3.410 \(\int \frac{1}{(a+b \log (c (d (e+f x)^m)^n))^3} \, dx\)
Optimal. Leaf size=169 \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{2 b^3 f m^3 n^3}-\frac{e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}-\frac{e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \]
[Out]
((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)])/(2*b^3*E^(a/(b*m*n))*f*m^3*n^3*(c*(d*(e +
f*x)^m)^n)^(1/(m*n))) - (e + f*x)/(2*b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n])^2) - (e + f*x)/(2*b^2*f*m^2*n^2*
(a + b*Log[c*(d*(e + f*x)^m)^n]))
________________________________________________________________________________________
Rubi [A] time = 0.222723, antiderivative size = 169, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{2389, 2297, 2300, 2178, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{2 b^3 f m^3 n^3}-\frac{e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}-\frac{e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-3),x]
[Out]
((e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)])/(2*b^3*E^(a/(b*m*n))*f*m^3*n^3*(c*(d*(e +
f*x)^m)^n)^(1/(m*n))) - (e + f*x)/(2*b*f*m*n*(a + b*Log[c*(d*(e + f*x)^m)^n])^2) - (e + f*x)/(2*b^2*f*m^2*n^2*
(a + b*Log[c*(d*(e + f*x)^m)^n]))
Rule 2389
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]
Rule 2297
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]
Rule 2300
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[E^(x/n)*(a +
b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]
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 2445
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^3} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^3} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}+\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^2} \, dx,x,e+f x\right )}{2 b f m n},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}-\frac{e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}+\operatorname{Subst}\left (\frac{\operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^n x^{m n}\right )} \, dx,x,e+f x\right )}{2 b^2 f m^2 n^2},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=-\frac{e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}-\frac{e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}+\operatorname{Subst}\left (\frac{\left ((e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac{1}{m n}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{m n}}}{a+b x} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{2 b^2 f m^3 n^3},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right )\\ &=\frac{e^{-\frac{a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )}{2 b^3 f m^3 n^3}-\frac{e+f x}{2 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2}-\frac{e+f x}{2 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.180611, size = 189, normalized size = 1.12 \[ -\frac{(e+f x) e^{-\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac{1}{m n}} \left (b m n e^{\frac{a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac{1}{m n}} \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )+b m n\right )-\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2 \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right )\right )}{2 b^3 f m^3 n^3 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-3),x]
[Out]
-((e + f*x)*(-(ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n)]*(a + b*Log[c*(d*(e + f*x)^m)^n])^2) + b
*E^(a/(b*m*n))*m*n*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(a + b*m*n + b*Log[c*(d*(e + f*x)^m)^n])))/(2*b^3*E^(a/(b*m
*n))*f*m^3*n^3*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(a + b*Log[c*(d*(e + f*x)^m)^n])^2)
________________________________________________________________________________________
Maple [F] time = 0.083, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{m} \right ) ^{n} \right ) \right ) ^{-3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^3,x)
[Out]
int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^3,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (e m n + e \log \left (c\right ) + e \log \left (d^{n}\right )\right )} b + a e +{\left ({\left (f m n + f \log \left (c\right ) + f \log \left (d^{n}\right )\right )} b + a f\right )} x +{\left (b f x + b e\right )} \log \left ({\left ({\left (f x + e\right )}^{m}\right )}^{n}\right )}{2 \,{\left (b^{4} f m^{2} n^{2} \log \left ({\left ({\left (f x + e\right )}^{m}\right )}^{n}\right )^{2} + a^{2} b^{2} f m^{2} n^{2} + 2 \,{\left (f m^{2} n^{2} \log \left (c\right ) + f m^{2} n^{2} \log \left (d^{n}\right )\right )} a b^{3} +{\left (f m^{2} n^{2} \log \left (c\right )^{2} + 2 \, f m^{2} n^{2} \log \left (c\right ) \log \left (d^{n}\right ) + f m^{2} n^{2} \log \left (d^{n}\right )^{2}\right )} b^{4} + 2 \,{\left (a b^{3} f m^{2} n^{2} +{\left (f m^{2} n^{2} \log \left (c\right ) + f m^{2} n^{2} \log \left (d^{n}\right )\right )} b^{4}\right )} \log \left ({\left ({\left (f x + e\right )}^{m}\right )}^{n}\right )\right )}} + \int \frac{1}{2 \,{\left (b^{3} m^{2} n^{2} \log \left ({\left ({\left (f x + e\right )}^{m}\right )}^{n}\right ) + a b^{2} m^{2} n^{2} +{\left (m^{2} n^{2} \log \left (c\right ) + m^{2} n^{2} \log \left (d^{n}\right )\right )} b^{3}\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="maxima")
[Out]
-1/2*((e*m*n + e*log(c) + e*log(d^n))*b + a*e + ((f*m*n + f*log(c) + f*log(d^n))*b + a*f)*x + (b*f*x + b*e)*lo
g(((f*x + e)^m)^n))/(b^4*f*m^2*n^2*log(((f*x + e)^m)^n)^2 + a^2*b^2*f*m^2*n^2 + 2*(f*m^2*n^2*log(c) + f*m^2*n^
2*log(d^n))*a*b^3 + (f*m^2*n^2*log(c)^2 + 2*f*m^2*n^2*log(c)*log(d^n) + f*m^2*n^2*log(d^n)^2)*b^4 + 2*(a*b^3*f
*m^2*n^2 + (f*m^2*n^2*log(c) + f*m^2*n^2*log(d^n))*b^4)*log(((f*x + e)^m)^n)) + integrate(1/2/(b^3*m^2*n^2*log
(((f*x + e)^m)^n) + a*b^2*m^2*n^2 + (m^2*n^2*log(c) + m^2*n^2*log(d^n))*b^3), x)
________________________________________________________________________________________
Fricas [B] time = 2.27516, size = 1053, normalized size = 6.23 \begin{align*} -\frac{{\left ({\left (b^{2} e m^{2} n^{2} + a b e m n +{\left (b^{2} f m^{2} n^{2} + a b f m n\right )} x +{\left (b^{2} f m^{2} n^{2} x + b^{2} e m^{2} n^{2}\right )} \log \left (f x + e\right ) +{\left (b^{2} f m n x + b^{2} e m n\right )} \log \left (c\right ) +{\left (b^{2} f m n^{2} x + b^{2} e m n^{2}\right )} \log \left (d\right )\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )} -{\left (b^{2} m^{2} n^{2} \log \left (f x + e\right )^{2} + b^{2} n^{2} \log \left (d\right )^{2} + b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right ) + a^{2} + 2 \,{\left (b^{2} m n^{2} \log \left (d\right ) + b^{2} m n \log \left (c\right ) + a b m n\right )} \log \left (f x + e\right ) + 2 \,{\left (b^{2} n \log \left (c\right ) + a b n\right )} \log \left (d\right )\right )} \logintegral \left ({\left (f x + e\right )} e^{\left (\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}\right )\right )} e^{\left (-\frac{b n \log \left (d\right ) + b \log \left (c\right ) + a}{b m n}\right )}}{2 \,{\left (b^{5} f m^{5} n^{5} \log \left (f x + e\right )^{2} + b^{5} f m^{3} n^{5} \log \left (d\right )^{2} + b^{5} f m^{3} n^{3} \log \left (c\right )^{2} + 2 \, a b^{4} f m^{3} n^{3} \log \left (c\right ) + a^{2} b^{3} f m^{3} n^{3} + 2 \,{\left (b^{5} f m^{4} n^{5} \log \left (d\right ) + b^{5} f m^{4} n^{4} \log \left (c\right ) + a b^{4} f m^{4} n^{4}\right )} \log \left (f x + e\right ) + 2 \,{\left (b^{5} f m^{3} n^{4} \log \left (c\right ) + a b^{4} f m^{3} n^{4}\right )} \log \left (d\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="fricas")
[Out]
-1/2*((b^2*e*m^2*n^2 + a*b*e*m*n + (b^2*f*m^2*n^2 + a*b*f*m*n)*x + (b^2*f*m^2*n^2*x + b^2*e*m^2*n^2)*log(f*x +
e) + (b^2*f*m*n*x + b^2*e*m*n)*log(c) + (b^2*f*m*n^2*x + b^2*e*m*n^2)*log(d))*e^((b*n*log(d) + b*log(c) + a)/
(b*m*n)) - (b^2*m^2*n^2*log(f*x + e)^2 + b^2*n^2*log(d)^2 + b^2*log(c)^2 + 2*a*b*log(c) + a^2 + 2*(b^2*m*n^2*l
og(d) + b^2*m*n*log(c) + a*b*m*n)*log(f*x + e) + 2*(b^2*n*log(c) + a*b*n)*log(d))*log_integral((f*x + e)*e^((b
*n*log(d) + b*log(c) + a)/(b*m*n))))*e^(-(b*n*log(d) + b*log(c) + a)/(b*m*n))/(b^5*f*m^5*n^5*log(f*x + e)^2 +
b^5*f*m^3*n^5*log(d)^2 + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3 + 2*(b^5*f*m^4*
n^5*log(d) + b^5*f*m^4*n^4*log(c) + a*b^4*f*m^4*n^4)*log(f*x + e) + 2*(b^5*f*m^3*n^4*log(c) + a*b^4*f*m^3*n^4)
*log(d))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \log{\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*ln(c*(d*(f*x+e)**m)**n))**3,x)
[Out]
Integral((a + b*log(c*(d*(e + f*x)**m)**n))**(-3), x)
________________________________________________________________________________________
Giac [B] time = 1.40324, size = 4699, normalized size = 27.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^3,x, algorithm="giac")
[Out]
-1/2*(f*x + e)*b^2*m^2*n^2*log(f*x + e)/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) +
2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^
4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3
*n^3) + 1/2*b^2*m^2*n^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)^2/(
(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^
5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2
+ 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) - 1/2*(f*x +
e)*b^2*m^2*n^2/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x +
e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m
^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3) - 1/2*(f*x + e)*b^2
*m*n^2*log(d)/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x +
e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^
3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3) + b^2*m*n^2*Ei(log(d
)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)*log(d)/((b^5*f*m^5*n^5*log(f*x + e)
^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*
b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d)
+ 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) - 1/2*(f*x + e)*b^2*m*n*log(c)/(b^5*f*m^
5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n
^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^
4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3) + b^2*m*n*Ei(log(d)/m + log(c)/(m*n) + a/(b
*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)*log(c)/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(
f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e
) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(
c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + 1/2*b^2*n^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x +
e))*e^(-a/(b*m*n))*log(d)^2/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4
*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*lo
g(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/
(m*n))*d^(1/m)) - 1/2*(f*x + e)*a*b*m*n/(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) +
2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^
4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3
*n^3) + a*b*m*n*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(f*x + e)/((b^5*f*m^5
*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^
5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4
*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + b^2*n*Ei(log(d)/m + l
og(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(c)*log(d)/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*
m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4
*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*
m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + 1/2*b^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + lo
g(f*x + e))*e^(-a/(b*m*n))*log(c)^2/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b
^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*l
og(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^
3)*c^(1/(m*n))*d^(1/m)) + a*b*n*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(d)/(
(b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^
5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2
+ 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + a*b*Ei(log(
d)/m + log(c)/(m*n) + a/(b*m*n) + log(f*x + e))*e^(-a/(b*m*n))*log(c)/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f
*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^
4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*log(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f
*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/(m*n))*d^(1/m)) + 1/2*a^2*Ei(log(d)/m + log(c)/(m*n) + a/(b*m*n) + l
og(f*x + e))*e^(-a/(b*m*n))/((b^5*f*m^5*n^5*log(f*x + e)^2 + 2*b^5*f*m^4*n^5*log(f*x + e)*log(d) + 2*b^5*f*m^4
*n^4*log(f*x + e)*log(c) + b^5*f*m^3*n^5*log(d)^2 + 2*a*b^4*f*m^4*n^4*log(f*x + e) + 2*b^5*f*m^3*n^4*log(c)*lo
g(d) + b^5*f*m^3*n^3*log(c)^2 + 2*a*b^4*f*m^3*n^4*log(d) + 2*a*b^4*f*m^3*n^3*log(c) + a^2*b^3*f*m^3*n^3)*c^(1/
(m*n))*d^(1/m))