3.456 \(\int \frac{g+h x}{(a+b \log (c (d (e+f x)^p)^q))^3} \, dx\)

Optimal. Leaf size=322 \[ \frac{(e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^2 p^3 q^3}+\frac{2 h (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Ei}\left (\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^2 p^3 q^3}+\frac{(e+f x) (f g-e h)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \]

[Out]

((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(2*b^3*E^(a/(b*p*q))*f^2*p^3*q
^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p
*q)])/(b^3*E^((2*a)/(b*p*q))*f^2*p^3*q^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - ((e + f*x)*(g + h*x))/(2*b*f*p*q*(
a + b*Log[c*(d*(e + f*x)^p)^q])^2) + ((f*g - e*h)*(e + f*x))/(2*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q
])) - ((e + f*x)*(g + h*x))/(b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))

________________________________________________________________________________________

Rubi [A]  time = 0.925535, antiderivative size = 322, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {2400, 2399, 2389, 2300, 2178, 2390, 2310, 2297, 2445} \[ \frac{(e+f x) e^{-\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^2 p^3 q^3}+\frac{2 h (e+f x)^2 e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Ei}\left (\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^2 p^3 q^3}+\frac{(e+f x) (f g-e h)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)/(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

((f*g - e*h)*(e + f*x)*ExpIntegralEi[(a + b*Log[c*(d*(e + f*x)^p)^q])/(b*p*q)])/(2*b^3*E^(a/(b*p*q))*f^2*p^3*q
^3*(c*(d*(e + f*x)^p)^q)^(1/(p*q))) + (2*h*(e + f*x)^2*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(b*p
*q)])/(b^3*E^((2*a)/(b*p*q))*f^2*p^3*q^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))) - ((e + f*x)*(g + h*x))/(2*b*f*p*q*(
a + b*Log[c*(d*(e + f*x)^p)^q])^2) + ((f*g - e*h)*(e + f*x))/(2*b^2*f^2*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q
])) - ((e + f*x)*(g + h*x))/(b^2*f*p^2*q^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))

Rule 2400

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((
d + e*x)*(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1))/(b*e*n*(p + 1)), x] + (-Dist[(q + 1)/(b*n*(p + 1)), I
nt[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Dist[(q*(e*f - d*g))/(b*e*n*(p + 1)), Int[(f + g*x
)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && LtQ[p, -1] && GtQ[q, 0]

Rule 2399

Int[((f_.) + (g_.)*(x_))^(q_.)/((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.)), x_Symbol] :> Int[ExpandIn
tegrand[(f + g*x)^q/(a + b*Log[c*(d + e*x)^n]), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g,
0] && IGtQ[q, 0]

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 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 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, 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 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{g+h x}{\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3} \, dx &=\operatorname{Subst}\left (\int \frac{g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\operatorname{Subst}\left (\frac{\int \frac{g+h x}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2} \, dx}{b p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(f g-e h) \int \frac{1}{\left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^2} \, dx}{2 b f p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{2 \int \frac{g+h x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(f g-e h) \int \frac{1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(f g-e h) \operatorname{Subst}\left (\int \frac{1}{\left (a+b \log \left (c d^q x^{p q}\right )\right )^2} \, dx,x,e+f x\right )}{2 b f^2 p q},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac{(f g-e h) (e+f x)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{2 \int \left (\frac{f g-e h}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}+\frac{h (e+f x)}{f \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}\right ) \, dx}{b^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(f g-e h) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{2 b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(f g-e h) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac{(f g-e h) (e+f x)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{(2 h) \int \frac{e+f x}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 (f g-e h)) \int \frac{1}{a+b \log \left (c d^q (e+f x)^{p q}\right )} \, dx}{b^2 f p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left ((f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{2 b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left ((f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 e^{-\frac{a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^2 p^3 q^3}-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac{(f g-e h) (e+f x)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{(2 h) \operatorname{Subst}\left (\int \frac{x}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 (f g-e h)) \operatorname{Subst}\left (\int \frac{1}{a+b \log \left (c d^q x^{p q}\right )} \, dx,x,e+f x\right )}{b^2 f^2 p^2 q^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 e^{-\frac{a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^2 p^3 q^3}-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac{(f g-e h) (e+f x)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}+\operatorname{Subst}\left (\frac{\left (2 h (e+f x)^2 \left (c d^q (e+f x)^{p q}\right )^{-\frac{2}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{2 x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (2 (f g-e h) (e+f x) \left (c d^q (e+f x)^{p q}\right )^{-\frac{1}{p q}}\right ) \operatorname{Subst}\left (\int \frac{e^{\frac{x}{p q}}}{a+b x} \, dx,x,\log \left (c d^q (e+f x)^{p q}\right )\right )}{b^2 f^2 p^3 q^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{e^{-\frac{a}{b p q}} (f g-e h) (e+f x) \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{1}{p q}} \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )}{2 b^3 f^2 p^3 q^3}+\frac{2 e^{-\frac{2 a}{b p q}} h (e+f x)^2 \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \text{Ei}\left (\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )}{b^3 f^2 p^3 q^3}-\frac{(e+f x) (g+h x)}{2 b f p q \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}+\frac{(f g-e h) (e+f x)}{2 b^2 f^2 p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}-\frac{(e+f x) (g+h x)}{b^2 f p^2 q^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.735854, size = 322, normalized size = 1. \[ -\frac{(e+f x) e^{-\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{-\frac{2}{p q}} \left (-e^{\frac{a}{b p q}} (f g-e h) \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{1}{p q}} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Ei}\left (\frac{a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{b p q}\right )-4 h (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \text{Ei}\left (\frac{2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{b p q}\right )+b p q e^{\frac{2 a}{b p q}} \left (c \left (d (e+f x)^p\right )^q\right )^{\frac{2}{p q}} \left (a (e h+f g+2 f h x)+b (e h+f (g+2 h x)) \log \left (c \left (d (e+f x)^p\right )^q\right )+b f p q (g+h x)\right )\right )}{2 b^3 f^2 p^3 q^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)/(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

-((e + f*x)*(-(E^(a/(b*p*q))*(f*g - e*h)*(c*(d*(e + f*x)^p)^q)^(1/(p*q))*ExpIntegralEi[(a + b*Log[c*(d*(e + f*
x)^p)^q])/(b*p*q)]*(a + b*Log[c*(d*(e + f*x)^p)^q])^2) - 4*h*(e + f*x)*ExpIntegralEi[(2*(a + b*Log[c*(d*(e + f
*x)^p)^q]))/(b*p*q)]*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + b*E^((2*a)/(b*p*q))*p*q*(c*(d*(e + f*x)^p)^q)^(2/(p*
q))*(b*f*p*q*(g + h*x) + a*(f*g + e*h + 2*f*h*x) + b*(e*h + f*(g + 2*h*x))*Log[c*(d*(e + f*x)^p)^q])))/(2*b^3*
E^((2*a)/(b*p*q))*f^2*p^3*q^3*(c*(d*(e + f*x)^p)^q)^(2/(p*q))*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)

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Maple [F]  time = 0.267, size = 0, normalized size = 0. \begin{align*} \int{\frac{hx+g}{ \left ( a+b\ln \left ( c \left ( d \left ( fx+e \right ) ^{p} \right ) ^{q} \right ) \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((h*x+g)/(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

[Out]

int((h*x+g)/(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left (2 \, a f^{2} h +{\left (f^{2} h p q + 2 \, f^{2} h \log \left (c\right ) + 2 \, f^{2} h \log \left (d^{q}\right )\right )} b\right )} x^{2} +{\left (e f g + e^{2} h\right )} a +{\left (e f g p q +{\left (e f g + e^{2} h\right )} \log \left (c\right ) +{\left (e f g + e^{2} h\right )} \log \left (d^{q}\right )\right )} b +{\left ({\left (f^{2} g + 3 \, e f h\right )} a +{\left (f^{2} g p q + e f h p q +{\left (f^{2} g + 3 \, e f h\right )} \log \left (c\right ) +{\left (f^{2} g + 3 \, e f h\right )} \log \left (d^{q}\right )\right )} b\right )} x +{\left (2 \, b f^{2} h x^{2} +{\left (f^{2} g + 3 \, e f h\right )} b x +{\left (e f g + e^{2} h\right )} b\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )}{2 \,{\left (b^{4} f^{2} p^{2} q^{2} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )^{2} + a^{2} b^{2} f^{2} p^{2} q^{2} + 2 \,{\left (f^{2} p^{2} q^{2} \log \left (c\right ) + f^{2} p^{2} q^{2} \log \left (d^{q}\right )\right )} a b^{3} +{\left (f^{2} p^{2} q^{2} \log \left (c\right )^{2} + 2 \, f^{2} p^{2} q^{2} \log \left (c\right ) \log \left (d^{q}\right ) + f^{2} p^{2} q^{2} \log \left (d^{q}\right )^{2}\right )} b^{4} + 2 \,{\left (a b^{3} f^{2} p^{2} q^{2} +{\left (f^{2} p^{2} q^{2} \log \left (c\right ) + f^{2} p^{2} q^{2} \log \left (d^{q}\right )\right )} b^{4}\right )} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right )\right )}} + \int \frac{4 \, f h x + f g + 3 \, e h}{2 \,{\left (b^{3} f p^{2} q^{2} \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + a b^{2} f p^{2} q^{2} +{\left (f p^{2} q^{2} \log \left (c\right ) + f p^{2} q^{2} \log \left (d^{q}\right )\right )} b^{3}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)/(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="maxima")

[Out]

-1/2*((2*a*f^2*h + (f^2*h*p*q + 2*f^2*h*log(c) + 2*f^2*h*log(d^q))*b)*x^2 + (e*f*g + e^2*h)*a + (e*f*g*p*q + (
e*f*g + e^2*h)*log(c) + (e*f*g + e^2*h)*log(d^q))*b + ((f^2*g + 3*e*f*h)*a + (f^2*g*p*q + e*f*h*p*q + (f^2*g +
3*e*f*h)*log(c) + (f^2*g + 3*e*f*h)*log(d^q))*b)*x + (2*b*f^2*h*x^2 + (f^2*g + 3*e*f*h)*b*x + (e*f*g + e^2*h)
*b)*log(((f*x + e)^p)^q))/(b^4*f^2*p^2*q^2*log(((f*x + e)^p)^q)^2 + a^2*b^2*f^2*p^2*q^2 + 2*(f^2*p^2*q^2*log(c
) + f^2*p^2*q^2*log(d^q))*a*b^3 + (f^2*p^2*q^2*log(c)^2 + 2*f^2*p^2*q^2*log(c)*log(d^q) + f^2*p^2*q^2*log(d^q)
^2)*b^4 + 2*(a*b^3*f^2*p^2*q^2 + (f^2*p^2*q^2*log(c) + f^2*p^2*q^2*log(d^q))*b^4)*log(((f*x + e)^p)^q)) + inte
grate(1/2*(4*f*h*x + f*g + 3*e*h)/(b^3*f*p^2*q^2*log(((f*x + e)^p)^q) + a*b^2*f*p^2*q^2 + (f*p^2*q^2*log(c) +
f*p^2*q^2*log(d^q))*b^3), x)

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Fricas [B]  time = 2.19165, size = 2130, normalized size = 6.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)/(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="fricas")

[Out]

1/2*(((b^2*f*g - b^2*e*h)*p^2*q^2*log(f*x + e)^2 + (b^2*f*g - b^2*e*h)*q^2*log(d)^2 + a^2*f*g - a^2*e*h + (b^2
*f*g - b^2*e*h)*log(c)^2 + 2*((b^2*f*g - b^2*e*h)*p*q^2*log(d) + (b^2*f*g - b^2*e*h)*p*q*log(c) + (a*b*f*g - a
*b*e*h)*p*q)*log(f*x + e) + 2*(a*b*f*g - a*b*e*h)*log(c) + 2*((b^2*f*g - b^2*e*h)*q*log(c) + (a*b*f*g - a*b*e*
h)*q)*log(d))*e^((b*q*log(d) + b*log(c) + a)/(b*p*q))*log_integral((f*x + e)*e^((b*q*log(d) + b*log(c) + a)/(b
*p*q))) - (b^2*e*f*g*p^2*q^2 + (a*b*e*f*g + a*b*e^2*h)*p*q + (b^2*f^2*h*p^2*q^2 + 2*a*b*f^2*h*p*q)*x^2 + ((b^2
*f^2*g + b^2*e*f*h)*p^2*q^2 + (a*b*f^2*g + 3*a*b*e*f*h)*p*q)*x + (2*b^2*f^2*h*p^2*q^2*x^2 + (b^2*f^2*g + 3*b^2
*e*f*h)*p^2*q^2*x + (b^2*e*f*g + b^2*e^2*h)*p^2*q^2)*log(f*x + e) + (2*b^2*f^2*h*p*q*x^2 + (b^2*f^2*g + 3*b^2*
e*f*h)*p*q*x + (b^2*e*f*g + b^2*e^2*h)*p*q)*log(c) + (2*b^2*f^2*h*p*q^2*x^2 + (b^2*f^2*g + 3*b^2*e*f*h)*p*q^2*
x + (b^2*e*f*g + b^2*e^2*h)*p*q^2)*log(d))*e^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q)) + 4*(b^2*h*p^2*q^2*log(f*
x + e)^2 + b^2*h*q^2*log(d)^2 + b^2*h*log(c)^2 + 2*a*b*h*log(c) + a^2*h + 2*(b^2*h*p*q^2*log(d) + b^2*h*p*q*lo
g(c) + a*b*h*p*q)*log(f*x + e) + 2*(b^2*h*q*log(c) + a*b*h*q)*log(d))*log_integral((f^2*x^2 + 2*e*f*x + e^2)*e
^(2*(b*q*log(d) + b*log(c) + a)/(b*p*q))))*e^(-2*(b*q*log(d) + b*log(c) + a)/(b*p*q))/(b^5*f^2*p^5*q^5*log(f*x
+ e)^2 + b^5*f^2*p^3*q^5*log(d)^2 + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q
^3 + 2*(b^5*f^2*p^4*q^5*log(d) + b^5*f^2*p^4*q^4*log(c) + a*b^4*f^2*p^4*q^4)*log(f*x + e) + 2*(b^5*f^2*p^3*q^4
*log(c) + a*b^4*f^2*p^3*q^4)*log(d))

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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((h*x+g)/(a+b*ln(c*(d*(f*x+e)**p)**q))**3,x)

[Out]

Timed out

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Giac [B]  time = 2.15156, size = 15570, normalized size = 48.35 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((h*x+g)/(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="giac")

[Out]

-1/2*(f*x + e)*b^2*f*g*p^2*q^2*log(f*x + e)/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*l
og(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) +
2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*
log(c) + a^2*b^3*f^2*p^3*q^3) - (f*x + e)^2*b^2*h*p^2*q^2*log(f*x + e)/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5
*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*
f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*lo
g(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) + 1/2*(f*x + e)*b^2*h*p^2*q^2*e*log(f*x + e)/(b^5*f^2
*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*
f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*lo
g(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) + 1/2*b^2*f*g*p^2*q^2*
Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)^2/((b^5*f^2*p^5*q^5*log(f*x
+ e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(
d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4
*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) - 1/2*(f*x + e)*b
^2*f*g*p^2*q^2/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log
(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(
d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)
- 1/2*(f*x + e)^2*b^2*h*p^2*q^2/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b
^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p
^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^
2*b^3*f^2*p^3*q^3) + 1/2*(f*x + e)*b^2*h*p^2*q^2*e/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x
+ e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x
+ e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p
^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) - 1/2*b^2*h*p^2*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e)
)*e^(-a/(b*p*q) + 1)*log(f*x + e)^2/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) +
2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f
^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c)
+ a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) + 2*b^2*h*p^2*q^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2
*log(f*x + e))*e^(-2*a/(b*p*q))*log(f*x + e)^2/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x +
e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e
) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*
q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(2/(p*q))*d^(2/p)) - 1/2*(f*x + e)*b^2*f*g*p*q^2*log(d)/(b^5*f^2*p^5*q^5*l
og(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^
5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2
*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) - (f*x + e)^2*b^2*h*p*q^2*log(d)
/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(
c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p
^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) + 1/2*(f*x +
e)*b^2*h*p*q^2*e*log(d)/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^
4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*lo
g(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2
*p^3*q^3) + b^2*f*g*p*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)*l
og(d)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e
)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5
*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p
*q))*d^(1/p)) - 1/2*(f*x + e)*b^2*f*g*p*q*log(c)/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x +
e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x +
e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3
*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) - (f*x + e)^2*b^2*h*p*q*log(c)/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*
p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p
^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d)
+ 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) + 1/2*(f*x + e)*b^2*h*p*q*e*log(c)/(b^5*f^2*p^5*q^5*log(f*
x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log
(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^
4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) + b^2*f*g*p*q*Ei(log(d)/p + log(c)/(p
*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x + e)*log(c)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^
2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2
*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d
) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) - b^2*h*p*q^2*Ei(log(d)/p + log(c)/
(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(f*x + e)*log(d)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*
b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b
^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4
*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) + 4*b^2*h*p*q^2*Ei(2*log(d)/p
+ 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(f*x + e)*log(d)/((b^5*f^2*p^5*q^5*log(f
*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*lo
g(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b
^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(2/(p*q))*d^(2/p)) + 1/2*b^2*f*g*q
^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(d)^2/((b^5*f^2*p^5*q^5*log(f*x +
e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^
2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^
2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) - 1/2*(f*x + e)*a*b*
f*g*p*q/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x +
e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^
5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3) - (f*x
+ e)^2*a*b*h*p*q/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*
log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*l
og(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q
^3) + 1/2*(f*x + e)*a*b*h*p*q*e/(b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^
5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^
3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2
*b^3*f^2*p^3*q^3) + a*b*f*g*p*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(f*x
+ e)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)
*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*
f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*
q))*d^(1/p)) - b^2*h*p*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(f*x + e
)*log(c)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x
+ e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) +
b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1
/(p*q))*d^(1/p)) + 4*b^2*h*p*q*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))
*log(f*x + e)*log(c)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*
q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(
c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p
^3*q^3)*c^(2/(p*q))*d^(2/p)) + b^2*f*g*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))
*log(c)*log(d)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*lo
g(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log
(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3
)*c^(1/(p*q))*d^(1/p)) - 1/2*b^2*h*q^2*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) +
1)*log(d)^2/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f
*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d)
+ b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c
^(1/(p*q))*d^(1/p)) + 2*b^2*h*q^2*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*
q))*log(d)^2/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(
f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d
) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*
c^(2/(p*q))*d^(2/p)) - a*b*h*p*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log
(f*x + e)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x
+ e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) +
b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(
1/(p*q))*d^(1/p)) + 4*a*b*h*p*q*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q)
)*log(f*x + e)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*lo
g(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log
(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3
)*c^(2/(p*q))*d^(2/p)) + 1/2*b^2*f*g*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log
(c)^2/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e
)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5
*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p
*q))*d^(1/p)) + a*b*f*g*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(d)/((b^5*f
^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^
5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*
log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p))
- b^2*h*q*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(c)*log(d)/((b^5*f^2*p
^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^
2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(
c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) + 4
*b^2*h*q*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(c)*log(d)/((b^5*f
^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^
5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*
log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(2/(p*q))*d^(2/p))
+ a*b*f*g*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))*log(c)/((b^5*f^2*p^5*q^5*log(
f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*l
og(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*
b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) - 1/2*b^2*h*Ei
(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(c)^2/((b^5*f^2*p^5*q^5*log(f*x + e
)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2
+ 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2
*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) + 2*b^2*h*Ei(2*log(d)
/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(c)^2/((b^5*f^2*p^5*q^5*log(f*x + e)^2
+ 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 +
2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^
3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(2/(p*q))*d^(2/p)) - a*b*h*q*Ei(log(d)/p +
log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q) + 1)*log(d)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^
2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2
*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d
) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) + 4*a*b*h*q*Ei(2*log(d)/p + 2*log(c
)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))*log(d)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p
^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^
4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) +
2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(2/(p*q))*d^(2/p)) + 1/2*a^2*f*g*Ei(log(d)/p + log(c)/(p*
q) + a/(b*p*q) + log(f*x + e))*e^(-a/(b*p*q))/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e
)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e)
+ 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q
^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) - a*b*h*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x
+ e))*e^(-a/(b*p*q) + 1)*log(c)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*
b^5*f^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*
p^3*q^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a
^2*b^3*f^2*p^3*q^3)*c^(1/(p*q))*d^(1/p)) + 4*a*b*h*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x +
e))*e^(-2*a/(b*p*q))*log(c)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f
^2*p^4*q^4*log(f*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q
^4*log(c)*log(d) + b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^
3*f^2*p^3*q^3)*c^(2/(p*q))*d^(2/p)) - 1/2*a^2*h*Ei(log(d)/p + log(c)/(p*q) + a/(b*p*q) + log(f*x + e))*e^(-a/(
b*p*q) + 1)/((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f
*x + e)*log(c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d)
+ b^5*f^2*p^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c
^(1/(p*q))*d^(1/p)) + 2*a^2*h*Ei(2*log(d)/p + 2*log(c)/(p*q) + 2*a/(b*p*q) + 2*log(f*x + e))*e^(-2*a/(b*p*q))/
((b^5*f^2*p^5*q^5*log(f*x + e)^2 + 2*b^5*f^2*p^4*q^5*log(f*x + e)*log(d) + 2*b^5*f^2*p^4*q^4*log(f*x + e)*log(
c) + b^5*f^2*p^3*q^5*log(d)^2 + 2*a*b^4*f^2*p^4*q^4*log(f*x + e) + 2*b^5*f^2*p^3*q^4*log(c)*log(d) + b^5*f^2*p
^3*q^3*log(c)^2 + 2*a*b^4*f^2*p^3*q^4*log(d) + 2*a*b^4*f^2*p^3*q^3*log(c) + a^2*b^3*f^2*p^3*q^3)*c^(2/(p*q))*d
^(2/p))