\(\int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx\) [80]

Optimal result

Integrand size = 25, antiderivative size = 294 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=-\frac {b^2 e^{-a-b x}}{d^3}+\frac {b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac {b^3 e^{-a-b x} x}{d^3}-\frac {(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac {b (b c-a d)^4 e^{-a-b x}}{2 d^6 (c+d x)}+\frac {6 b^2 (b c-a d)^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {4 b^2 (b c-a d)^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {b^2 (b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{2 d^7} \]

[Out]

-b^2*exp(-b*x-a)/d^3+b^2*(-4*a*d+3*b*c)*exp(-b*x-a)/d^4-b^3*exp(-b*x-a)*x/d^3-1/2*(-a*d+b*c)^4*exp(-b*x-a)/d^5
/(d*x+c)^2+4*b*(-a*d+b*c)^3*exp(-b*x-a)/d^5/(d*x+c)+1/2*b*(-a*d+b*c)^4*exp(-b*x-a)/d^6/(d*x+c)+6*b^2*(-a*d+b*c
)^2*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^5+4*b^2*(-a*d+b*c)^3*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^6+1/2*b^2*(-a*d+b*c
)^4*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^7

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2230, 2225, 2207, 2208, 2209} \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=-\frac {b^3 x e^{-a-b x}}{d^3}+\frac {b^2 e^{\frac {b c}{d}-a} (b c-a d)^4 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{2 d^7}+\frac {4 b^2 e^{\frac {b c}{d}-a} (b c-a d)^3 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {6 b^2 e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {b^2 e^{-a-b x} (3 b c-4 a d)}{d^4}-\frac {b^2 e^{-a-b x}}{d^3}+\frac {b e^{-a-b x} (b c-a d)^4}{2 d^6 (c+d x)}-\frac {e^{-a-b x} (b c-a d)^4}{2 d^5 (c+d x)^2}+\frac {4 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)} \]

[In]

Int[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^3,x]

[Out]

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

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 2208

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(c + d*x)^(m
+ 1)*((b*F^(g*(e + f*x)))^n/(d*(m + 1))), x] - Dist[f*g*n*(Log[F]/(d*(m + 1))), 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] && LtQ[m, -1] && IntegerQ[2*m] &&  !TrueQ[$UseGamm
a]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !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 2230

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !TrueQ[$UseGamma]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b^3 (3 b c-4 a d) e^{-a-b x}}{d^4}+\frac {b^4 e^{-a-b x} x}{d^3}+\frac {(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)^3}-\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^4 (c+d x)^2}+\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx \\ & = \frac {b^4 \int e^{-a-b x} x \, dx}{d^3}-\frac {\left (b^3 (3 b c-4 a d)\right ) \int e^{-a-b x} \, dx}{d^4}+\frac {\left (6 b^2 (b c-a d)^2\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^4}-\frac {\left (4 b (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{d^4}+\frac {(b c-a d)^4 \int \frac {e^{-a-b x}}{(c+d x)^3} \, dx}{d^4} \\ & = \frac {b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac {b^3 e^{-a-b x} x}{d^3}-\frac {(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac {6 b^2 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {b^3 \int e^{-a-b x} \, dx}{d^3}+\frac {\left (4 b^2 (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^5}-\frac {\left (b (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{2 d^5} \\ & = -\frac {b^2 e^{-a-b x}}{d^3}+\frac {b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac {b^3 e^{-a-b x} x}{d^3}-\frac {(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac {b (b c-a d)^4 e^{-a-b x}}{2 d^6 (c+d x)}+\frac {6 b^2 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {4 b^2 (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {\left (b^2 (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{2 d^6} \\ & = -\frac {b^2 e^{-a-b x}}{d^3}+\frac {b^2 (3 b c-4 a d) e^{-a-b x}}{d^4}-\frac {b^3 e^{-a-b x} x}{d^3}-\frac {(b c-a d)^4 e^{-a-b x}}{2 d^5 (c+d x)^2}+\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)}+\frac {b (b c-a d)^4 e^{-a-b x}}{2 d^6 (c+d x)}+\frac {6 b^2 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {4 b^2 (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}+\frac {b^2 (b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{2 d^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.89 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=\frac {e^{-a} \left (\frac {d e^{-b x} \left (-a^4 d^5+b^5 c^4 (c+d x)+a^3 b d^4 ((-4+a) c+(-8+a) d x)+b^4 c^3 d ((7-4 a) c-4 (-2+a) d x)-2 b^2 d^3 \left (\left (1+4 a-9 a^2+2 a^3\right ) c^2+2 \left (1+4 a-6 a^2+a^3\right ) c d x+(1+4 a) d^2 x^2\right )+2 b^3 d^2 \left (\left (3-10 a+3 a^2\right ) c^3+\left (5-12 a+3 a^2\right ) c^2 d x+c d^2 x^2-d^3 x^3\right )\right )}{(c+d x)^2}+b^2 (b c-a d)^2 \left (b^2 c^2-2 (-4+a) b c d+\left (12-8 a+a^2\right ) d^2\right ) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{2 d^7} \]

[In]

Integrate[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x)^3,x]

[Out]

((d*(-(a^4*d^5) + b^5*c^4*(c + d*x) + a^3*b*d^4*((-4 + a)*c + (-8 + a)*d*x) + b^4*c^3*d*((7 - 4*a)*c - 4*(-2 +
a)*d*x) - 2*b^2*d^3*((1 + 4*a - 9*a^2 + 2*a^3)*c^2 + 2*(1 + 4*a - 6*a^2 + a^3)*c*d*x + (1 + 4*a)*d^2*x^2) + 2
*b^3*d^2*((3 - 10*a + 3*a^2)*c^3 + (5 - 12*a + 3*a^2)*c^2*d*x + c*d^2*x^2 - d^3*x^3)))/(E^(b*x)*(c + d*x)^2) +
b^2*(b*c - a*d)^2*(b^2*c^2 - 2*(-4 + a)*b*c*d + (12 - 8*a + a^2)*d^2)*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x
))/d)])/(2*d^7*E^a)

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.42

method	result	size
derivativedivides	\(-\frac {\frac {3 b^{3} a \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {3 b^{4} c \,{\mathrm e}^{-b x -a}}{d^{4}}-\frac {b^{3} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\)	\(418\)
default	\(-\frac {\frac {3 b^{3} a \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {3 b^{4} c \,{\mathrm e}^{-b x -a}}{d^{4}}-\frac {b^{3} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{3} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\)	\(418\)
risch	\(\text {Expression too large to display}\)	\(1107\)

[In]

int(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^3,x,method=_RETURNVERBOSE)

[Out]

-1/b*(3*b^3/d^3*a*exp(-b*x-a)-3*b^4/d^4*c*exp(-b*x-a)-1/d^3*b^3*((-b*x-a)*exp(-b*x-a)-exp(-b*x-a))+4/d^6*(a^3*
d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*b^3*(-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))-(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^3/d^7*(-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))+6/d^5*(a^2
*d^2-2*a*b*c*d+b^2*c^2)*b^3*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.87 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=\frac {{\left (b^{6} c^{6} - 4 \, {\left (a - 2\right )} b^{5} c^{5} d + 6 \, {\left (a^{2} - 4 \, a + 2\right )} b^{4} c^{4} d^{2} - 4 \, {\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{3} c^{3} d^{3} + {\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, {\left (a - 2\right )} b^{5} c^{3} d^{3} + 6 \, {\left (a^{2} - 4 \, a + 2\right )} b^{4} c^{2} d^{4} - 4 \, {\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{3} c d^{5} + {\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} d^{6}\right )} x^{2} + 2 \, {\left (b^{6} c^{5} d - 4 \, {\left (a - 2\right )} b^{5} c^{4} d^{2} + 6 \, {\left (a^{2} - 4 \, a + 2\right )} b^{4} c^{3} d^{3} - 4 \, {\left (a^{3} - 6 \, a^{2} + 6 \, a\right )} b^{3} c^{2} d^{4} + {\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c d^{5}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (2 \, b^{3} d^{6} x^{3} - b^{5} c^{5} d + {\left (4 \, a - 7\right )} b^{4} c^{4} d^{2} - 2 \, {\left (3 \, a^{2} - 10 \, a + 3\right )} b^{3} c^{3} d^{3} + a^{4} d^{6} + 2 \, {\left (2 \, a^{3} - 9 \, a^{2} + 4 \, a + 1\right )} b^{2} c^{2} d^{4} - {\left (a^{4} - 4 \, a^{3}\right )} b c d^{5} - 2 \, {\left (b^{3} c d^{5} - {\left (4 \, a + 1\right )} b^{2} d^{6}\right )} x^{2} - {\left (b^{5} c^{4} d^{2} - 4 \, {\left (a - 2\right )} b^{4} c^{3} d^{3} + 2 \, {\left (3 \, a^{2} - 12 \, a + 5\right )} b^{3} c^{2} d^{4} - 4 \, {\left (a^{3} - 6 \, a^{2} + 4 \, a + 1\right )} b^{2} c d^{5} + {\left (a^{4} - 8 \, a^{3}\right )} b d^{6}\right )} x\right )} e^{\left (-b x - a\right )}}{2 \, {\left (d^{9} x^{2} + 2 \, c d^{8} x + c^{2} d^{7}\right )}} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^3,x, algorithm="fricas")

[Out]

1/2*((b^6*c^6 - 4*(a - 2)*b^5*c^5*d + 6*(a^2 - 4*a + 2)*b^4*c^4*d^2 - 4*(a^3 - 6*a^2 + 6*a)*b^3*c^3*d^3 + (a^4
- 8*a^3 + 12*a^2)*b^2*c^2*d^4 + (b^6*c^4*d^2 - 4*(a - 2)*b^5*c^3*d^3 + 6*(a^2 - 4*a + 2)*b^4*c^2*d^4 - 4*(a^3
- 6*a^2 + 6*a)*b^3*c*d^5 + (a^4 - 8*a^3 + 12*a^2)*b^2*d^6)*x^2 + 2*(b^6*c^5*d - 4*(a - 2)*b^5*c^4*d^2 + 6*(a^
2 - 4*a + 2)*b^4*c^3*d^3 - 4*(a^3 - 6*a^2 + 6*a)*b^3*c^2*d^4 + (a^4 - 8*a^3 + 12*a^2)*b^2*c*d^5)*x)*Ei(-(b*d*x
+ b*c)/d)*e^((b*c - a*d)/d) - (2*b^3*d^6*x^3 - b^5*c^5*d + (4*a - 7)*b^4*c^4*d^2 - 2*(3*a^2 - 10*a + 3)*b^3*c
^3*d^3 + a^4*d^6 + 2*(2*a^3 - 9*a^2 + 4*a + 1)*b^2*c^2*d^4 - (a^4 - 4*a^3)*b*c*d^5 - 2*(b^3*c*d^5 - (4*a + 1)*
b^2*d^6)*x^2 - (b^5*c^4*d^2 - 4*(a - 2)*b^4*c^3*d^3 + 2*(3*a^2 - 12*a + 5)*b^3*c^2*d^4 - 4*(a^3 - 6*a^2 + 4*a
+ 1)*b^2*c*d^5 + (a^4 - 8*a^3)*b*d^6)*x)*e^(-b*x - a))/(d^9*x^2 + 2*c*d^8*x + c^2*d^7)

Sympy [F]

\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=\left (\int \frac {a^{4}}{c^{3} e^{b x} + 3 c^{2} d x e^{b x} + 3 c d^{2} x^{2} e^{b x} + d^{3} x^{3} e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c^{3} e^{b x} + 3 c^{2} d x e^{b x} + 3 c d^{2} x^{2} e^{b x} + d^{3} x^{3} e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c^{3} e^{b x} + 3 c^{2} d x e^{b x} + 3 c d^{2} x^{2} e^{b x} + d^{3} x^{3} e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c^{3} e^{b x} + 3 c^{2} d x e^{b x} + 3 c d^{2} x^{2} e^{b x} + d^{3} x^{3} e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c^{3} e^{b x} + 3 c^{2} d x e^{b x} + 3 c d^{2} x^{2} e^{b x} + d^{3} x^{3} e^{b x}}\, dx\right ) e^{- a} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)**4/(d*x+c)**3,x)

[Out]

(Integral(a**4/(c**3*exp(b*x) + 3*c**2*d*x*exp(b*x) + 3*c*d**2*x**2*exp(b*x) + d**3*x**3*exp(b*x)), x) + Integ
ral(b**4*x**4/(c**3*exp(b*x) + 3*c**2*d*x*exp(b*x) + 3*c*d**2*x**2*exp(b*x) + d**3*x**3*exp(b*x)), x) + Integr
al(4*a*b**3*x**3/(c**3*exp(b*x) + 3*c**2*d*x*exp(b*x) + 3*c*d**2*x**2*exp(b*x) + d**3*x**3*exp(b*x)), x) + Int
egral(6*a**2*b**2*x**2/(c**3*exp(b*x) + 3*c**2*d*x*exp(b*x) + 3*c*d**2*x**2*exp(b*x) + d**3*x**3*exp(b*x)), x)
+ Integral(4*a**3*b*x/(c**3*exp(b*x) + 3*c**2*d*x*exp(b*x) + 3*c*d**2*x**2*exp(b*x) + d**3*x**3*exp(b*x)), x)
)*exp(-a)

Maxima [F]

\[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=\int { \frac {{\left (b x + a\right )}^{4} e^{\left (-b x - a\right )}}{{\left (d x + c\right )}^{3}} \,d x } \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^3,x, algorithm="maxima")

[Out]

-a^4*e^(-a + b*c/d)*exp_integral_e(3, (d*x + c)*b/d)/((d*x + c)^2*d) - (b^3*d^2*x^4 + (4*a*b^2*d^2 + b^2*d^2)*
x^3 + 3*(2*a^2*b*d^2 + b^2*c*d)*x^2 + (4*a^3*d^2 - 3*b^2*c^2 + 12*a*b*c*d - 6*a^2*d^2)*x)*e^(-b*x)/(d^5*x^3*e^
a + 3*c*d^4*x^2*e^a + 3*c^2*d^3*x*e^a + c^3*d^2*e^a) - integrate(-(4*a^3*c*d^2 - 3*b^2*c^3 + 12*a*b*c^2*d - 6*
a^2*c*d^2 + (3*b^3*c^3 - 8*a^3*d^3 + 12*b^2*c^2*d + 6*(3*b*c*d^2 + 2*d^3)*a^2 - 12*(b^2*c^2*d + 2*b*c*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), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1995 vs. \(2 (279) = 558\).

Time = 0.41 (sec) , antiderivative size = 1995, normalized size of antiderivative = 6.79 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=\text {Too large to display} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^3,x, algorithm="giac")

[Out]

1/2*(b^6*c^4*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^5*c^3*d^3*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*
c/d) + 6*a^2*b^4*c^2*d^4*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^3*c*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^
(-a + b*c/d) + a^4*b^2*d^6*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 2*b^6*c^5*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a
+ b*c/d) - 8*a*b^5*c^4*d^2*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*a^2*b^4*c^3*d^3*x*Ei(-(b*d*x + b*c)/d)*e
^(-a + b*c/d) - 8*a^3*b^3*c^2*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 2*a^4*b^2*c*d^5*x*Ei(-(b*d*x + b*c)/
d)*e^(-a + b*c/d) + 8*b^5*c^3*d^3*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 24*a*b^4*c^2*d^4*x^2*Ei(-(b*d*x +
b*c)/d)*e^(-a + b*c/d) + 24*a^2*b^3*c*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 8*a^3*b^2*d^6*x^2*Ei(-(b*d
*x + b*c)/d)*e^(-a + b*c/d) + b^6*c^6*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^5*c^5*d*Ei(-(b*d*x + b*c)/d)
*e^(-a + b*c/d) + 6*a^2*b^4*c^4*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^3*c^3*d^3*Ei(-(b*d*x + b*c)/
d)*e^(-a + b*c/d) + a^4*b^2*c^2*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 16*b^5*c^4*d^2*x*Ei(-(b*d*x + b*c)/d
)*e^(-a + b*c/d) - 48*a*b^4*c^3*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 48*a^2*b^3*c^2*d^4*x*Ei(-(b*d*x +
b*c)/d)*e^(-a + b*c/d) - 16*a^3*b^2*c*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*b^4*c^2*d^4*x^2*Ei(-(b*d*
x + b*c)/d)*e^(-a + b*c/d) - 24*a*b^3*c*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*a^2*b^2*d^6*x^2*Ei(-(
b*d*x + b*c)/d)*e^(-a + b*c/d) + b^5*c^4*d^2*x*e^(-b*x - a) - 4*a*b^4*c^3*d^3*x*e^(-b*x - a) + 6*a^2*b^3*c^2*d
^4*x*e^(-b*x - a) - 4*a^3*b^2*c*d^5*x*e^(-b*x - a) + a^4*b*d^6*x*e^(-b*x - a) - 2*b^3*d^6*x^3*e^(-b*x - a) + 8
*b^5*c^5*d*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 24*a*b^4*c^4*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 24*a^2
*b^3*c^3*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 8*a^3*b^2*c^2*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 24*
b^4*c^3*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 48*a*b^3*c^2*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 2
4*a^2*b^2*c*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^5*c^5*d*e^(-b*x - a) - 4*a*b^4*c^4*d^2*e^(-b*x - a)
+ 6*a^2*b^3*c^3*d^3*e^(-b*x - a) - 4*a^3*b^2*c^2*d^4*e^(-b*x - a) + a^4*b*c*d^5*e^(-b*x - a) + 8*b^4*c^3*d^3*x
*e^(-b*x - a) - 24*a*b^3*c^2*d^4*x*e^(-b*x - a) + 24*a^2*b^2*c*d^5*x*e^(-b*x - a) - 8*a^3*b*d^6*x*e^(-b*x - a)
+ 2*b^3*c*d^5*x^2*e^(-b*x - a) - 8*a*b^2*d^6*x^2*e^(-b*x - a) + 12*b^4*c^4*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b
*c/d) - 24*a*b^3*c^3*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*a^2*b^2*c^2*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a +
b*c/d) + 7*b^4*c^4*d^2*e^(-b*x - a) - 20*a*b^3*c^3*d^3*e^(-b*x - a) + 18*a^2*b^2*c^2*d^4*e^(-b*x - a) - 4*a^3
*b*c*d^5*e^(-b*x - a) - a^4*d^6*e^(-b*x - a) + 10*b^3*c^2*d^4*x*e^(-b*x - a) - 16*a*b^2*c*d^5*x*e^(-b*x - a) -
2*b^2*d^6*x^2*e^(-b*x - a) + 6*b^3*c^3*d^3*e^(-b*x - a) - 8*a*b^2*c^2*d^4*e^(-b*x - a) - 4*b^2*c*d^5*x*e^(-b*
x - a) - 2*b^2*c^2*d^4*e^(-b*x - a))/(d^9*x^2 + 2*c*d^8*x + c^2*d^7)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{{\left (c+d\,x\right )}^3} \,d x \]

[In]

int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x)^3,x)

[Out]

int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x)^3, x)