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]

(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)

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Rubi [A]  time = 1.03656, antiderivative size = 240, normalized size of antiderivative = 1., 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 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 6742

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

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 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 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 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]

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*}

Mathematica [F]  time = 0.401324, size = 0, normalized size = 0. \[ \int \frac{e^{\frac{e}{c+d x}}}{(a+b x)^3} \, dx \]

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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Maple [A]  time = 0.01, size = 240, normalized size = 1. \begin{align*} -{\frac{e}{d} \left ( -{\frac{be{d}^{3}}{ \left ( ad-bc \right ) ^{4}} \left ( -{\frac{1}{2}{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-bc}} \right ) ^{-2}}-{\frac{1}{2}{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-bc}} \right ) ^{-1}}-{\frac{1}{2}{{\rm e}^{-{\frac{be}{ad-bc}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-bc}} \right ) } \right ) }+{\frac{{d}^{3}}{ \left ( ad-bc \right ) ^{3}} \left ( -{{{\rm e}^{{\frac{e}{dx+c}}}} \left ({\frac{e}{dx+c}}+{\frac{be}{ad-bc}} \right ) ^{-1}}-{{\rm e}^{-{\frac{be}{ad-bc}}}}{\it Ei} \left ( 1,-{\frac{e}{dx+c}}-{\frac{be}{ad-bc}} \right ) \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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Fricas [B]  time = 1.6449, size = 1007, normalized size = 4.2 \begin{align*} \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 )}} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\frac{e}{c + d x}}}{\left (a + b x\right )^{3}}\, dx \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\frac{e}{d x + c}\right )}}{{\left (b x + a\right )}^{3}}\,{d x} \end{align*}

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]

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