^2\right ) \int e^{-a-b x} (a+b x)^4 \, dx}{b^2}+\frac {(40 d (b c-a d)) \int e^{-a-b x} (a+b x)^3 \, dx}{b^2}+\frac {\left (12 (b c-a d)^2\right ) \int e^{-a-b x} (a+b x)^2 \, dx}{b^2} \\ & = -\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (120 d^2\right ) \int e^{-a-b x} (a+b x)^3 \, dx}{b^2}+\frac {(120 d (b c-a d)) \int e^{-a-b x} (a+b x)^2 \, dx}{b^2}+\frac {\left (24 (b c-a d)^2\right ) \int e^{-a-b x} (a+b x) \, dx}{b^2} \\ & = -\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (360 d^2\right ) \int e^{-a-b x} (a+b x)^2 \, dx}{b^2}+\frac {(240 d (b c-a d)) \int e^{-a-b x} (a+b x) \, dx}{b^2}+\frac {\left (24 (b c-a d)^2\right ) \int e^{-a-b x} \, dx}{b^2} \\ & = -\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (720 d^2\right ) \int e^{-a-b x} (a+b x) \, dx}{b^2}+\frac {(240 d (b c-a d)) \int e^{-a-b x} \, dx}{b^2} \\ & = -\frac {240 d (b c-a d) e^{-a-b x}}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3}+\frac {\left (720 d^2\right ) \int e^{-a-b x} \, dx}{b^2} \\ & = -\frac {720 d^2 e^{-a-b x}}{b^3}-\frac {240 d (b c-a d) e^{-a-b x}}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x}}{b^3}-\frac {720 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {240 d (b c-a d) e^{-a-b x} (a+b x)}{b^3}-\frac {24 (b c-a d)^2 e^{-a-b x} (a+b x)}{b^3}-\frac {360 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d (b c-a d) e^{-a-b x} (a+b x)^2}{b^3}-\frac {12 (b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {120 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {40 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^3}-\frac {4 (b c-a d)^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {30 d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {10 d (b c-a d) e^{-a-b x} (a+b x)^4}{b^3}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {6 d^2 e^{-a-b x} (a+b x)^5}{b^3}-\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^5}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^6}{b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.07 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.65 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=\frac {e^{-a-b x} \left (-2 \left (360+240 a+72 a^2+12 a^3+a^4\right ) d^2-b^6 x^4 (c+d x)^2-2 b^5 x^3 (c+d x) (2 (1+a) c+(3+2 a) d x)-2 b d \left (\left (120+96 a+36 a^2+8 a^3+a^4\right ) c+\left (360+240 a+72 a^2+12 a^3+a^4\right ) d x\right )-2 b^4 x^2 \left (3 \left (2+2 a+a^2\right ) c^2+2 \left (10+8 a+3 a^2\right ) c d x+\left (15+10 a+3 a^2\right ) d^2 x^2\right )-4 b^3 x \left (\left (6+6 a+3 a^2+a^3\right ) c^2+\left (30+24 a+9 a^2+2 a^3\right ) c d x+\left (30+20 a+6 a^2+a^3\right ) d^2 x^2\right )-b^2 \left (\left (24+24 a+12 a^2+4 a^3+a^4\right ) c^2+2 \left (120+96 a+36 a^2+8 a^3+a^4\right ) c d x+\left (360+240 a+72 a^2+12 a^3+a^4\right ) d^2 x^2\right )\right )}{b^3} \]

[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 + (30 + 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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.13

method	result	size
norman	\(\left (-4 a \,b^{2} d^{2}-2 c d \,b^{3}-6 b^{2} d^{2}\right ) x^{5} {\mathrm e}^{-b x -a}+\left (-6 a^{2} b \,d^{2}-8 a \,b^{2} c d -c^{2} b^{3}-20 a b \,d^{2}-10 b^{2} c d -30 b \,d^{2}\right ) x^{4} {\mathrm e}^{-b x -a}+\left (-4 a^{3} d^{2}-12 a^{2} b c d -4 a \,b^{2} c^{2}-24 a^{2} d^{2}-32 a b c d -4 b^{2} c^{2}-80 a \,d^{2}-40 b c d -120 d^{2}\right ) x^{3} {\mathrm e}^{-b x -a}-\frac {\left (c^{2} a^{4} b^{2}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}-d^{2} b^{3} x^{6} {\mathrm e}^{-b x -a}-\frac {\left (d^{2} a^{4}+8 c d \,a^{3} b +6 c^{2} a^{2} b^{2}+12 a^{3} d^{2}+36 a^{2} b c d +12 a \,b^{2} c^{2}+72 a^{2} d^{2}+96 a b c d +12 b^{2} c^{2}+240 a \,d^{2}+120 b c d +360 d^{2}\right ) x^{2} {\mathrm e}^{-b x -a}}{b}-\frac {2 \left (c d \,a^{4} b +2 c^{2} a^{3} b^{2}+d^{2} a^{4}+8 c d \,a^{3} b +6 c^{2} a^{2} b^{2}+12 a^{3} d^{2}+36 a^{2} b c d +12 a \,b^{2} c^{2}+72 a^{2} d^{2}+96 a b c d +12 b^{2} c^{2}+240 a \,d^{2}+120 b c d +360 d^{2}\right ) x \,{\mathrm e}^{-b x -a}}{b^{2}}\)	\(560\)
gosper	\(-\frac {\left (d^{2} b^{6} x^{6}+4 a \,b^{5} d^{2} x^{5}+2 b^{6} c d \,x^{5}+6 a^{2} b^{4} d^{2} x^{4}+8 a \,b^{5} c d \,x^{4}+b^{6} c^{2} x^{4}+6 b^{5} d^{2} x^{5}+4 a^{3} b^{3} d^{2} x^{3}+12 a^{2} b^{4} c d \,x^{3}+4 a \,b^{5} c^{2} x^{3}+20 a \,b^{4} d^{2} x^{4}+10 b^{5} c d \,x^{4}+a^{4} b^{2} d^{2} x^{2}+8 a^{3} b^{3} c d \,x^{2}+6 a^{2} b^{4} c^{2} x^{2}+24 a^{2} b^{3} d^{2} x^{3}+32 a \,b^{4} c d \,x^{3}+4 b^{5} c^{2} x^{3}+30 b^{4} d^{2} x^{4}+2 a^{4} b^{2} c d x +4 a^{3} b^{3} c^{2} x +12 a^{3} b^{2} d^{2} x^{2}+36 a^{2} b^{3} c d \,x^{2}+12 a \,b^{4} c^{2} x^{2}+80 a \,b^{3} d^{2} x^{3}+40 b^{4} c d \,x^{3}+c^{2} a^{4} b^{2}+2 a^{4} b \,d^{2} x +16 a^{3} b^{2} c d x +12 a^{2} b^{3} c^{2} x +72 a^{2} b^{2} d^{2} x^{2}+96 a \,b^{3} c d \,x^{2}+12 b^{4} c^{2} x^{2}+120 b^{3} d^{2} x^{3}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+24 a^{3} b \,d^{2} x +72 a^{2} b^{2} c d x +24 a \,b^{3} c^{2} x +240 a \,b^{2} d^{2} x^{2}+120 b^{3} c d \,x^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+144 a^{2} b \,d^{2} x +192 a \,b^{2} c d x +24 b^{3} c^{2} x +360 b^{2} d^{2} x^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+480 a b \,d^{2} x +240 x \,b^{2} d c +144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+720 b \,d^{2} x +480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(640\)
risch	\(-\frac {\left (d^{2} b^{6} x^{6}+4 a \,b^{5} d^{2} x^{5}+2 b^{6} c d \,x^{5}+6 a^{2} b^{4} d^{2} x^{4}+8 a \,b^{5} c d \,x^{4}+b^{6} c^{2} x^{4}+6 b^{5} d^{2} x^{5}+4 a^{3} b^{3} d^{2} x^{3}+12 a^{2} b^{4} c d \,x^{3}+4 a \,b^{5} c^{2} x^{3}+20 a \,b^{4} d^{2} x^{4}+10 b^{5} c d \,x^{4}+a^{4} b^{2} d^{2} x^{2}+8 a^{3} b^{3} c d \,x^{2}+6 a^{2} b^{4} c^{2} x^{2}+24 a^{2} b^{3} d^{2} x^{3}+32 a \,b^{4} c d \,x^{3}+4 b^{5} c^{2} x^{3}+30 b^{4} d^{2} x^{4}+2 a^{4} b^{2} c d x +4 a^{3} b^{3} c^{2} x +12 a^{3} b^{2} d^{2} x^{2}+36 a^{2} b^{3} c d \,x^{2}+12 a \,b^{4} c^{2} x^{2}+80 a \,b^{3} d^{2} x^{3}+40 b^{4} c d \,x^{3}+c^{2} a^{4} b^{2}+2 a^{4} b \,d^{2} x +16 a^{3} b^{2} c d x +12 a^{2} b^{3} c^{2} x +72 a^{2} b^{2} d^{2} x^{2}+96 a \,b^{3} c d \,x^{2}+12 b^{4} c^{2} x^{2}+120 b^{3} d^{2} x^{3}+2 c d \,a^{4} b +4 c^{2} a^{3} b^{2}+24 a^{3} b \,d^{2} x +72 a^{2} b^{2} c d x +24 a \,b^{3} c^{2} x +240 a \,b^{2} d^{2} x^{2}+120 b^{3} c d \,x^{2}+2 d^{2} a^{4}+16 c d \,a^{3} b +12 c^{2} a^{2} b^{2}+144 a^{2} b \,d^{2} x +192 a \,b^{2} c d x +24 b^{3} c^{2} x +360 b^{2} d^{2} x^{2}+24 a^{3} d^{2}+72 a^{2} b c d +24 a \,b^{2} c^{2}+480 a b \,d^{2} x +240 x \,b^{2} d c +144 a^{2} d^{2}+192 a b c d +24 b^{2} c^{2}+720 b \,d^{2} x +480 a \,d^{2}+240 b c d +720 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(640\)
meijerg	\(\frac {{\mathrm e}^{-a} d^{2} \left (720-\frac {\left (7 b^{6} x^{6}+42 b^{5} x^{5}+210 b^{4} x^{4}+840 b^{3} x^{3}+2520 b^{2} x^{2}+5040 b x +5040\right ) {\mathrm e}^{-b x}}{7}\right )}{b^{3}}+\frac {2 \,{\mathrm e}^{-a} d c \left (120-\frac {\left (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 ) {\mathrm e}^{-b x}}{6}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} c^{2} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a \,d^{2} \left (120-\frac {\left (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 ) {\mathrm e}^{-b x}}{6}\right )}{b^{3}}+\frac {8 \,{\mathrm e}^{-a} a d c \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{2}}+\frac {4 \,{\mathrm e}^{-a} a \,c^{2} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b}+\frac {6 \,{\mathrm e}^{-a} a^{2} d^{2} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{3}}+\frac {12 \,{\mathrm e}^{-a} a^{2} d c \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{2}}+\frac {6 \,{\mathrm e}^{-a} a^{2} c^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}+\frac {4 \,{\mathrm e}^{-a} a^{3} d^{2} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{3}}+\frac {8 \,{\mathrm e}^{-a} a^{3} d c \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{2}}+\frac {4 \,{\mathrm e}^{-a} a^{3} c^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b}+\frac {{\mathrm e}^{-a} a^{4} d^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {2 \,{\mathrm e}^{-a} a^{4} d c \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} a^{4} c^{2} \left (1-{\mathrm e}^{-b x}\right )}{b}\)	\(670\)
derivativedivides	\(-\frac {c^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+\frac {d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d c \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d^{2} a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d a c \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\)	\(694\)
default	\(-\frac {c^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )+\frac {d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{6}-6 \left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}+30 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+360 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-720 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+720 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d c \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d^{2} a \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d a c \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\)	\(694\)
parts	\(-d^{2} b^{3} x^{6} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} b^{2} a \,d^{2} x^{5}-2 \,{\mathrm e}^{-b x -a} b^{3} c d \,x^{5}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} d^{2} x^{4}-8 \,{\mathrm e}^{-b x -a} b^{2} a c d \,x^{4}-{\mathrm e}^{-b x -a} b^{3} c^{2} x^{4}-4 \,{\mathrm e}^{-b x -a} a^{3} d^{2} x^{3}-12 \,{\mathrm e}^{-b x -a} b \,a^{2} c d \,x^{3}-4 \,{\mathrm e}^{-b x -a} b^{2} a \,c^{2} x^{3}-\frac {{\mathrm e}^{-b x -a} a^{4} d^{2} x^{2}}{b}-8 \,{\mathrm e}^{-b x -a} a^{3} c d \,x^{2}-6 \,{\mathrm e}^{-b x -a} b \,a^{2} c^{2} x^{2}-\frac {2 \,{\mathrm e}^{-b x -a} a^{4} c d x}{b}-4 \,{\mathrm e}^{-b x -a} a^{3} c^{2} x -\frac {{\mathrm e}^{-b x -a} c^{2} a^{4}}{b}-\frac {2 \left (-2 b \,c^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+5 c d \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )-\frac {3 d^{2} \left (\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d^{2} a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}+4 d a c \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )-\frac {5 d^{2} a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b}\right )}{b^{2}}\)	\(870\)
parallelrisch	\(\text {Expression too large to display}\)	\(1171\)

[In]

int(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.72 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {{\left (b^{6} d^{2} x^{6} + 2 \, {\left (b^{6} c d + {\left (2 \, a + 3\right )} b^{5} d^{2}\right )} x^{5} + {\left (a^{4} + 4 \, a^{3} + 12 \, a^{2} + 24 \, a + 24\right )} b^{2} c^{2} + {\left (b^{6} c^{2} + 2 \, {\left (4 \, a + 5\right )} b^{5} c d + 2 \, {\left (3 \, a^{2} + 10 \, a + 15\right )} b^{4} d^{2}\right )} x^{4} + 2 \, {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b c d + 4 \, {\left ({\left (a + 1\right )} b^{5} c^{2} + {\left (3 \, a^{2} + 8 \, a + 10\right )} b^{4} c d + {\left (a^{3} + 6 \, a^{2} + 20 \, a + 30\right )} b^{3} d^{2}\right )} x^{3} + 2 \, {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} d^{2} + {\left (6 \, {\left (a^{2} + 2 \, a + 2\right )} b^{4} c^{2} + 4 \, {\left (2 \, a^{3} + 9 \, a^{2} + 24 \, a + 30\right )} b^{3} c d + {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b^{2} d^{2}\right )} x^{2} + 2 \, {\left (2 \, {\left (a^{3} + 3 \, a^{2} + 6 \, a + 6\right )} b^{3} c^{2} + {\left (a^{4} + 8 \, a^{3} + 36 \, a^{2} + 96 \, a + 120\right )} b^{2} c d + {\left (a^{4} + 12 \, a^{3} + 72 \, a^{2} + 240 \, a + 360\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )}}{b^{3}} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,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

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 899, normalized size of antiderivative = 1.82 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=\begin {cases} \frac {\left (- a^{4} b^{2} c^{2} - 2 a^{4} b^{2} c d x - a^{4} b^{2} d^{2} x^{2} - 2 a^{4} b c d - 2 a^{4} b d^{2} x - 2 a^{4} d^{2} - 4 a^{3} b^{3} c^{2} x - 8 a^{3} b^{3} c d x^{2} - 4 a^{3} b^{3} d^{2} x^{3} - 4 a^{3} b^{2} c^{2} - 16 a^{3} b^{2} c d x - 12 a^{3} b^{2} d^{2} x^{2} - 16 a^{3} b c d - 24 a^{3} b d^{2} x - 24 a^{3} d^{2} - 6 a^{2} b^{4} c^{2} x^{2} - 12 a^{2} b^{4} c d x^{3} - 6 a^{2} b^{4} d^{2} x^{4} - 12 a^{2} b^{3} c^{2} x - 36 a^{2} b^{3} c d x^{2} - 24 a^{2} b^{3} d^{2} x^{3} - 12 a^{2} b^{2} c^{2} - 72 a^{2} b^{2} c d x - 72 a^{2} b^{2} d^{2} x^{2} - 72 a^{2} b c d - 144 a^{2} b d^{2} x - 144 a^{2} d^{2} - 4 a b^{5} c^{2} x^{3} - 8 a b^{5} c d x^{4} - 4 a b^{5} d^{2} x^{5} - 12 a b^{4} c^{2} x^{2} - 32 a b^{4} c d x^{3} - 20 a b^{4} d^{2} x^{4} - 24 a b^{3} c^{2} x - 96 a b^{3} c d x^{2} - 80 a b^{3} d^{2} x^{3} - 24 a b^{2} c^{2} - 192 a b^{2} c d x - 240 a b^{2} d^{2} x^{2} - 192 a b c d - 480 a b d^{2} x - 480 a d^{2} - b^{6} c^{2} x^{4} - 2 b^{6} c d x^{5} - b^{6} d^{2} x^{6} - 4 b^{5} c^{2} x^{3} - 10 b^{5} c d x^{4} - 6 b^{5} d^{2} x^{5} - 12 b^{4} c^{2} x^{2} - 40 b^{4} c d x^{3} - 30 b^{4} d^{2} x^{4} - 24 b^{3} c^{2} x - 120 b^{3} c d x^{2} - 120 b^{3} d^{2} x^{3} - 24 b^{2} c^{2} - 240 b^{2} c d x - 360 b^{2} d^{2} x^{2} - 240 b c d - 720 b d^{2} x - 720 d^{2}\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\a^{4} c^{2} x + \frac {b^{4} d^{2} x^{7}}{7} + x^{6} \cdot \left (\frac {2 a b^{3} d^{2}}{3} + \frac {b^{4} c d}{3}\right ) + x^{5} \cdot \left (\frac {6 a^{2} b^{2} d^{2}}{5} + \frac {8 a b^{3} c d}{5} + \frac {b^{4} c^{2}}{5}\right ) + x^{4} \left (a^{3} b d^{2} + 3 a^{2} b^{2} c d + a b^{3} c^{2}\right ) + x^{3} \left (\frac {a^{4} d^{2}}{3} + \frac {8 a^{3} b c d}{3} + 2 a^{2} b^{2} c^{2}\right ) + x^{2} \left (a^{4} c d + 2 a^{3} b c^{2}\right ) & \text {otherwise} \end {cases} \]

[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 - 1
92*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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.21 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {4 \, {\left (b x + 1\right )} a^{3} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {a^{4} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {2 \, {\left (b x + 1\right )} a^{4} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{2} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {8 \, {\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{4} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a c^{2} e^{\left (-b x - a\right )}}{b} - \frac {12 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{3} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} c^{2} e^{\left (-b x - a\right )}}{b} - \frac {8 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {6 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a^{2} d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {2 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} c d e^{\left (-b x - a\right )}}{b^{2}} - \frac {4 \, {\left (b^{5} x^{5} + 5 \, b^{4} x^{4} + 20 \, b^{3} x^{3} + 60 \, b^{2} x^{2} + 120 \, b x + 120\right )} a d^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {{\left (b^{6} x^{6} + 6 \, b^{5} x^{5} + 30 \, b^{4} x^{4} + 120 \, b^{3} x^{3} + 360 \, b^{2} x^{2} + 720 \, b x + 720\right )} d^{2} e^{\left (-b x - a\right )}}{b^{3}} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,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

Giac [A] (verification not implemented)

none

Time = 0.39 (sec) , antiderivative size = 674, normalized size of antiderivative = 1.36 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-\frac {{\left (b^{10} d^{2} x^{6} + 2 \, b^{10} c d x^{5} + 4 \, a b^{9} d^{2} x^{5} + b^{10} c^{2} x^{4} + 8 \, a b^{9} c d x^{4} + 6 \, a^{2} b^{8} d^{2} x^{4} + 6 \, b^{9} d^{2} x^{5} + 4 \, a b^{9} c^{2} x^{3} + 12 \, a^{2} b^{8} c d x^{3} + 4 \, a^{3} b^{7} d^{2} x^{3} + 10 \, b^{9} c d x^{4} + 20 \, a b^{8} d^{2} x^{4} + 6 \, a^{2} b^{8} c^{2} x^{2} + 8 \, a^{3} b^{7} c d x^{2} + a^{4} b^{6} d^{2} x^{2} + 4 \, b^{9} c^{2} x^{3} + 32 \, a b^{8} c d x^{3} + 24 \, a^{2} b^{7} d^{2} x^{3} + 30 \, b^{8} d^{2} x^{4} + 4 \, a^{3} b^{7} c^{2} x + 2 \, a^{4} b^{6} c d x + 12 \, a b^{8} c^{2} x^{2} + 36 \, a^{2} b^{7} c d x^{2} + 12 \, a^{3} b^{6} d^{2} x^{2} + 40 \, b^{8} c d x^{3} + 80 \, a b^{7} d^{2} x^{3} + a^{4} b^{6} c^{2} + 12 \, a^{2} b^{7} c^{2} x + 16 \, a^{3} b^{6} c d x + 2 \, a^{4} b^{5} d^{2} x + 12 \, b^{8} c^{2} x^{2} + 96 \, a b^{7} c d x^{2} + 72 \, a^{2} b^{6} d^{2} x^{2} + 120 \, b^{7} d^{2} x^{3} + 4 \, a^{3} b^{6} c^{2} + 2 \, a^{4} b^{5} c d + 24 \, a b^{7} c^{2} x + 72 \, a^{2} b^{6} c d x + 24 \, a^{3} b^{5} d^{2} x + 120 \, b^{7} c d x^{2} + 240 \, a b^{6} d^{2} x^{2} + 12 \, a^{2} b^{6} c^{2} + 16 \, a^{3} b^{5} c d + 2 \, a^{4} b^{4} d^{2} + 24 \, b^{7} c^{2} x + 192 \, a b^{6} c d x + 144 \, a^{2} b^{5} d^{2} x + 360 \, b^{6} d^{2} x^{2} + 24 \, a b^{6} c^{2} + 72 \, a^{2} b^{5} c d + 24 \, a^{3} b^{4} d^{2} + 240 \, b^{6} c d x + 480 \, a b^{5} d^{2} x + 24 \, b^{6} c^{2} + 192 \, a b^{5} c d + 144 \, a^{2} b^{4} d^{2} + 720 \, b^{5} d^{2} x + 240 \, b^{5} c d + 480 \, a b^{4} d^{2} + 720 \, b^{4} d^{2}\right )} e^{\left (-b x - a\right )}}{b^{7}} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4*(d*x+c)^2,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

Mupad [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.08 \[ \int e^{-a-b x} (a+b x)^4 (c+d x)^2 \, dx=-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (120\,c\,d+b\,\left (6\,a^2\,c^2+12\,a\,c^2+12\,c^2\right )+\frac {a^4\,d^2+12\,a^3\,d^2+72\,a^2\,d^2+240\,a\,d^2+360\,d^2}{b}+96\,a\,c\,d+36\,a^2\,c\,d+8\,a^3\,c\,d\right )-x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (4\,a^3\,d^2+12\,a^2\,b\,c\,d+24\,a^2\,d^2+4\,a\,b^2\,c^2+32\,a\,b\,c\,d+80\,a\,d^2+4\,b^2\,c^2+40\,b\,c\,d+120\,d^2\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (a^4\,b^2\,c^2+2\,a^4\,b\,c\,d+2\,a^4\,d^2+4\,a^3\,b^2\,c^2+16\,a^3\,b\,c\,d+24\,a^3\,d^2+12\,a^2\,b^2\,c^2+72\,a^2\,b\,c\,d+144\,a^2\,d^2+24\,a\,b^2\,c^2+192\,a\,b\,c\,d+480\,a\,d^2+24\,b^2\,c^2+240\,b\,c\,d+720\,d^2\right )}{b^3}-b^3\,d^2\,x^6\,{\mathrm {e}}^{-a-b\,x}-b\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (6\,a^2\,d^2+8\,a\,b\,c\,d+20\,a\,d^2+b^2\,c^2+10\,b\,c\,d+30\,d^2\right )-\frac {2\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^4\,b\,c\,d+a^4\,d^2+2\,a^3\,b^2\,c^2+8\,a^3\,b\,c\,d+12\,a^3\,d^2+6\,a^2\,b^2\,c^2+36\,a^2\,b\,c\,d+72\,a^2\,d^2+12\,a\,b^2\,c^2+96\,a\,b\,c\,d+240\,a\,d^2+12\,b^2\,c^2+120\,b\,c\,d+360\,d^2\right )}{b^2}-2\,b^2\,d\,x^5\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,d+2\,a\,d+b\,c\right ) \]

[In]

int(exp(- a - b*x)*(a + b*x)^4*(c + d*x)^2,x)

[Out]

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