Integrand size = 25, antiderivative size = 277 \[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=-\frac {6 e^{-a-b x}}{d}+\frac {2 (b c-a d) e^{-a-b x}}{d^2}-\frac {(b c-a d)^2 e^{-a-b x}}{d^3}+\frac {(b c-a d)^3 e^{-a-b x}}{d^4}-\frac {6 e^{-a-b x} (a+b x)}{d}+\frac {2 (b c-a d) e^{-a-b x} (a+b x)}{d^2}-\frac {(b c-a d)^2 e^{-a-b x} (a+b x)}{d^3}-\frac {3 e^{-a-b x} (a+b x)^2}{d}+\frac {(b c-a d) e^{-a-b x} (a+b x)^2}{d^2}-\frac {e^{-a-b x} (a+b x)^3}{d}+\frac {(b c-a d)^4 e^{-a+\frac {b c}{d}} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5} \] Output:
-6*exp(-b*x-a)/d+2*(-a*d+b*c)*exp(-b*x-a)/d^2-(-a*d+b*c)^2*exp(-b*x-a)/d^3 +(-a*d+b*c)^3*exp(-b*x-a)/d^4-6*exp(-b*x-a)*(b*x+a)/d+2*(-a*d+b*c)*exp(-b* x-a)*(b*x+a)/d^2-(-a*d+b*c)^2*exp(-b*x-a)*(b*x+a)/d^3-3*exp(-b*x-a)*(b*x+a )^2/d+(-a*d+b*c)*exp(-b*x-a)*(b*x+a)^2/d^2-exp(-b*x-a)*(b*x+a)^3/d+(-a*d+b *c)^4*exp(-a+b*c/d)*Ei(-b*(d*x+c)/d)/d^5
Time = 1.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.63 \[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=\frac {e^{-a-b x} \left (-d \left (2 \left (3+4 a+3 a^2+2 a^3\right ) d^3+2 b d^2 \left (-\left (\left (1+2 a+3 a^2\right ) c\right )+\left (3+4 a+3 a^2\right ) d x\right )+b^2 d \left ((1+4 a) c^2-2 (1+2 a) c d x+(3+4 a) d^2 x^2\right )+b^3 \left (-c^3+c^2 d x-c d^2 x^2+d^3 x^3\right )\right )+(b c-a d)^4 e^{b \left (\frac {c}{d}+x\right )} \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )\right )}{d^5} \] Input:
Integrate[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x),x]
Output:
(E^(-a - b*x)*(-(d*(2*(3 + 4*a + 3*a^2 + 2*a^3)*d^3 + 2*b*d^2*(-((1 + 2*a + 3*a^2)*c) + (3 + 4*a + 3*a^2)*d*x) + b^2*d*((1 + 4*a)*c^2 - 2*(1 + 2*a)* c*d*x + (3 + 4*a)*d^2*x^2) + b^3*(-c^3 + c^2*d*x - c*d^2*x^2 + d^3*x^3))) + (b*c - a*d)^4*E^(b*(c/d + x))*ExpIntegralEi[-((b*(c + d*x))/d)]))/d^5
Time = 1.00 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2629, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle \int \left (\frac {e^{-a-b x} (a d-b c)^4}{d^4 (c+d x)}-\frac {b e^{-a-b x} (b c-a d)^3}{d^4}+\frac {b e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}-\frac {b e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac {b e^{-a-b x} (a+b x)^3}{d}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^{\frac {b c}{d}-a} (b c-a d)^4 \operatorname {ExpIntegralEi}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {e^{-a-b x} (b c-a d)^3}{d^4}-\frac {e^{-a-b x} (b c-a d)^2}{d^3}-\frac {e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}+\frac {2 e^{-a-b x} (b c-a d)}{d^2}+\frac {e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac {2 e^{-a-b x} (a+b x) (b c-a d)}{d^2}-\frac {6 e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)^3}{d}-\frac {3 e^{-a-b x} (a+b x)^2}{d}-\frac {6 e^{-a-b x} (a+b x)}{d}\) |
Input:
Int[(E^(-a - b*x)*(a + b*x)^4)/(c + d*x),x]
Output:
(-6*E^(-a - b*x))/d + (2*(b*c - a*d)*E^(-a - b*x))/d^2 - ((b*c - a*d)^2*E^ (-a - b*x))/d^3 + ((b*c - a*d)^3*E^(-a - b*x))/d^4 - (6*E^(-a - b*x)*(a + b*x))/d + (2*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/d^2 - ((b*c - a*d)^2*E^(- a - b*x)*(a + b*x))/d^3 - (3*E^(-a - b*x)*(a + b*x)^2)/d + ((b*c - a*d)*E^ (-a - b*x)*(a + b*x)^2)/d^2 - (E^(-a - b*x)*(a + b*x)^3)/d + ((b*c - a*d)^ 4*E^(-a + (b*c)/d)*ExpIntegralEi[-((b*(c + d*x))/d)])/d^5
Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.11 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.77
| method | result | size |
| derivativedivides | \(-\frac {\frac {b \,a^{3} {\mathrm e}^{-b x -a}}{d}-\frac {3 b^{2} a^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \,a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d}+\frac {3 b^{3} a \,c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {2 b^{2} a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {b a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d}-\frac {b^{4} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {b^{3} c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}-\frac {b^{2} c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}-\frac {b \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{d}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b \,{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{d^{5}}}{b}\) | \(489\) |
| default | \(-\frac {\frac {b \,a^{3} {\mathrm e}^{-b x -a}}{d}-\frac {3 b^{2} a^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b \,a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d}+\frac {3 b^{3} a \,c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {2 b^{2} a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{2}}+\frac {b a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d}-\frac {b^{4} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {b^{3} c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{d^{3}}-\frac {b^{2} c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{d^{2}}-\frac {b \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{d}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b \,{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right )}{d^{5}}}{b}\) | \(489\) |
| risch | \(-\frac {6 \,{\mathrm e}^{-b x -a}}{d}-\frac {8 a \,{\mathrm e}^{-b x -a}}{d}-\frac {6 a^{2} {\mathrm e}^{-b x -a}}{d}-\frac {4 a^{3} {\mathrm e}^{-b x -a}}{d}+\frac {4 b^{2} c a \,{\mathrm e}^{-b x -a} x}{d^{2}}+\frac {4 b \,{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right ) a^{3} c}{d^{2}}-\frac {6 b^{2} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right ) a^{2} c^{2}}{d^{3}}+\frac {4 b^{3} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right ) a \,c^{3}}{d^{4}}-\frac {b^{2} c^{2} {\mathrm e}^{-b x -a}}{d^{3}}+\frac {2 b c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {b^{3} {\mathrm e}^{-b x -a} x^{3}}{d}-\frac {3 b^{2} {\mathrm e}^{-b x -a} x^{2}}{d}+\frac {b^{3} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {6 b \,{\mathrm e}^{-b x -a} x}{d}-\frac {{\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right ) a^{4}}{d}+\frac {6 b \,a^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {4 b^{2} a \,c^{2} {\mathrm e}^{-b x -a}}{d^{3}}-\frac {b^{4} {\mathrm e}^{-\frac {d a -b c}{d}} \operatorname {expIntegral}_{1}\left (b x +a -\frac {d a -b c}{d}\right ) c^{4}}{d^{5}}-\frac {6 b \,a^{2} {\mathrm e}^{-b x -a} x}{d}+\frac {4 b c a \,{\mathrm e}^{-b x -a}}{d^{2}}-\frac {4 b^{2} a \,{\mathrm e}^{-b x -a} x^{2}}{d}-\frac {8 b a \,{\mathrm e}^{-b x -a} x}{d}-\frac {b^{3} c^{2} {\mathrm e}^{-b x -a} x}{d^{3}}+\frac {2 b^{2} c \,{\mathrm e}^{-b x -a} x}{d^{2}}+\frac {b^{3} c \,{\mathrm e}^{-b x -a} x^{2}}{d^{2}}\) | \(595\) |
Input:
int(exp(-b*x-a)*(b*x+a)^4/(d*x+c),x,method=_RETURNVERBOSE)
Output:
-1/b*(b/d*a^3*exp(-b*x-a)-3*b^2/d^2*a^2*c*exp(-b*x-a)-b/d*a^2*((-b*x-a)*ex p(-b*x-a)-exp(-b*x-a))+3*b^3/d^3*a*c^2*exp(-b*x-a)+2*b^2/d^2*a*c*((-b*x-a) *exp(-b*x-a)-exp(-b*x-a))+b/d*a*((-b*x-a)^2*exp(-b*x-a)-2*(-b*x-a)*exp(-b* x-a)+2*exp(-b*x-a))-b^4/d^4*c^3*exp(-b*x-a)-b^3/d^3*c^2*((-b*x-a)*exp(-b*x -a)-exp(-b*x-a))-b^2/d^2*c*((-b*x-a)^2*exp(-b*x-a)-2*(-b*x-a)*exp(-b*x-a)+ 2*exp(-b*x-a))-1/d*b*(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))+(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/d^5*exp(-(a*d-b*c)/d)*Ei(1,b*x+a-(a*d-b*c)/d))
Time = 0.07 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=\frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (b^{3} d^{4} x^{3} - b^{3} c^{3} d + {\left (4 \, a + 1\right )} b^{2} c^{2} d^{2} - 2 \, {\left (3 \, a^{2} + 2 \, a + 1\right )} b c d^{3} + 2 \, {\left (2 \, a^{3} + 3 \, a^{2} + 4 \, a + 3\right )} d^{4} - {\left (b^{3} c d^{3} - {\left (4 \, a + 3\right )} b^{2} d^{4}\right )} x^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, {\left (2 \, a + 1\right )} b^{2} c d^{3} + 2 \, {\left (3 \, a^{2} + 4 \, a + 3\right )} b d^{4}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{5}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c),x, algorithm="fricas")
Output:
((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*E i(-(b*d*x + b*c)/d)*e^((b*c - a*d)/d) - (b^3*d^4*x^3 - b^3*c^3*d + (4*a + 1)*b^2*c^2*d^2 - 2*(3*a^2 + 2*a + 1)*b*c*d^3 + 2*(2*a^3 + 3*a^2 + 4*a + 3) *d^4 - (b^3*c*d^3 - (4*a + 3)*b^2*d^4)*x^2 + (b^3*c^2*d^2 - 2*(2*a + 1)*b^ 2*c*d^3 + 2*(3*a^2 + 4*a + 3)*b*d^4)*x)*e^(-b*x - a))/d^5
\[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=\left (\int \frac {a^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c e^{b x} + d x e^{b x}}\, dx\right ) e^{- a} \] Input:
integrate(exp(-b*x-a)*(b*x+a)**4/(d*x+c),x)
Output:
(Integral(a**4/(c*exp(b*x) + d*x*exp(b*x)), x) + Integral(b**4*x**4/(c*exp (b*x) + d*x*exp(b*x)), x) + Integral(4*a*b**3*x**3/(c*exp(b*x) + d*x*exp(b *x)), x) + Integral(6*a**2*b**2*x**2/(c*exp(b*x) + d*x*exp(b*x)), x) + Int egral(4*a**3*b*x/(c*exp(b*x) + d*x*exp(b*x)), x))*exp(-a)
\[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=\int { \frac {{\left (b x + a\right )}^{4} e^{\left (-b x - a\right )}}{d x + c} \,d x } \] Input:
integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c),x, algorithm="maxima")
Output:
-a^4*e^(-a + b*c/d)*exp_integral_e(1, (d*x + c)*b/d)/d - (b^3*d^2*x^4 + (4 *a*b^2*d^2 + 3*b^2*d^2)*x^3 + (6*a^2*b*d^2 + b^2*c*d + 8*a*b*d^2 + 6*b*d^2 )*x^2 + (4*a^3*d^2 - b^2*c^2 + 6*a^2*d^2 + 4*b*c*d + 4*(b*c*d + 2*d^2)*a + 6*d^2)*x)*e^(-b*x)/(d^3*x*e^a + c*d^2*e^a) + integrate((4*a^3*c*d^2 - b^2 *c^3 + 6*a^2*c*d^2 + 4*b*c^2*d + 6*c*d^2 + 4*(b*c^2*d + 2*c*d^2)*a + (b^3* c^3 + 6*a^2*b*c*d^2 - 2*b^2*c^2*d + 6*b*c*d^2 - 4*(b^2*c^2*d - 2*b*c*d^2)* a)*x)*e^(-b*x)/(d^4*x^2*e^a + 2*c*d^3*x*e^a + c^2*d^2*e^a), x)
Leaf count of result is larger than twice the leaf count of optimal. 546 vs. \(2 (266) = 532\).
Time = 0.13 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.97 \[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=-\frac {b^{3} d^{4} x^{3} e^{\left (-b x - a\right )} - b^{3} c d^{3} x^{2} e^{\left (-b x - a\right )} + 4 \, a b^{2} d^{4} x^{2} e^{\left (-b x - a\right )} - b^{4} c^{4} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 4 \, a b^{3} c^{3} d {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 6 \, a^{2} b^{2} c^{2} d^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 4 \, a^{3} b c d^{3} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - a^{4} d^{4} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{3} c^{2} d^{2} x e^{\left (-b x - a\right )} - 4 \, a b^{2} c d^{3} x e^{\left (-b x - a\right )} + 6 \, a^{2} b d^{4} x e^{\left (-b x - a\right )} + 3 \, b^{2} d^{4} x^{2} e^{\left (-b x - a\right )} - b^{3} c^{3} d e^{\left (-b x - a\right )} + 4 \, a b^{2} c^{2} d^{2} e^{\left (-b x - a\right )} - 6 \, a^{2} b c d^{3} e^{\left (-b x - a\right )} + 4 \, a^{3} d^{4} e^{\left (-b x - a\right )} - 2 \, b^{2} c d^{3} x e^{\left (-b x - a\right )} + 8 \, a b d^{4} x e^{\left (-b x - a\right )} + b^{2} c^{2} d^{2} e^{\left (-b x - a\right )} - 4 \, a b c d^{3} e^{\left (-b x - a\right )} + 6 \, a^{2} d^{4} e^{\left (-b x - a\right )} + 6 \, b d^{4} x e^{\left (-b x - a\right )} - 2 \, b c d^{3} e^{\left (-b x - a\right )} + 8 \, a d^{4} e^{\left (-b x - a\right )} + 6 \, d^{4} e^{\left (-b x - a\right )}}{d^{5}} \] Input:
integrate(exp(-b*x-a)*(b*x+a)^4/(d*x+c),x, algorithm="giac")
Output:
-(b^3*d^4*x^3*e^(-b*x - a) - b^3*c*d^3*x^2*e^(-b*x - a) + 4*a*b^2*d^4*x^2* e^(-b*x - a) - b^4*c^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + 4*a*b^3*c^3*d *Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - 6*a^2*b^2*c^2*d^2*Ei(-(b*d*x + b*c) /d)*e^(-a + b*c/d) + 4*a^3*b*c*d^3*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) - a ^4*d^4*Ei(-(b*d*x + b*c)/d)*e^(-a + b*c/d) + b^3*c^2*d^2*x*e^(-b*x - a) - 4*a*b^2*c*d^3*x*e^(-b*x - a) + 6*a^2*b*d^4*x*e^(-b*x - a) + 3*b^2*d^4*x^2* e^(-b*x - a) - b^3*c^3*d*e^(-b*x - a) + 4*a*b^2*c^2*d^2*e^(-b*x - a) - 6*a ^2*b*c*d^3*e^(-b*x - a) + 4*a^3*d^4*e^(-b*x - a) - 2*b^2*c*d^3*x*e^(-b*x - a) + 8*a*b*d^4*x*e^(-b*x - a) + b^2*c^2*d^2*e^(-b*x - a) - 4*a*b*c*d^3*e^ (-b*x - a) + 6*a^2*d^4*e^(-b*x - a) + 6*b*d^4*x*e^(-b*x - a) - 2*b*c*d^3*e ^(-b*x - a) + 8*a*d^4*e^(-b*x - a) + 6*d^4*e^(-b*x - a))/d^5
Timed out. \[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=\int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{c+d\,x} \,d x \] Input:
int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x),x)
Output:
int((exp(- a - b*x)*(a + b*x)^4)/(c + d*x), x)
\[ \int \frac {e^{-a-b x} (a+b x)^4}{c+d x} \, dx=\frac {6 a^{2} b c \,d^{2}-6 a^{2} b \,d^{3} x -4 a \,b^{2} c^{2} d -4 a \,b^{2} d^{3} x^{2}+4 a b c \,d^{2}-8 a b \,d^{3} x -b^{3} c^{2} d x +b^{3} c \,d^{2} x^{2}+2 b^{2} c \,d^{2} x +e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a^{4} d^{4}+e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) b^{4} c^{4}+6 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a^{2} b^{2} c^{2} d^{2}-4 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a \,b^{3} c^{3} d -b^{3} d^{3} x^{3}-b^{2} c^{2} d -3 b^{2} d^{3} x^{2}+2 b c \,d^{2}-6 b \,d^{3} x -4 a^{3} d^{3}-6 a^{2} d^{3}-8 a \,d^{3}+b^{3} c^{3}-6 d^{3}-4 e^{b x} \left (\int \frac {1}{e^{b x} c +e^{b x} d x}d x \right ) a^{3} b c \,d^{3}+4 a \,b^{2} c \,d^{2} x}{e^{b x +a} d^{4}} \] Input:
int(exp(-b*x-a)*(b*x+a)^4/(d*x+c),x)
Output:
(e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*a**4*d**4 - 4*e**(b*x)*int( 1/(e**(b*x)*c + e**(b*x)*d*x),x)*a**3*b*c*d**3 + 6*e**(b*x)*int(1/(e**(b*x )*c + e**(b*x)*d*x),x)*a**2*b**2*c**2*d**2 - 4*e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d*x),x)*a*b**3*c**3*d + e**(b*x)*int(1/(e**(b*x)*c + e**(b*x)*d *x),x)*b**4*c**4 - 4*a**3*d**3 + 6*a**2*b*c*d**2 - 6*a**2*b*d**3*x - 6*a** 2*d**3 - 4*a*b**2*c**2*d + 4*a*b**2*c*d**2*x - 4*a*b**2*d**3*x**2 + 4*a*b* c*d**2 - 8*a*b*d**3*x - 8*a*d**3 + b**3*c**3 - b**3*c**2*d*x + b**3*c*d**2 *x**2 - b**3*d**3*x**3 - b**2*c**2*d + 2*b**2*c*d**2*x - 3*b**2*d**3*x**2 + 2*b*c*d**2 - 6*b*d**3*x - 6*d**3)/(e**(a + b*x)*d**4)