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

Optimal. Leaf size=277 \[ \frac {e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {e^{-a-b x} (b c-a d)^3}{d^4}-\frac {e^{-a-b x} (b c-a d)^2}{d^3}-\frac {e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}+\frac {2 e^{-a-b x} (b c-a d)}{d^2}+\frac {e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac {2 e^{-a-b x} (a+b x) (b c-a d)}{d^2}-\frac {6 e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)^3}{d}-\frac {3 e^{-a-b x} (a+b x)^2}{d}-\frac {6 e^{-a-b x} (a+b x)}{d} \]

[Out]

-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

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Rubi [A]  time = 0.34, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2199, 2194, 2176, 2178} \[ \frac {e^{\frac {b c}{d}-a} (b c-a d)^4 \text {Ei}\left (-\frac {b (c+d x)}{d}\right )}{d^5}+\frac {e^{-a-b x} (b c-a d)^3}{d^4}-\frac {e^{-a-b x} (b c-a d)^2}{d^3}-\frac {e^{-a-b x} (a+b x) (b c-a d)^2}{d^3}+\frac {2 e^{-a-b x} (b c-a d)}{d^2}+\frac {e^{-a-b x} (a+b x)^2 (b c-a d)}{d^2}+\frac {2 e^{-a-b x} (a+b x) (b c-a d)}{d^2}-\frac {6 e^{-a-b x}}{d}-\frac {e^{-a-b x} (a+b x)^3}{d}-\frac {3 e^{-a-b x} (a+b x)^2}{d}-\frac {6 e^{-a-b x} (a+b x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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

Rule 2176

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] &&  !$UseGamma === True

Rule 2178

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

Rule 2194

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 2199

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] &&  !$UseGamma === True

Rubi steps

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

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Mathematica [A]  time = 0.31, size = 175, normalized size = 0.63 \[ \frac {e^{-a-b x} \left ((b c-a d)^4 e^{b \left (\frac {c}{d}+x\right )} \text {Ei}\left (-\frac {b (c+d x)}{d}\right )-d \left (2 b d^2 \left (\left (3 a^2+4 a+3\right ) d x-\left (3 a^2+2 a+1\right ) c\right )+2 \left (2 a^3+3 a^2+4 a+3\right ) d^3+b^2 d \left ((4 a+1) c^2-2 (2 a+1) c d x+(4 a+3) d^2 x^2\right )+b^3 \left (-c^3+c^2 d x-c d^2 x^2+d^3 x^3\right )\right )\right )}{d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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

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fricas [A]  time = 0.41, size = 235, normalized size = 0.85 \[ \frac {{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (\frac {b c - a d}{d}\right )} - {\left (b^{3} d^{4} x^{3} - b^{3} c^{3} d + {\left (4 \, a + 1\right )} b^{2} c^{2} d^{2} - 2 \, {\left (3 \, a^{2} + 2 \, a + 1\right )} b c d^{3} + 2 \, {\left (2 \, a^{3} + 3 \, a^{2} + 4 \, a + 3\right )} d^{4} - {\left (b^{3} c d^{3} - {\left (4 \, a + 3\right )} b^{2} d^{4}\right )} x^{2} + {\left (b^{3} c^{2} d^{2} - 2 \, {\left (2 \, a + 1\right )} b^{2} c d^{3} + 2 \, {\left (3 \, a^{2} + 4 \, a + 3\right )} b d^{4}\right )} x\right )} e^{\left (-b x - a\right )}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((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)*Ei(-(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

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giac [B]  time = 0.40, size = 546, normalized size = 1.97 \[ -\frac {b^{3} d^{4} x^{3} e^{\left (-b x - a\right )} - b^{3} c d^{3} x^{2} e^{\left (-b x - a\right )} + 4 \, a b^{2} d^{4} x^{2} e^{\left (-b x - a\right )} - b^{4} c^{4} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 4 \, a b^{3} c^{3} d {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - 6 \, a^{2} b^{2} c^{2} d^{2} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + 4 \, a^{3} b c d^{3} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} - a^{4} d^{4} {\rm Ei}\left (-\frac {b d x + b c}{d}\right ) e^{\left (-a + \frac {b c}{d}\right )} + b^{3} c^{2} d^{2} x e^{\left (-b x - a\right )} - 4 \, a b^{2} c d^{3} x e^{\left (-b x - a\right )} + 6 \, a^{2} b d^{4} x e^{\left (-b x - a\right )} + 3 \, b^{2} d^{4} x^{2} e^{\left (-b x - a\right )} - b^{3} c^{3} d e^{\left (-b x - a\right )} + 4 \, a b^{2} c^{2} d^{2} e^{\left (-b x - a\right )} - 6 \, a^{2} b c d^{3} e^{\left (-b x - a\right )} + 4 \, a^{3} d^{4} e^{\left (-b x - a\right )} - 2 \, b^{2} c d^{3} x e^{\left (-b x - a\right )} + 8 \, a b d^{4} x e^{\left (-b x - a\right )} + b^{2} c^{2} d^{2} e^{\left (-b x - a\right )} - 4 \, a b c d^{3} e^{\left (-b x - a\right )} + 6 \, a^{2} d^{4} e^{\left (-b x - a\right )} + 6 \, b d^{4} x e^{\left (-b x - a\right )} - 2 \, b c d^{3} e^{\left (-b x - a\right )} + 8 \, a d^{4} e^{\left (-b x - a\right )} + 6 \, d^{4} e^{\left (-b x - a\right )}}{d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(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

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maple [A]  time = 0.02, size = 489, normalized size = 1.77 \[ -\frac {\frac {a^{3} b \,{\mathrm e}^{-b x -a}}{d}-\frac {3 a^{2} b^{2} c \,{\mathrm e}^{-b x -a}}{d^{2}}+\frac {3 a \,b^{3} c^{2} {\mathrm e}^{-b x -a}}{d^{3}}-\frac {b^{4} c^{3} {\mathrm e}^{-b x -a}}{d^{4}}-\frac {\left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) a^{2} b}{d}+\frac {2 \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) a \,b^{2} c}{d^{2}}-\frac {\left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right ) b^{3} c^{2}}{d^{3}}+\frac {\left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right ) a b}{d}-\frac {\left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right ) b^{2} c}{d^{2}}-\frac {\left (\left (-b x -a \right )^{3} {\mathrm e}^{-b x -a}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right ) b}{d}+\frac {\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) b \Ei \left (1, b x +a -\frac {a d -b c}{d}\right ) {\mathrm e}^{-\frac {a d -b c}{d}}}{d^{5}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{4} e^{\left (-a + \frac {b c}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{d} - \frac {{\left (b^{3} d^{2} x^{4} + {\left (4 \, a b^{2} d^{2} + 3 \, b^{2} d^{2}\right )} x^{3} + {\left (6 \, a^{2} b d^{2} + b^{2} c d + 8 \, a b d^{2} + 6 \, b d^{2}\right )} x^{2} + {\left (4 \, a^{3} d^{2} - b^{2} c^{2} + 6 \, a^{2} d^{2} + 4 \, b c d + 4 \, {\left (b c d + 2 \, d^{2}\right )} a + 6 \, d^{2}\right )} x\right )} e^{\left (-b x\right )}}{d^{3} x e^{a} + c d^{2} e^{a}} + \int \frac {{\left (4 \, a^{3} c d^{2} - b^{2} c^{3} + 6 \, a^{2} c d^{2} + 4 \, b c^{2} d + 6 \, c d^{2} + 4 \, {\left (b c^{2} d + 2 \, c d^{2}\right )} a + {\left (b^{3} c^{3} + 6 \, a^{2} b c d^{2} - 2 \, b^{2} c^{2} d + 6 \, b c d^{2} - 4 \, {\left (b^{2} c^{2} d - 2 \, b c d^{2}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{d^{4} x^{2} e^{a} + 2 \, c d^{3} x e^{a} + c^{2} d^{2} e^{a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{-a-b\,x}\,{\left (a+b\,x\right )}^4}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \left (\int \frac {a^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {b^{4} x^{4}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {4 a b^{3} x^{3}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {6 a^{2} b^{2} x^{2}}{c e^{b x} + d x e^{b x}}\, dx + \int \frac {4 a^{3} b x}{c e^{b x} + d x e^{b x}}\, dx\right ) e^{- a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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