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

Optimal. Leaf size=240 \[ \frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}+\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 1.04, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2223, 6742, 2222, 2210, 2228, 2178, 2209} \[ \frac {b d^2 e^2 e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{2 (b c-a d)^4}+\frac {d^2 e e^{\frac {b e}{b c-a d}} \text {Ei}\left (-\frac {d e (a+b x)}{(b c-a d) (c+d x)}\right )}{(b c-a d)^3}+\frac {d^2 e e^{\frac {e}{c+d x}}}{2 (b c-a d)^3}+\frac {d^2 e^{\frac {e}{c+d x}}}{2 b (b c-a d)^2}+\frac {d e e^{\frac {e}{c+d x}}}{2 (a+b x) (b c-a d)^2}-\frac {e^{\frac {e}{c+d x}}}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[E^(e/(c + d*x))/(a + b*x)^3,x]

[Out]

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

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 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2210

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Simp[(F^a*ExpIntegralEi[
b*(c + d*x)^n*Log[F]])/(f*n), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]

Rule 2222

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[d/f, Int[F^(a + b/(c + d
*x))/(c + d*x), x], x] - Dist[(d*e - c*f)/f, Int[F^(a + b/(c + d*x))/((c + d*x)*(e + f*x)), x], x] /; FreeQ[{F
, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0]

Rule 2223

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))*((e_.) + (f_.)*(x_))^(m_), x_Symbol] :> Simp[((e + f*x)^(m + 1)*
F^(a + b/(c + d*x)))/(f*(m + 1)), x] + Dist[(b*d*Log[F])/(f*(m + 1)), Int[((e + f*x)^(m + 1)*F^(a + b/(c + d*x
)))/(c + d*x)^2, x], x] /; FreeQ[{F, a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && ILtQ[m, -1]

Rule 2228

Int[(F_)^((a_.) + (b_.)/((c_.) + (d_.)*(x_)))/(((e_.) + (f_.)*(x_))*((g_.) + (h_.)*(x_))), x_Symbol] :> -Dist[
d/(f*(d*g - c*h)), Subst[Int[F^(a - (b*h)/(d*g - c*h) + (d*b*x)/(d*g - c*h))/x, x], x, (g + h*x)/(c + d*x)], x
] /; FreeQ[{F, a, b, c, d, e, f}, x] && EqQ[d*e - c*f, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [F]  time = 0.48, size = 0, normalized size = 0.00 \[ \int \frac {e^{\frac {e}{c+d x}}}{(a+b x)^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[E^(e/(c + d*x))/(a + b*x)^3,x]

[Out]

Integrate[E^(e/(c + d*x))/(a + b*x)^3, x]

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fricas [B]  time = 0.42, size = 517, normalized size = 2.15 \[ \frac {{\left (a^{2} b d^{2} e^{2} + {\left (b^{3} d^{2} e^{2} + 2 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} + 2 \, {\left (a^{2} b c d^{2} - a^{3} d^{3}\right )} e + 2 \, {\left (a b^{2} d^{2} e^{2} + 2 \, {\left (a b^{2} c d^{2} - a^{2} b d^{3}\right )} e\right )} x\right )} {\rm Ei}\left (-\frac {b d e x + a d e}{b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x}\right ) e^{\left (\frac {b e}{b c - a d}\right )} - {\left (b^{3} c^{4} - 4 \, a b^{2} c^{3} d + 5 \, a^{2} b c^{2} d^{2} - 2 \, a^{3} c d^{3} - {\left (b^{3} c^{2} d^{2} - 2 \, a b^{2} c d^{3} + a^{2} b d^{4} + {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e\right )} x^{2} - {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} e - {\left (2 \, a b^{2} c^{2} d^{2} - 4 \, a^{2} b c d^{3} + 2 \, a^{3} d^{4} + {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} e\right )} x\right )} e^{\left (\frac {e}{d x + c}\right )}}{2 \, {\left (a^{2} b^{4} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + a^{6} d^{4} + {\left (b^{6} c^{4} - 4 \, a b^{5} c^{3} d + 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{3} b^{3} c d^{3} + a^{4} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b^{5} c^{4} - 4 \, a^{2} b^{4} c^{3} d + 6 \, a^{3} b^{3} c^{2} d^{2} - 4 \, a^{4} b^{2} c d^{3} + a^{5} b d^{4}\right )} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.53, size = 1759, normalized size = 7.33 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="giac")

[Out]

1/2*(2*b^3*c*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4) - 4*b^3*c^2*
d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c) + 2*b^3*c^3*d*E
i(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 - 2*a*b^2*d^2*Ei
(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4) + 8*a*b^2*c*d^2*Ei(-(b*e - b*
c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c) - 6*a*b^2*c^2*d^2*Ei(-(b*e - b
*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 - 4*a^2*b*d^3*Ei(-(b*e - b*
c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c) + 6*a^2*b*c*d^3*Ei(-(b*e - b*c
*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 - 2*a^3*d^4*Ei(-(b*e - b*c*e/
(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 4)/(d*x + c)^2 + b^3*c^2*d*e^(e/(d*x + c) + 3)
- 2*b^3*c^3*d*e^(e/(d*x + c) + 3)/(d*x + c) - 2*a*b^2*c*d^2*e^(e/(d*x + c) + 3) + 6*a*b^2*c^2*d^2*e^(e/(d*x +
c) + 3)/(d*x + c) + a^2*b*d^3*e^(e/(d*x + c) + 3) - 6*a^2*b*c*d^3*e^(e/(d*x + c) + 3)/(d*x + c) + 2*a^3*d^4*e^
(e/(d*x + c) + 3)/(d*x + c) + b^3*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a
*d) + 5) - 2*b^3*c*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x +
c) + b^3*c^2*d*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c)^2
+ 2*a*b^2*d^2*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c) -
2*a*b^2*c*d^2*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c)^2 +
a^2*b*d^3*Ei(-(b*e - b*c*e/(d*x + c) + a*d*e/(d*x + c))/(b*c - a*d))*e^(b*e/(b*c - a*d) + 5)/(d*x + c)^2 + b^
3*c*d*e^(e/(d*x + c) + 4) - b^3*c^2*d*e^(e/(d*x + c) + 4)/(d*x + c) - a*b^2*d^2*e^(e/(d*x + c) + 4) + 2*a*b^2*
c*d^2*e^(e/(d*x + c) + 4)/(d*x + c) - a^2*b*d^3*e^(e/(d*x + c) + 4)/(d*x + c))*d*e^(-1)/(b^6*c^4*e^2 - 2*b^6*c
^5*e^2/(d*x + c) + b^6*c^6*e^2/(d*x + c)^2 - 4*a*b^5*c^3*d*e^2 + 10*a*b^5*c^4*d*e^2/(d*x + c) - 6*a*b^5*c^5*d*
e^2/(d*x + c)^2 + 6*a^2*b^4*c^2*d^2*e^2 - 20*a^2*b^4*c^3*d^2*e^2/(d*x + c) + 15*a^2*b^4*c^4*d^2*e^2/(d*x + c)^
2 - 4*a^3*b^3*c*d^3*e^2 + 20*a^3*b^3*c^2*d^3*e^2/(d*x + c) - 20*a^3*b^3*c^3*d^3*e^2/(d*x + c)^2 + a^4*b^2*d^4*
e^2 - 10*a^4*b^2*c*d^4*e^2/(d*x + c) + 15*a^4*b^2*c^2*d^4*e^2/(d*x + c)^2 + 2*a^5*b*d^5*e^2/(d*x + c) - 6*a^5*
b*c*d^5*e^2/(d*x + c)^2 + a^6*d^6*e^2/(d*x + c)^2)

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maple [A]  time = 0.02, size = 240, normalized size = 1.00 \[ -\frac {\left (-\frac {\left (-\frac {\Ei \left (1, -\frac {b e}{a d -b c}-\frac {e}{d x +c}\right ) {\mathrm e}^{-\frac {b e}{a d -b c}}}{2}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {b e}{a d -b c}+\frac {e}{d x +c}\right )^{2}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{2 \left (\frac {b e}{a d -b c}+\frac {e}{d x +c}\right )}\right ) b \,d^{3} e}{\left (a d -b c \right )^{4}}+\frac {\left (-\Ei \left (1, -\frac {b e}{a d -b c}-\frac {e}{d x +c}\right ) {\mathrm e}^{-\frac {b e}{a d -b c}}-\frac {{\mathrm e}^{\frac {e}{d x +c}}}{\frac {b e}{a d -b c}+\frac {e}{d x +c}}\right ) d^{3}}{\left (a d -b c \right )^{3}}\right ) e}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1/(d*x+c)*e)/(b*x+a)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(e/(d*x+c))/(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(e^(e/(d*x + c))/(b*x + a)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{\frac {e}{c+d\,x}}}{{\left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(e/(c + d*x))/(a + b*x)^3,x)

[Out]

int(exp(e/(c + d*x))/(a + b*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(e/(c + d*x))/(a + b*x)**3, x)

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