\(\int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx\) [79]

Optimal result

Integrand size = 25, antiderivative size = 258 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=-\frac {2 b e^{-a-b x}}{d^2}+\frac {4 b (b c-a d) e^{-a-b x}}{d^3}-\frac {6 b (b c-a d)^2 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}+\frac {4 b^2 (b c-a d) e^{-a-b x} (c+d x)}{d^4}-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}-\frac {4 b (b c-a d)^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b (b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 258, 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, 2208, 2209, 2207} \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=-\frac {b^3 e^{-a-b x} (c+d x)^2}{d^4}+\frac {4 b^2 e^{-a-b x} (c+d x) (b c-a d)}{d^4}-\frac {2 b^2 e^{-a-b x} (c+d x)}{d^3}-\frac {b e^{\frac {b c}{d}-a} (b c-a d)^4 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b e^{\frac {b c}{d}-a} (b c-a d)^3 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {e^{-a-b x} (b c-a d)^4}{d^5 (c+d x)}-\frac {6 b e^{-a-b x} (b c-a d)^2}{d^4}+\frac {4 b e^{-a-b x} (b c-a d)}{d^3}-\frac {2 b e^{-a-b x}}{d^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\frac {e^{-a} \left (-\frac {d e^{-b x} \left ((b c-a d)^4+b d \left (3 b^2 c^2-2 (1+4 a) b c d+2 \left (1+2 a+3 a^2\right ) d^2\right ) (c+d x)-2 b^2 d^2 (b c-(1+2 a) d) x (c+d x)+b^3 d^3 x^2 (c+d x)\right )}{c+d x}-b (b c-(-4+a) d) (b c-a d)^3 e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{d^6} \]

[In]

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

[Out]

(-((d*((b*c - a*d)^4 + b*d*(3*b^2*c^2 - 2*(1 + 4*a)*b*c*d + 2*(1 + 2*a + 3*a^2)*d^2)*(c + d*x) - 2*b^2*d^2*(b*
c - (1 + 2*a)*d)*x*(c + d*x) + b^3*d^3*x^2*(c + d*x)))/(E^(b*x)*(c + d*x))) - b*(b*c - (-4 + a)*d)*(b*c - a*d)
^3*E^((b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(d^6*E^a)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.57

method	result	size
derivativedivides	\(-\frac {\frac {3 b^{2} a^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {6 b^{3} a c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {2 b^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {3 b^{4} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {2 b^{3} c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}+\frac {b^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\)	\(406\)
default	\(-\frac {\frac {3 b^{2} a^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {6 b^{3} a c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {2 b^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {3 b^{4} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {2 b^{3} c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}+\frac {b^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{2} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}+\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\)	\(406\)
risch	\(-\frac {4 b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{3}}{d^{2}}+\frac {4 b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{3}}{d^{5}}-\frac {2 b \,{\mathrm e}^{-b x -a}}{d^{2}}+\frac {2 b^{3} c \,{\mathrm e}^{-b x -a} x}{d^{3}}+\frac {b \,{\mathrm e}^{-b x -a} a^{4}}{d^{2} \left (-b x -\frac {b c}{d}\right )}+\frac {b^{5} {\mathrm e}^{-b x -a} c^{4}}{d^{6} \left (-b x -\frac {b c}{d}\right )}+\frac {b \,{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{4}}{d^{2}}+\frac {b^{5} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) c^{4}}{d^{6}}-\frac {4 b^{2} a \,{\mathrm e}^{-b x -a} x}{d^{2}}-\frac {4 b a \,{\mathrm e}^{-b x -a}}{d^{2}}+\frac {2 b^{2} c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {b^{3} {\mathrm e}^{-b x -a} x^{2}}{d^{2}}-\frac {2 b^{2} {\mathrm e}^{-b x -a} x}{d^{2}}+\frac {12 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2} c}{d^{3}}-\frac {12 b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a \,c^{2}}{d^{4}}+\frac {6 b^{3} {\mathrm e}^{-b x -a} a^{2} c^{2}}{d^{4} \left (-b x -\frac {b c}{d}\right )}-\frac {4 b^{4} {\mathrm e}^{-b x -a} a \,c^{3}}{d^{5} \left (-b x -\frac {b c}{d}\right )}-\frac {4 b^{2} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{3} c}{d^{3}}+\frac {6 b^{3} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a^{2} c^{2}}{d^{4}}-\frac {4 b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right ) a \,c^{3}}{d^{5}}-\frac {4 b^{2} {\mathrm e}^{-b x -a} a^{3} c}{d^{3} \left (-b x -\frac {b c}{d}\right )}+\frac {8 b^{2} a c \,{\mathrm e}^{-b x -a}}{d^{3}}-\frac {6 b \,a^{2} {\mathrm e}^{-b x -a}}{d^{2}}-\frac {3 b^{3} c^{2} {\mathrm e}^{-b x -a}}{d^{4}}\)	\(761\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.37 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=-\frac {{\left (b^{5} c^{5} - 4 \, {\left (a - 1\right )} b^{4} c^{4} d + 6 \, {\left (a^{2} - 2 \, a\right )} b^{3} c^{3} d^{2} - 4 \, {\left (a^{3} - 3 \, a^{2}\right )} b^{2} c^{2} d^{3} + {\left (a^{4} - 4 \, a^{3}\right )} b c d^{4} + {\left (b^{5} c^{4} d - 4 \, {\left (a - 1\right )} b^{4} c^{3} d^{2} + 6 \, {\left (a^{2} - 2 \, a\right )} b^{3} c^{2} d^{3} - 4 \, {\left (a^{3} - 3 \, a^{2}\right )} b^{2} c d^{4} + {\left (a^{4} - 4 \, a^{3}\right )} b d^{5}\right )} x\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} + {\left (b^{3} d^{5} x^{3} + b^{4} c^{4} d - {\left (4 \, a - 3\right )} b^{3} c^{3} d^{2} + a^{4} d^{5} + 2 \, {\left (3 \, a^{2} - 4 \, a - 1\right )} b^{2} c^{2} d^{3} - 2 \, {\left (2 \, a^{3} - 3 \, a^{2} - 2 \, a - 1\right )} b c d^{4} - {\left (b^{3} c d^{4} - 2 \, {\left (2 \, a + 1\right )} b^{2} d^{5}\right )} x^{2} + {\left (b^{3} c^{2} d^{3} - 4 \, a b^{2} c d^{4} + 2 \, {\left (3 \, a^{2} + 2 \, a + 1\right )} b d^{5}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{7} x + c d^{6}} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

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

[Out]

(Integral(a**4/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x) + Integral(b**4*x**4/(c**2*exp(b*x)
+ 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x) + Integral(4*a*b**3*x**3/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**
2*x**2*exp(b*x)), x) + Integral(6*a**2*b**2*x**2/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x) +
Integral(4*a**3*b*x/(c**2*exp(b*x) + 2*c*d*x*exp(b*x) + d**2*x**2*exp(b*x)), x))*exp(-a)

Maxima [F]

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

[In]

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2861 vs. \(2 (249) = 498\).

Time = 0.37 (sec) , antiderivative size = 2861, normalized size of antiderivative = 11.09 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^2} \, dx=\text {Too large to display} \]

[In]

integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c)^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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