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

Optimal. Leaf size=294 \[ \frac{b^2 e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{2 d^7}+\frac{4 b^2 e^{\frac{b c}{d}-a} (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{6 b^2 e^{\frac{b c}{d}-a} (b c-a d)^2 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^2 e^{-a-b x} (3 b c-4 a d)}{d^4}-\frac{b^2 e^{-a-b x}}{d^3}-\frac{b^3 x e^{-a-b x}}{d^3}+\frac{b e^{-a-b x} (b c-a d)^4}{2 d^6 (c+d x)}-\frac{e^{-a-b x} (b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{4 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)} \]

[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)

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Rubi [A]  time = 0.407777, antiderivative size = 294, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2199, 2194, 2176, 2177, 2178} \[ \frac{b^2 e^{\frac{b c}{d}-a} (b c-a d)^4 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{2 d^7}+\frac{4 b^2 e^{\frac{b c}{d}-a} (b c-a d)^3 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^6}+\frac{6 b^2 e^{\frac{b c}{d}-a} (b c-a d)^2 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )}{d^5}+\frac{b^2 e^{-a-b x} (3 b c-4 a d)}{d^4}-\frac{b^2 e^{-a-b x}}{d^3}-\frac{b^3 x e^{-a-b x}}{d^3}+\frac{b e^{-a-b x} (b c-a d)^4}{2 d^6 (c+d x)}-\frac{e^{-a-b x} (b c-a d)^4}{2 d^5 (c+d x)^2}+\frac{4 b e^{-a-b x} (b c-a d)^3}{d^5 (c+d x)} \]

Antiderivative was successfully verified.

[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 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

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 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 2177

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] &&  !$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

Rubi steps

\begin{align*} \int \frac{e^{-a-b x} (a+b x)^4}{(c+d x)^3} \, dx &=\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]  time = 0.652042, size = 267, normalized size = 0.91 \[ \frac{e^{-a} \left (b^2 e^{\frac{b c}{d}} \left (\left (a^2-8 a+12\right ) d^2-2 (a-4) b c d+b^2 c^2\right ) (b c-a d)^2 \text{Ei}\left (-\frac{b (c+d x)}{d}\right )+\frac{d e^{-b x} \left (2 b^3 d^2 \left (\left (3 a^2-12 a+5\right ) c^2 d x+\left (3 a^2-10 a+3\right ) c^3+c d^2 x^2-d^3 x^3\right )-2 b^2 d^3 \left (\left (2 a^3-9 a^2+4 a+1\right ) c^2+2 \left (a^3-6 a^2+4 a+1\right ) c d x+(4 a+1) d^2 x^2\right )+a^3 b d^4 ((a-4) c+(a-8) d x)-a^4 d^5+b^4 c^3 d ((7-4 a) c-4 (a-2) d x)+b^5 c^4 (c+d x)\right )}{(c+d x)^2}\right )}{2 d^7} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.015, size = 418, normalized size = 1.4 \begin{align*} -{\frac{1}{b} \left ( -{\frac{{b}^{3} \left ( \left ( -bx-a \right ){{\rm e}^{-bx-a}}-{{\rm e}^{-bx-a}} \right ) }{{d}^{3}}}+3\,{\frac{a{b}^{3}{{\rm e}^{-bx-a}}}{{d}^{3}}}-3\,{\frac{{b}^{4}c{{\rm e}^{-bx-a}}}{{d}^{4}}}-{\frac{ \left ({a}^{4}{d}^{4}-4\,{a}^{3}bc{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}}{{d}^{7}} \left ( -{\frac{{{\rm e}^{-bx-a}}}{2} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-2}}-{\frac{{{\rm e}^{-bx-a}}}{2} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{\frac{1}{2}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) } \right ) }+6\,{\frac{ \left ({a}^{2}{d}^{2}-2\,abcd+{b}^{2}{c}^{2} \right ){b}^{3}}{{d}^{5}}{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) }+4\,{\frac{ \left ({a}^{3}{d}^{3}-3\,{a}^{2}bc{d}^{2}+3\,a{b}^{2}{c}^{2}d-{b}^{3}{c}^{3} \right ){b}^{3}}{{d}^{6}} \left ( -{{{\rm e}^{-bx-a}} \left ( -bx-a+{\frac{ad-bc}{d}} \right ) ^{-1}}-{{\rm e}^{-{\frac{ad-bc}{d}}}}{\it Ei} \left ( 1,bx+a-{\frac{ad-bc}{d}} \right ) \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{a^{4} e^{\left (-a + \frac{b c}{d}\right )} E_{3}\left (\frac{{\left (d x + c\right )} b}{d}\right )}{{\left (d x + c\right )}^{2} d} - \frac{{\left (b^{3} d^{2} x^{4} +{\left (4 \, a b^{2} d^{2} + b^{2} d^{2}\right )} x^{3} + 3 \,{\left (2 \, a^{2} b d^{2} + b^{2} c d\right )} x^{2} +{\left (4 \, a^{3} d^{2} - 3 \, b^{2} c^{2} + 12 \, a b c d - 6 \, a^{2} d^{2}\right )} x\right )} e^{\left (-b x\right )}}{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}} - \int -\frac{{\left (4 \, a^{3} c d^{2} - 3 \, b^{2} c^{3} + 12 \, a b c^{2} d - 6 \, a^{2} c d^{2} +{\left (3 \, b^{3} c^{3} - 8 \, a^{3} d^{3} + 12 \, b^{2} c^{2} d + 6 \,{\left (3 \, b c d^{2} + 2 \, d^{3}\right )} a^{2} - 12 \,{\left (b^{2} c^{2} d + 2 \, b c d^{2}\right )} a\right )} x\right )} e^{\left (-b x\right )}}{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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.56864, size = 1158, normalized size = 3.94 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.312, size = 2419, normalized size = 8.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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) + 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) + 24*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) + 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))/(d^9*x^2 + 2*c*d^8*x + c^2*d^7)