\(\int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx\) [81]

Optimal result

Integrand size = 25, antiderivative size = 396 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=-\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{6 d^7 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {2 b^3 (b c-a d)^3 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {b^3 (b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2230, 2225, 2208, 2209} \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=-\frac {b^3 e^{\frac {b c}{d}-a} (b c-a d)^4 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8}-\frac {2 b^3 e^{\frac {b c}{d}-a} (b c-a d)^3 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {6 b^3 e^{\frac {b c}{d}-a} (b c-a d)^2 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {4 b^3 e^{\frac {b c}{d}-a} (b c-a d) \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {b^3 e^{-a-b x}}{d^4}-\frac {b^2 e^{-a-b x} (b c-a d)^4}{6 d^7 (c+d x)}-\frac {2 b^2 e^{-a-b x} (b c-a d)^3}{d^6 (c+d x)}-\frac {6 b^2 e^{-a-b x} (b c-a d)^2}{d^5 (c+d x)}+\frac {b e^{-a-b x} (b c-a d)^4}{6 d^6 (c+d x)^2}-\frac {e^{-a-b x} (b c-a d)^4}{3 d^5 (c+d x)^3}+\frac {2 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)^2} \]

[In]

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

[Out]

-((b^3*E^(-a - b*x))/d^4) - ((b*c - a*d)^4*E^(-a - b*x))/(3*d^5*(c + d*x)^3) + (2*b*(b*c - a*d)^3*E^(-a - b*x)
)/(d^5*(c + d*x)^2) + (b*(b*c - a*d)^4*E^(-a - b*x))/(6*d^6*(c + d*x)^2) - (6*b^2*(b*c - a*d)^2*E^(-a - b*x))/
(d^5*(c + d*x)) - (2*b^2*(b*c - a*d)^3*E^(-a - b*x))/(d^6*(c + d*x)) - (b^2*(b*c - a*d)^4*E^(-a - b*x))/(6*d^7
*(c + d*x)) - (4*b^3*(b*c - a*d)*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5 - (6*b^3*(b*c - a*d)^
2*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^6 - (2*b^3*(b*c - a*d)^3*E^(-a + (b*c)/d)*ExpIntegralE
i[-((b*(c + d*x))/d)])/d^7 - (b^3*(b*c - a*d)^4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/(6*d^8)

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^4 e^{-a-b x}}{d^4}+\frac {(-b c+a d)^4 e^{-a-b x}}{d^4 (c+d x)^4}-\frac {4 b (b c-a d)^3 e^{-a-b x}}{d^4 (c+d x)^3}+\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^4 (c+d x)^2}-\frac {4 b^3 (b c-a d) e^{-a-b x}}{d^4 (c+d x)}\right ) \, dx \\ & = \frac {b^4 \int e^{-a-b x} \, dx}{d^4}-\frac {\left (4 b^3 (b c-a d)\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^4}+\frac {\left (6 b^2 (b c-a d)^2\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{d^4}-\frac {\left (4 b (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{(c+d x)^3} \, dx}{d^4}+\frac {(b c-a d)^4 \int \frac {e^{-a-b x}}{(c+d x)^4} \, dx}{d^4} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {\left (6 b^3 (b c-a d)^2\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^5}+\frac {\left (2 b^2 (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{d^5}-\frac {\left (b (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{(c+d x)^3} \, dx}{3 d^5} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {\left (2 b^3 (b c-a d)^3\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{d^6}+\frac {\left (b^2 (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{(c+d x)^2} \, dx}{6 d^6} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{6 d^7 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {2 b^3 (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {\left (b^3 (b c-a d)^4\right ) \int \frac {e^{-a-b x}}{c+d x} \, dx}{6 d^7} \\ & = -\frac {b^3 e^{-a-b x}}{d^4}-\frac {(b c-a d)^4 e^{-a-b x}}{3 d^5 (c+d x)^3}+\frac {2 b (b c-a d)^3 e^{-a-b x}}{d^5 (c+d x)^2}+\frac {b (b c-a d)^4 e^{-a-b x}}{6 d^6 (c+d x)^2}-\frac {6 b^2 (b c-a d)^2 e^{-a-b x}}{d^5 (c+d x)}-\frac {2 b^2 (b c-a d)^3 e^{-a-b x}}{d^6 (c+d x)}-\frac {b^2 (b c-a d)^4 e^{-a-b x}}{6 d^7 (c+d x)}-\frac {4 b^3 (b c-a d) e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}-\frac {6 b^3 (b c-a d)^2 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^6}-\frac {2 b^3 (b c-a d)^3 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^7}-\frac {b^3 (b c-a d)^4 e^{-a+\frac {b c}{d}} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{6 d^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.26 (sec) , antiderivative size = 389, normalized size of antiderivative = 0.98 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\frac {e^{-a} \left (-\frac {d e^{-b x} \left (2 a^4 d^6+b^6 c^4 (c+d x)^2-a^3 b d^5 ((-4+a) c+(-12+a) d x)-b^5 c^3 d (c+d x) ((-11+4 a) c+4 (-3+a) d x)+a^2 b^2 d^4 \left (\left (12-8 a+a^2\right ) c^2+2 \left (18-10 a+a^2\right ) c d x+(-6+a)^2 d^2 x^2\right )+2 b^4 c^2 d^2 \left (\left (13-16 a+3 a^2\right ) c^2+2 \left (15-17 a+3 a^2\right ) c d x+3 \left (6-6 a+a^2\right ) d^2 x^2\right )+2 b^3 d^3 \left (\left (3-22 a+15 a^2-2 a^3\right ) c^3+\left (9-54 a+33 a^2-4 a^3\right ) c^2 d x+\left (9-36 a+18 a^2-2 a^3\right ) c d^2 x^2+3 d^3 x^3\right )\right )}{(c+d x)^3}-b^3 \left (b^4 c^4-4 (-3+a) b^3 c^3 d+6 \left (6-6 a+a^2\right ) b^2 c^2 d^2-4 \left (-6+18 a-9 a^2+a^3\right ) b c d^3+a \left (-24+36 a-12 a^2+a^3\right ) d^4\right ) e^{\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{6 d^8} \]

[In]

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

[Out]

(-((d*(2*a^4*d^6 + b^6*c^4*(c + d*x)^2 - a^3*b*d^5*((-4 + a)*c + (-12 + a)*d*x) - b^5*c^3*d*(c + d*x)*((-11 +
4*a)*c + 4*(-3 + a)*d*x) + a^2*b^2*d^4*((12 - 8*a + a^2)*c^2 + 2*(18 - 10*a + a^2)*c*d*x + (-6 + a)^2*d^2*x^2)
+ 2*b^4*c^2*d^2*((13 - 16*a + 3*a^2)*c^2 + 2*(15 - 17*a + 3*a^2)*c*d*x + 3*(6 - 6*a + a^2)*d^2*x^2) + 2*b^3*d
^3*((3 - 22*a + 15*a^2 - 2*a^3)*c^3 + (9 - 54*a + 33*a^2 - 4*a^3)*c^2*d*x + (9 - 36*a + 18*a^2 - 2*a^3)*c*d^2*
x^2 + 3*d^3*x^3)))/(E^(b*x)*(c + d*x)^3)) - b^3*(b^4*c^4 - 4*(-3 + a)*b^3*c^3*d + 6*(6 - 6*a + a^2)*b^2*c^2*d^
2 - 4*(-6 + 18*a - 9*a^2 + a^3)*b*c*d^3 + a*(-24 + 36*a - 12*a^2 + a^3)*d^4)*E^((b*c)/d)*ExpIntegralEi[-((b*(c
+ d*x))/d)])/(6*d^8*E^a)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.29

method	result	size
derivativedivides	\(-\frac {\frac {b^{4} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{8}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {4 \left (a d -c b \right ) b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\)	\(511\)
default	\(-\frac {\frac {b^{4} {\mathrm e}^{-b x -a}}{d^{4}}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{3 \left (-b x -a +\frac {a d -c b}{d}\right )^{3}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{6 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{6}\right )}{d^{8}}+\frac {6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{-b x -a +\frac {a d -c b}{d}}-{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )\right )}{d^{6}}-\frac {4 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{4} \left (-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )^{2}}-\frac {{\mathrm e}^{-b x -a}}{2 \left (-b x -a +\frac {a d -c b}{d}\right )}-\frac {{\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{2}\right )}{d^{7}}+\frac {4 \left (a d -c b \right ) b^{4} {\mathrm e}^{-\frac {a d -c b}{d}} \operatorname {Ei}_{1}\left (b x +a -\frac {a d -c b}{d}\right )}{d^{5}}}{b}\)	\(511\)
risch	\(\text {Expression too large to display}\)	\(1550\)

[In]

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

[Out]

-1/b*(b^4/d^4*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^4/d^8*(-1/3*exp(-b
*x-a)/(-b*x-a+(a*d-b*c)/d)^3-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)^2-1/6*exp(-b*x-a)/(-b*x-a+(a*d-b*c)/d)-1/6*e
xp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))+6/d^6*(a^2*d^2-2*a*b*c*d+b^2*c^2)*b^4*(-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^7*(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)*b^4*(-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))+4/d^5*(a*d-b*c)*b^4*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (377) = 754\).

Time = 0.28 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.00 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=-\frac {{\left (b^{7} c^{7} - 4 \, {\left (a - 3\right )} b^{6} c^{6} d + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{5} d^{2} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c^{4} d^{3} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c^{3} d^{4} + {\left (b^{7} c^{4} d^{3} - 4 \, {\left (a - 3\right )} b^{6} c^{3} d^{4} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{2} d^{5} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c d^{6} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} d^{7}\right )} x^{3} + 3 \, {\left (b^{7} c^{5} d^{2} - 4 \, {\left (a - 3\right )} b^{6} c^{4} d^{3} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{3} d^{4} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c^{2} d^{5} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c d^{6}\right )} x^{2} + 3 \, {\left (b^{7} c^{6} d - 4 \, {\left (a - 3\right )} b^{6} c^{5} d^{2} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{5} c^{4} d^{3} - 4 \, {\left (a^{3} - 9 \, a^{2} + 18 \, a - 6\right )} b^{4} c^{3} d^{4} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2} - 24 \, a\right )} b^{3} c^{2} 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^{6} c^{6} d - {\left (4 \, a - 11\right )} b^{5} c^{5} d^{2} + 6 \, b^{3} d^{7} x^{3} + 2 \, {\left (3 \, a^{2} - 16 \, a + 13\right )} b^{4} c^{4} d^{3} - 2 \, {\left (2 \, a^{3} - 15 \, a^{2} + 22 \, a - 3\right )} b^{3} c^{3} d^{4} + 2 \, a^{4} d^{7} + {\left (a^{4} - 8 \, a^{3} + 12 \, a^{2}\right )} b^{2} c^{2} d^{5} - {\left (a^{4} - 4 \, a^{3}\right )} b c d^{6} + {\left (b^{6} c^{4} d^{3} - 4 \, {\left (a - 3\right )} b^{5} c^{3} d^{4} + 6 \, {\left (a^{2} - 6 \, a + 6\right )} b^{4} c^{2} d^{5} - 2 \, {\left (2 \, a^{3} - 18 \, a^{2} + 36 \, a - 9\right )} b^{3} c d^{6} + {\left (a^{4} - 12 \, a^{3} + 36 \, a^{2}\right )} b^{2} d^{7}\right )} x^{2} + {\left (2 \, b^{6} c^{5} d^{2} - {\left (8 \, a - 23\right )} b^{5} c^{4} d^{3} + 4 \, {\left (3 \, a^{2} - 17 \, a + 15\right )} b^{4} c^{3} d^{4} - 2 \, {\left (4 \, a^{3} - 33 \, a^{2} + 54 \, a - 9\right )} b^{3} c^{2} d^{5} + 2 \, {\left (a^{4} - 10 \, a^{3} + 18 \, a^{2}\right )} b^{2} c d^{6} - {\left (a^{4} - 12 \, a^{3}\right )} b d^{7}\right )} x\right )} e^{\left (-b x - a\right )}}{6 \, {\left (d^{11} x^{3} + 3 \, c d^{10} x^{2} + 3 \, c^{2} d^{9} x + c^{3} d^{8}\right )}} \]

[In]

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

[Out]

-1/6*((b^7*c^7 - 4*(a - 3)*b^6*c^6*d + 6*(a^2 - 6*a + 6)*b^5*c^5*d^2 - 4*(a^3 - 9*a^2 + 18*a - 6)*b^4*c^4*d^3
+ (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*c^3*d^4 + (b^7*c^4*d^3 - 4*(a - 3)*b^6*c^3*d^4 + 6*(a^2 - 6*a + 6)*b^5*c^
2*d^5 - 4*(a^3 - 9*a^2 + 18*a - 6)*b^4*c*d^6 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*d^7)*x^3 + 3*(b^7*c^5*d^2 -
4*(a - 3)*b^6*c^4*d^3 + 6*(a^2 - 6*a + 6)*b^5*c^3*d^4 - 4*(a^3 - 9*a^2 + 18*a - 6)*b^4*c^2*d^5 + (a^4 - 12*a^3
+ 36*a^2 - 24*a)*b^3*c*d^6)*x^2 + 3*(b^7*c^6*d - 4*(a - 3)*b^6*c^5*d^2 + 6*(a^2 - 6*a + 6)*b^5*c^4*d^3 - 4*(a
^3 - 9*a^2 + 18*a - 6)*b^4*c^3*d^4 + (a^4 - 12*a^3 + 36*a^2 - 24*a)*b^3*c^2*d^5)*x)*Ei(-(b*d*x + b*c)/d)*e^((b
*c - a*d)/d) + (b^6*c^6*d - (4*a - 11)*b^5*c^5*d^2 + 6*b^3*d^7*x^3 + 2*(3*a^2 - 16*a + 13)*b^4*c^4*d^3 - 2*(2*
a^3 - 15*a^2 + 22*a - 3)*b^3*c^3*d^4 + 2*a^4*d^7 + (a^4 - 8*a^3 + 12*a^2)*b^2*c^2*d^5 - (a^4 - 4*a^3)*b*c*d^6
+ (b^6*c^4*d^3 - 4*(a - 3)*b^5*c^3*d^4 + 6*(a^2 - 6*a + 6)*b^4*c^2*d^5 - 2*(2*a^3 - 18*a^2 + 36*a - 9)*b^3*c*d
^6 + (a^4 - 12*a^3 + 36*a^2)*b^2*d^7)*x^2 + (2*b^6*c^5*d^2 - (8*a - 23)*b^5*c^4*d^3 + 4*(3*a^2 - 17*a + 15)*b^
4*c^3*d^4 - 2*(4*a^3 - 33*a^2 + 54*a - 9)*b^3*c^2*d^5 + 2*(a^4 - 10*a^3 + 18*a^2)*b^2*c*d^6 - (a^4 - 12*a^3)*b
*d^7)*x)*e^(-b*x - a))/(d^11*x^3 + 3*c*d^10*x^2 + 3*c^2*d^9*x + c^3*d^8)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-a^4*e^(-a + b*c/d)*exp_integral_e(4, (d*x + c)*b/d)/((d*x + c)^3*d) - (b^3*d^2*x^4 + 4*a*b^2*d^2*x^3 + 2*(3*a
^2*b*d^2 + 2*b^2*c*d - 2*a*b*d^2)*x^2 + 4*(a^3*d^2 - b^2*c^2 - 3*a^2*d^2 - 2*b*c*d + 2*(2*b*c*d + 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) - integrate(-4*(a^
3*c*d^2 - b^2*c^3 - 3*a^2*c*d^2 - 2*b*c^2*d + 2*(2*b*c^2*d + c*d^2)*a + (b^3*c^3 - 3*a^3*d^3 + 7*b^2*c^2*d + 6
*b*c*d^2 + 3*(2*b*c*d^2 + 3*d^3)*a^2 - 2*(2*b^2*c^2*d + 8*b*c*d^2 + 3*d^3)*a)*x)*e^(-b*x)/(d^7*x^5*e^a + 5*c*d
^6*x^4*e^a + 10*c^2*d^5*x^3*e^a + 10*c^3*d^4*x^2*e^a + 5*c^4*d^3*x*e^a + c^5*d^2*e^a), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3178 vs. \(2 (377) = 754\).

Time = 0.40 (sec) , antiderivative size = 3178, normalized size of antiderivative = 8.03 \[ \int \frac {e^{-a-b x} (a+b x)^4}{(c+d x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*(b^7*c^4*d^3*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^6*c^3*d^4*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b
*c/d) + 6*a^2*b^5*c^2*d^5*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^4*c*d^6*x^3*Ei(-(b*d*x + b*c)/d)*e
^(-a + b*c/d) + a^4*b^3*d^7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*b^7*c^5*d^2*x^2*Ei(-(b*d*x + b*c)/d)*e
^(-a + b*c/d) - 12*a*b^6*c^4*d^3*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 18*a^2*b^5*c^3*d^4*x^2*Ei(-(b*d*x +
b*c)/d)*e^(-a + b*c/d) - 12*a^3*b^4*c^2*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*a^4*b^3*c*d^6*x^2*Ei(
-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 12*b^6*c^3*d^4*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 36*a*b^5*c^2*d^5*x
^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 36*a^2*b^4*c*d^6*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 12*a^3*b^3
*d^7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*b^7*c^6*d*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 12*a*b^6*c^
5*d^2*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 18*a^2*b^5*c^4*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 12*a^
3*b^4*c^3*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 3*a^4*b^3*c^2*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d)
+ 36*b^6*c^4*d^3*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 108*a*b^5*c^3*d^4*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a +
b*c/d) + 108*a^2*b^4*c^2*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 36*a^3*b^3*c*d^6*x^2*Ei(-(b*d*x + b*c)/
d)*e^(-a + b*c/d) + 36*b^5*c^2*d^5*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 72*a*b^4*c*d^6*x^3*Ei(-(b*d*x + b
*c)/d)*e^(-a + b*c/d) + 36*a^2*b^3*d^7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^6*c^4*d^3*x^2*e^(-b*x - a)
- 4*a*b^5*c^3*d^4*x^2*e^(-b*x - a) + 6*a^2*b^4*c^2*d^5*x^2*e^(-b*x - a) - 4*a^3*b^3*c*d^6*x^2*e^(-b*x - a) + a
^4*b^2*d^7*x^2*e^(-b*x - a) + b^7*c^7*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a*b^6*c^6*d*Ei(-(b*d*x + b*c)/d)
*e^(-a + b*c/d) + 6*a^2*b^5*c^5*d^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 4*a^3*b^4*c^4*d^3*Ei(-(b*d*x + b*c)/
d)*e^(-a + b*c/d) + a^4*b^3*c^3*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 36*b^6*c^5*d^2*x*Ei(-(b*d*x + b*c)/d
)*e^(-a + b*c/d) - 108*a*b^5*c^4*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 108*a^2*b^4*c^3*d^4*x*Ei(-(b*d*x
+ b*c)/d)*e^(-a + b*c/d) - 36*a^3*b^3*c^2*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 108*b^5*c^3*d^4*x^2*Ei(-
(b*d*x + b*c)/d)*e^(-a + b*c/d) - 216*a*b^4*c^2*d^5*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 108*a^2*b^3*c*d^
6*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 24*b^4*c*d^6*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 24*a*b^3*d^
7*x^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 2*b^6*c^5*d^2*x*e^(-b*x - a) - 8*a*b^5*c^4*d^3*x*e^(-b*x - a) + 12
*a^2*b^4*c^3*d^4*x*e^(-b*x - a) - 8*a^3*b^3*c^2*d^5*x*e^(-b*x - a) + 2*a^4*b^2*c*d^6*x*e^(-b*x - a) + 12*b^5*c
^3*d^4*x^2*e^(-b*x - a) - 36*a*b^4*c^2*d^5*x^2*e^(-b*x - a) + 36*a^2*b^3*c*d^6*x^2*e^(-b*x - a) - 12*a^3*b^2*d
^7*x^2*e^(-b*x - a) + 12*b^6*c^6*d*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 36*a*b^5*c^5*d^2*Ei(-(b*d*x + b*c)/d)
*e^(-a + b*c/d) + 36*a^2*b^4*c^4*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 12*a^3*b^3*c^3*d^4*Ei(-(b*d*x + b*c
)/d)*e^(-a + b*c/d) + 108*b^5*c^4*d^3*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 216*a*b^4*c^3*d^4*x*Ei(-(b*d*x +
b*c)/d)*e^(-a + b*c/d) + 108*a^2*b^3*c^2*d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 72*b^4*c^2*d^5*x^2*Ei(-(
b*d*x + b*c)/d)*e^(-a + b*c/d) - 72*a*b^3*c*d^6*x^2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^6*c^6*d*e^(-b*x -
a) - 4*a*b^5*c^5*d^2*e^(-b*x - a) + 6*a^2*b^4*c^4*d^3*e^(-b*x - a) - 4*a^3*b^3*c^3*d^4*e^(-b*x - a) + a^4*b^2*
c^2*d^5*e^(-b*x - a) + 23*b^5*c^4*d^3*x*e^(-b*x - a) - 68*a*b^4*c^3*d^4*x*e^(-b*x - a) + 66*a^2*b^3*c^2*d^5*x*
e^(-b*x - a) - 20*a^3*b^2*c*d^6*x*e^(-b*x - a) - a^4*b*d^7*x*e^(-b*x - a) + 36*b^4*c^2*d^5*x^2*e^(-b*x - a) -
72*a*b^3*c*d^6*x^2*e^(-b*x - a) + 36*a^2*b^2*d^7*x^2*e^(-b*x - a) + 6*b^3*d^7*x^3*e^(-b*x - a) + 36*b^5*c^5*d^
2*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 72*a*b^4*c^4*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 36*a^2*b^3*c^3*
d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 72*b^4*c^3*d^4*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 72*a*b^3*c^2*
d^5*x*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 11*b^5*c^5*d^2*e^(-b*x - a) - 32*a*b^4*c^4*d^3*e^(-b*x - a) + 30*a
^2*b^3*c^3*d^4*e^(-b*x - a) - 8*a^3*b^2*c^2*d^5*e^(-b*x - a) - a^4*b*c*d^6*e^(-b*x - a) + 60*b^4*c^3*d^4*x*e^(
-b*x - a) - 108*a*b^3*c^2*d^5*x*e^(-b*x - a) + 36*a^2*b^2*c*d^6*x*e^(-b*x - a) + 12*a^3*b*d^7*x*e^(-b*x - a) +
18*b^3*c*d^6*x^2*e^(-b*x - a) + 24*b^4*c^4*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 24*a*b^3*c^3*d^4*Ei(-(b*
d*x + b*c)/d)*e^(-a + b*c/d) + 26*b^4*c^4*d^3*e^(-b*x - a) - 44*a*b^3*c^3*d^4*e^(-b*x - a) + 12*a^2*b^2*c^2*d^
5*e^(-b*x - a) + 4*a^3*b*c*d^6*e^(-b*x - a) + 2*a^4*d^7*e^(-b*x - a) + 18*b^3*c^2*d^5*x*e^(-b*x - a) + 6*b^3*c
^3*d^4*e^(-b*x - a))/(d^11*x^3 + 3*c*d^10*x^2 + 3*c^2*d^9*x + c^3*d^8)

Mupad [F(-1)]

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

[In]

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

[Out]

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