3.1.57 \(\int e^{-a-b x} x^2 (a+b x)^3 \, dx\) [57]

Optimal. Leaf size=318 \[ -\frac {120 e^{-a-b x}}{b^3}-\frac {72 a e^{-a-b x}}{b^3}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x} x}{b^2}-\frac {72 a e^{-a-b x} x}{b^2}-\frac {18 a^2 e^{-a-b x} x}{b^2}-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {60 e^{-a-b x} x^2}{b}-\frac {36 a e^{-a-b x} x^2}{b}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5 \]

[Out]

-120*exp(-b*x-a)/b^3-72*a*exp(-b*x-a)/b^3-18*a^2*exp(-b*x-a)/b^3-2*a^3*exp(-b*x-a)/b^3-120*exp(-b*x-a)*x/b^2-7
2*a*exp(-b*x-a)*x/b^2-18*a^2*exp(-b*x-a)*x/b^2-2*a^3*exp(-b*x-a)*x/b^2-60*exp(-b*x-a)*x^2/b-36*a*exp(-b*x-a)*x
^2/b-9*a^2*exp(-b*x-a)*x^2/b-a^3*exp(-b*x-a)*x^2/b-20*exp(-b*x-a)*x^3-12*a*exp(-b*x-a)*x^3-3*a^2*exp(-b*x-a)*x
^3-5*b*exp(-b*x-a)*x^4-3*a*b*exp(-b*x-a)*x^4-b^2*exp(-b*x-a)*x^5

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Rubi [A]
time = 0.29, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2227, 2207, 2225} \begin {gather*} -\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {2 a^3 x e^{-a-b x}}{b^2}-\frac {a^3 x^2 e^{-a-b x}}{b}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {18 a^2 x e^{-a-b x}}{b^2}-3 a^2 x^3 e^{-a-b x}-\frac {9 a^2 x^2 e^{-a-b x}}{b}-\frac {72 a e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x}}{b^3}-b^2 x^5 e^{-a-b x}-\frac {72 a x e^{-a-b x}}{b^2}-\frac {120 x e^{-a-b x}}{b^2}-3 a b x^4 e^{-a-b x}-5 b x^4 e^{-a-b x}-12 a x^3 e^{-a-b x}-20 x^3 e^{-a-b x}-\frac {36 a x^2 e^{-a-b x}}{b}-\frac {60 x^2 e^{-a-b x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-a - b*x)*x^2*(a + b*x)^3,x]

[Out]

(-120*E^(-a - b*x))/b^3 - (72*a*E^(-a - b*x))/b^3 - (18*a^2*E^(-a - b*x))/b^3 - (2*a^3*E^(-a - b*x))/b^3 - (12
0*E^(-a - b*x)*x)/b^2 - (72*a*E^(-a - b*x)*x)/b^2 - (18*a^2*E^(-a - b*x)*x)/b^2 - (2*a^3*E^(-a - b*x)*x)/b^2 -
(60*E^(-a - b*x)*x^2)/b - (36*a*E^(-a - b*x)*x^2)/b - (9*a^2*E^(-a - b*x)*x^2)/b - (a^3*E^(-a - b*x)*x^2)/b -
20*E^(-a - b*x)*x^3 - 12*a*E^(-a - b*x)*x^3 - 3*a^2*E^(-a - b*x)*x^3 - 5*b*E^(-a - b*x)*x^4 - 3*a*b*E^(-a - b
*x)*x^4 - b^2*E^(-a - b*x)*x^5

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*} \int e^{-a-b x} x^2 (a+b x)^3 \, dx &=\int \left (a^3 e^{-a-b x} x^2+3 a^2 b e^{-a-b x} x^3+3 a b^2 e^{-a-b x} x^4+b^3 e^{-a-b x} x^5\right ) \, dx\\ &=a^3 \int e^{-a-b x} x^2 \, dx+\left (3 a^2 b\right ) \int e^{-a-b x} x^3 \, dx+\left (3 a b^2\right ) \int e^{-a-b x} x^4 \, dx+b^3 \int e^{-a-b x} x^5 \, dx\\ &=-\frac {a^3 e^{-a-b x} x^2}{b}-3 a^2 e^{-a-b x} x^3-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+\left (9 a^2\right ) \int e^{-a-b x} x^2 \, dx+\frac {\left (2 a^3\right ) \int e^{-a-b x} x \, dx}{b}+(12 a b) \int e^{-a-b x} x^3 \, dx+\left (5 b^2\right ) \int e^{-a-b x} x^4 \, dx\\ &=-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+(36 a) \int e^{-a-b x} x^2 \, dx+\frac {\left (2 a^3\right ) \int e^{-a-b x} \, dx}{b^2}+\frac {\left (18 a^2\right ) \int e^{-a-b x} x \, dx}{b}+(20 b) \int e^{-a-b x} x^3 \, dx\\ &=-\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {18 a^2 e^{-a-b x} x}{b^2}-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {36 a e^{-a-b x} x^2}{b}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+60 \int e^{-a-b x} x^2 \, dx+\frac {\left (18 a^2\right ) \int e^{-a-b x} \, dx}{b^2}+\frac {(72 a) \int e^{-a-b x} x \, dx}{b}\\ &=-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {72 a e^{-a-b x} x}{b^2}-\frac {18 a^2 e^{-a-b x} x}{b^2}-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {60 e^{-a-b x} x^2}{b}-\frac {36 a e^{-a-b x} x^2}{b}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+\frac {(72 a) \int e^{-a-b x} \, dx}{b^2}+\frac {120 \int e^{-a-b x} x \, dx}{b}\\ &=-\frac {72 a e^{-a-b x}}{b^3}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x} x}{b^2}-\frac {72 a e^{-a-b x} x}{b^2}-\frac {18 a^2 e^{-a-b x} x}{b^2}-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {60 e^{-a-b x} x^2}{b}-\frac {36 a e^{-a-b x} x^2}{b}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5+\frac {120 \int e^{-a-b x} \, dx}{b^2}\\ &=-\frac {120 e^{-a-b x}}{b^3}-\frac {72 a e^{-a-b x}}{b^3}-\frac {18 a^2 e^{-a-b x}}{b^3}-\frac {2 a^3 e^{-a-b x}}{b^3}-\frac {120 e^{-a-b x} x}{b^2}-\frac {72 a e^{-a-b x} x}{b^2}-\frac {18 a^2 e^{-a-b x} x}{b^2}-\frac {2 a^3 e^{-a-b x} x}{b^2}-\frac {60 e^{-a-b x} x^2}{b}-\frac {36 a e^{-a-b x} x^2}{b}-\frac {9 a^2 e^{-a-b x} x^2}{b}-\frac {a^3 e^{-a-b x} x^2}{b}-20 e^{-a-b x} x^3-12 a e^{-a-b x} x^3-3 a^2 e^{-a-b x} x^3-5 b e^{-a-b x} x^4-3 a b e^{-a-b x} x^4-b^2 e^{-a-b x} x^5\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 130, normalized size = 0.41 \begin {gather*} e^{-b x} \left (-\frac {2 \left (60+36 a+9 a^2+a^3\right ) e^{-a}}{b^3}-\frac {2 \left (60+36 a+9 a^2+a^3\right ) e^{-a} x}{b^2}-\frac {\left (60+36 a+9 a^2+a^3\right ) e^{-a} x^2}{b}-\left (20+12 a+3 a^2\right ) e^{-a} x^3-(5+3 a) b e^{-a} x^4-b^2 e^{-a} x^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-a - b*x)*x^2*(a + b*x)^3,x]

[Out]

((-2*(60 + 36*a + 9*a^2 + a^3))/(b^3*E^a) - (2*(60 + 36*a + 9*a^2 + a^3)*x)/(b^2*E^a) - ((60 + 36*a + 9*a^2 +
a^3)*x^2)/(b*E^a) - ((20 + 12*a + 3*a^2)*x^3)/E^a - ((5 + 3*a)*b*x^4)/E^a - (b^2*x^5)/E^a)/E^(b*x)

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Maple [A]
time = 0.06, size = 291, normalized size = 0.92

method	result	size
gosper	\(-\frac {\left (b^{5} x^{5}+3 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+5 b^{4} x^{4}+a^{3} b^{2} x^{2}+12 a \,b^{3} x^{3}+9 a^{2} b^{2} x^{2}+20 b^{3} x^{3}+2 a^{3} b x +36 a \,b^{2} x^{2}+18 a^{2} b x +60 b^{2} x^{2}+2 a^{3}+72 a b x +18 a^{2}+120 b x +72 a +120\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(143\)
risch	\(-\frac {\left (b^{5} x^{5}+3 a \,b^{4} x^{4}+3 a^{2} b^{3} x^{3}+5 b^{4} x^{4}+a^{3} b^{2} x^{2}+12 a \,b^{3} x^{3}+9 a^{2} b^{2} x^{2}+20 b^{3} x^{3}+2 a^{3} b x +36 a \,b^{2} x^{2}+18 a^{2} b x +60 b^{2} x^{2}+2 a^{3}+72 a b x +18 a^{2}+120 b x +72 a +120\right ) {\mathrm e}^{-b x -a}}{b^{3}}\)	\(143\)
norman	\(\left (-3 a b -5 b \right ) x^{4} {\mathrm e}^{-b x -a}+\left (-3 a^{2}-12 a -20\right ) x^{3} {\mathrm e}^{-b x -a}-b^{2} {\mathrm e}^{-b x -a} x^{5}-\frac {2 \left (a^{3}+9 a^{2}+36 a +60\right ) {\mathrm e}^{-b x -a}}{b^{3}}-\frac {2 \left (a^{3}+9 a^{2}+36 a +60\right ) x \,{\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a^{3}+9 a^{2}+36 a +60\right ) x^{2} {\mathrm e}^{-b x -a}}{b}\)	\(148\)
meijerg	\(\frac {{\mathrm e}^{-a} \left (120-\frac {\left (6 b^{5} x^{5}+30 b^{4} x^{4}+120 b^{3} x^{3}+360 b^{2} x^{2}+720 b x +720\right ) {\mathrm e}^{-b x}}{6}\right )}{b^{3}}+\frac {3 \,{\mathrm e}^{-a} a \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{3}}+\frac {3 \,{\mathrm e}^{-a} a^{2} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{3}}+\frac {{\mathrm e}^{-a} a^{3} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}\)	\(183\)
derivativedivides	\(\frac {\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}+a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+2 a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}\)	\(291\)
default	\(\frac {\left (-b x -a \right )^{5} {\mathrm e}^{-b x -a}-5 \left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}+20 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-60 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+120 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-120 \,{\mathrm e}^{-b x -a}+a^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )+2 a \left (\left (-b x -a \right )^{4} {\mathrm e}^{-b x -a}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{3}}\)	\(291\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-b*x-a)*x^2*(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.44, size = 164, normalized size = 0.52 \begin {gather*} -\frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} a^{3} e^{\left (-b x - a\right )}}{b^{3}} - \frac {3 \, {\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} a^{2} e^{\left (-b x - a\right )}}{b^{3}} - \frac {3 \, {\left (b^{4} x^{4} + 4 \, b^{3} x^{3} + 12 \, b^{2} x^{2} + 24 \, b x + 24\right )} a e^{\left (-b x - a\right )}}{b^{3}} - \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 )} e^{\left (-b x - a\right )}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*x^2*(b*x+a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.38, size = 102, normalized size = 0.32 \begin {gather*} -\frac {{\left (b^{5} x^{5} + {\left (3 \, a + 5\right )} b^{4} x^{4} + {\left (3 \, a^{2} + 12 \, a + 20\right )} b^{3} x^{3} + {\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b^{2} x^{2} + 2 \, a^{3} + 2 \, {\left (a^{3} + 9 \, a^{2} + 36 \, a + 60\right )} b x + 18 \, a^{2} + 72 \, a + 120\right )} e^{\left (-b x - a\right )}}{b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*x^2*(b*x+a)^3,x, algorithm="fricas")

[Out]

-(b^5*x^5 + (3*a + 5)*b^4*x^4 + (3*a^2 + 12*a + 20)*b^3*x^3 + (a^3 + 9*a^2 + 36*a + 60)*b^2*x^2 + 2*a^3 + 2*(a
^3 + 9*a^2 + 36*a + 60)*b*x + 18*a^2 + 72*a + 120)*e^(-b*x - a)/b^3

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Sympy [A]
time = 0.10, size = 196, normalized size = 0.62 \begin {gather*} \begin {cases} \frac {\left (- a^{3} b^{2} x^{2} - 2 a^{3} b x - 2 a^{3} - 3 a^{2} b^{3} x^{3} - 9 a^{2} b^{2} x^{2} - 18 a^{2} b x - 18 a^{2} - 3 a b^{4} x^{4} - 12 a b^{3} x^{3} - 36 a b^{2} x^{2} - 72 a b x - 72 a - b^{5} x^{5} - 5 b^{4} x^{4} - 20 b^{3} x^{3} - 60 b^{2} x^{2} - 120 b x - 120\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\\frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{4}}{4} + \frac {3 a b^{2} x^{5}}{5} + \frac {b^{3} x^{6}}{6} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*x**2*(b*x+a)**3,x)

[Out]

Piecewise(((-a**3*b**2*x**2 - 2*a**3*b*x - 2*a**3 - 3*a**2*b**3*x**3 - 9*a**2*b**2*x**2 - 18*a**2*b*x - 18*a**
2 - 3*a*b**4*x**4 - 12*a*b**3*x**3 - 36*a*b**2*x**2 - 72*a*b*x - 72*a - b**5*x**5 - 5*b**4*x**4 - 20*b**3*x**3
- 60*b**2*x**2 - 120*b*x - 120)*exp(-a - b*x)/b**3, Ne(b**3, 0)), (a**3*x**3/3 + 3*a**2*b*x**4/4 + 3*a*b**2*x
**5/5 + b**3*x**6/6, True))

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Giac [A]
time = 2.28, size = 163, normalized size = 0.51 \begin {gather*} -\frac {{\left (b^{8} x^{5} + 3 \, a b^{7} x^{4} + 3 \, a^{2} b^{6} x^{3} + 5 \, b^{7} x^{4} + a^{3} b^{5} x^{2} + 12 \, a b^{6} x^{3} + 9 \, a^{2} b^{5} x^{2} + 20 \, b^{6} x^{3} + 2 \, a^{3} b^{4} x + 36 \, a b^{5} x^{2} + 18 \, a^{2} b^{4} x + 60 \, b^{5} x^{2} + 2 \, a^{3} b^{3} + 72 \, a b^{4} x + 18 \, a^{2} b^{3} + 120 \, b^{4} x + 72 \, a b^{3} + 120 \, b^{3}\right )} e^{\left (-b x - a\right )}}{b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(-b*x-a)*x^2*(b*x+a)^3,x, algorithm="giac")

[Out]

-(b^8*x^5 + 3*a*b^7*x^4 + 3*a^2*b^6*x^3 + 5*b^7*x^4 + a^3*b^5*x^2 + 12*a*b^6*x^3 + 9*a^2*b^5*x^2 + 20*b^6*x^3
+ 2*a^3*b^4*x + 36*a*b^5*x^2 + 18*a^2*b^4*x + 60*b^5*x^2 + 2*a^3*b^3 + 72*a*b^4*x + 18*a^2*b^3 + 120*b^4*x + 7
2*a*b^3 + 120*b^3)*e^(-b*x - a)/b^6

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Mupad [B]
time = 3.52, size = 126, normalized size = 0.40 \begin {gather*} -x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2+3\,a\,b\,x+12\,a+b^2\,x^2+5\,b\,x+20\right )-\frac {2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b^3}-\frac {2\,x\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b^2}-\frac {x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (a^3+9\,a^2+36\,a+60\right )}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*exp(- a - b*x)*(a + b*x)^3,x)

[Out]

- x^3*exp(- a - b*x)*(12*a + 5*b*x + 3*a^2 + b^2*x^2 + 3*a*b*x + 20) - (2*exp(- a - b*x)*(36*a + 9*a^2 + a^3 +
60))/b^3 - (2*x*exp(- a - b*x)*(36*a + 9*a^2 + a^3 + 60))/b^2 - (x^2*exp(- a - b*x)*(36*a + 9*a^2 + a^3 + 60)
)/b

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