3.435 \(\int (g+h x)^2 (a+b \log (c (d (e+f x)^p)^q))^3 \, dx\)

Optimal. Leaf size=492 \[ \frac{3 b^2 h p^2 q^2 (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^3}+\frac{2 b^2 h^2 p^2 q^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}+\frac{6 a b^2 p^2 q^2 x (f g-e h)^2}{f^2}-\frac{3 b h p q (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac{3 b p q (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}+\frac{h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{(e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}-\frac{b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac{h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}+\frac{6 b^3 p^2 q^2 (e+f x) (f g-e h)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^3}-\frac{3 b^3 h p^3 q^3 (e+f x)^2 (f g-e h)}{4 f^3}-\frac{6 b^3 p^3 q^3 x (f g-e h)^2}{f^2}-\frac{2 b^3 h^2 p^3 q^3 (e+f x)^3}{27 f^3} \]

[Out]

(6*a*b^2*(f*g - e*h)^2*p^2*q^2*x)/f^2 - (6*b^3*(f*g - e*h)^2*p^3*q^3*x)/f^2 - (3*b^3*h*(f*g - e*h)*p^3*q^3*(e
+ f*x)^2)/(4*f^3) - (2*b^3*h^2*p^3*q^3*(e + f*x)^3)/(27*f^3) + (6*b^3*(f*g - e*h)^2*p^2*q^2*(e + f*x)*Log[c*(d
*(e + f*x)^p)^q])/f^3 + (3*b^2*h*(f*g - e*h)*p^2*q^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^3) + (
2*b^2*h^2*p^2*q^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*f^3) - (3*b*(f*g - e*h)^2*p*q*(e + f*x)*(a
+ b*Log[c*(d*(e + f*x)^p)^q])^2)/f^3 - (3*b*h*(f*g - e*h)*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/
(2*f^3) - (b*h^2*p*q*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*f^3) + ((f*g - e*h)^2*(e + f*x)*(a + b
*Log[c*(d*(e + f*x)^p)^q])^3)/f^3 + (h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f^3 + (h^2*
(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(3*f^3)

________________________________________________________________________________________

Rubi [A]  time = 0.94834, antiderivative size = 492, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2401, 2389, 2296, 2295, 2390, 2305, 2304, 2445} \[ \frac{3 b^2 h p^2 q^2 (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^3}+\frac{2 b^2 h^2 p^2 q^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}+\frac{6 a b^2 p^2 q^2 x (f g-e h)^2}{f^2}-\frac{3 b h p q (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac{3 b p q (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}+\frac{h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{(e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}-\frac{b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac{h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}+\frac{6 b^3 p^2 q^2 (e+f x) (f g-e h)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^3}-\frac{3 b^3 h p^3 q^3 (e+f x)^2 (f g-e h)}{4 f^3}-\frac{6 b^3 p^3 q^3 x (f g-e h)^2}{f^2}-\frac{2 b^3 h^2 p^3 q^3 (e+f x)^3}{27 f^3} \]

Antiderivative was successfully verified.

[In]

Int[(g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

(6*a*b^2*(f*g - e*h)^2*p^2*q^2*x)/f^2 - (6*b^3*(f*g - e*h)^2*p^3*q^3*x)/f^2 - (3*b^3*h*(f*g - e*h)*p^3*q^3*(e
+ f*x)^2)/(4*f^3) - (2*b^3*h^2*p^3*q^3*(e + f*x)^3)/(27*f^3) + (6*b^3*(f*g - e*h)^2*p^2*q^2*(e + f*x)*Log[c*(d
*(e + f*x)^p)^q])/f^3 + (3*b^2*h*(f*g - e*h)*p^2*q^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(2*f^3) + (
2*b^2*h^2*p^2*q^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(9*f^3) - (3*b*(f*g - e*h)^2*p*q*(e + f*x)*(a
+ b*Log[c*(d*(e + f*x)^p)^q])^2)/f^3 - (3*b*h*(f*g - e*h)*p*q*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/
(2*f^3) - (b*h^2*p*q*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(3*f^3) + ((f*g - e*h)^2*(e + f*x)*(a + b
*Log[c*(d*(e + f*x)^p)^q])^3)/f^3 + (h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/f^3 + (h^2*
(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^3)/(3*f^3)

Rule 2401

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Int[Exp
andIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, 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 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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 (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx &=\operatorname{Subst}\left (\int (g+h x)^2 \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 )\\ &=\operatorname{Subst}\left (\int \left (\frac{(f g-e h)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{f^2}+\frac{2 h (f g-e h) (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{f^2}+\frac{h^2 (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3}{f^2}\right ) \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h^2 \int (e+f x)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 h (f g-e h)) \int (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{f^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)^2 \int \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )^3 \, dx}{f^2},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\operatorname{Subst}\left (\frac{h^2 \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{(2 h (f g-e h)) \operatorname{Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f^3},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)^2 \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^3 \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}-\operatorname{Subst}\left (\frac{\left (b h^2 p q\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{(3 b h (f g-e h) p q) \operatorname{Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname{Subst}\left (\frac{\left (3 b (f g-e h)^2 p q\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right )^2 \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=-\frac{3 b (f g-e h)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}-\frac{3 b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac{b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac{(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}+\operatorname{Subst}\left (\frac{\left (2 b^2 h^2 p^2 q^2\right ) \operatorname{Subst}\left (\int x^2 \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{3 f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (3 b^2 h (f g-e h) p^2 q^2\right ) \operatorname{Subst}\left (\int x \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname{Subst}\left (\frac{\left (6 b^2 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \left (a+b \log \left (c d^q x^{p q}\right )\right ) \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{6 a b^2 (f g-e h)^2 p^2 q^2 x}{f^2}-\frac{3 b^3 h (f g-e h) p^3 q^3 (e+f x)^2}{4 f^3}-\frac{2 b^3 h^2 p^3 q^3 (e+f x)^3}{27 f^3}+\frac{3 b^2 h (f g-e h) p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^3}+\frac{2 b^2 h^2 p^2 q^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac{3 b (f g-e h)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}-\frac{3 b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac{b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac{(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}+\operatorname{Subst}\left (\frac{\left (6 b^3 (f g-e h)^2 p^2 q^2\right ) \operatorname{Subst}\left (\int \log \left (c d^q x^{p q}\right ) \, dx,x,e+f x\right )}{f^3},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac{6 a b^2 (f g-e h)^2 p^2 q^2 x}{f^2}-\frac{6 b^3 (f g-e h)^2 p^3 q^3 x}{f^2}-\frac{3 b^3 h (f g-e h) p^3 q^3 (e+f x)^2}{4 f^3}-\frac{2 b^3 h^2 p^3 q^3 (e+f x)^3}{27 f^3}+\frac{6 b^3 (f g-e h)^2 p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^3}+\frac{3 b^2 h (f g-e h) p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^3}+\frac{2 b^2 h^2 p^2 q^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac{3 b (f g-e h)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}-\frac{3 b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac{b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac{(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac{h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}\\ \end{align*}

Mathematica [A]  time = 0.27518, size = 378, normalized size = 0.77 \[ \frac{-4 b h^2 p q \left (2 b p q \left (b f p q x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+9 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2\right )+108 h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+108 (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-324 b p q (f g-e h)^2 \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f x (a-b p q)+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )-81 b h p q (f g-e h) \left (2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )+36 h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{108 f^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(g + h*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^3,x]

[Out]

(108*(f*g - e*h)^2*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^3 + 108*h*(f*g - e*h)*(e + f*x)^2*(a + b*Log[c*(
d*(e + f*x)^p)^q])^3 + 36*h^2*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^3 - 324*b*(f*g - e*h)^2*p*q*((e + f
*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 - 2*b*p*q*(f*(a - b*p*q)*x + b*(e + f*x)*Log[c*(d*(e + f*x)^p)^q])) - 8
1*b*h*(f*g - e*h)*p*q*(2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + b*p*q*(b*f*p*q*x*(2*e + f*x) - 2*(e
+ f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q]))) - 4*b*h^2*p*q*(9*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 +
2*b*p*q*(b*f*p*q*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 3*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q]))))/(108*f^3)

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Maple [F]  time = 0.507, size = 0, normalized size = 0. \begin{align*} \int \left ( hx+g \right ) ^{2} \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)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

[Out]

int((h*x+g)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))^3,x)

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Maxima [B]  time = 1.41119, size = 1681, normalized size = 3.42 \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)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="maxima")

[Out]

1/3*b^3*h^2*x^3*log(((f*x + e)^p*d)^q*c)^3 + a*b^2*h^2*x^3*log(((f*x + e)^p*d)^q*c)^2 + b^3*g*h*x^2*log(((f*x
+ e)^p*d)^q*c)^3 - 3*a^2*b*f*g^2*p*q*(x/f - e*log(f*x + e)/f^2) + 1/6*a^2*b*f*h^2*p*q*(6*e^3*log(f*x + e)/f^4
- (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/f^3) - 3/2*a^2*b*f*g*h*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^2 - 2*e*x)/f^2)
+ a^2*b*h^2*x^3*log(((f*x + e)^p*d)^q*c) + 3*a*b^2*g*h*x^2*log(((f*x + e)^p*d)^q*c)^2 + b^3*g^2*x*log(((f*x +
e)^p*d)^q*c)^3 + 1/3*a^3*h^2*x^3 + 3*a^2*b*g*h*x^2*log(((f*x + e)^p*d)^q*c) + 3*a*b^2*g^2*x*log(((f*x + e)^p*d
)^q*c)^2 + a^3*g*h*x^2 + 3*a^2*b*g^2*x*log(((f*x + e)^p*d)^q*c) - 3*(2*f*p*q*(x/f - e*log(f*x + e)/f^2)*log(((
f*x + e)^p*d)^q*c) + (e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*p^2*q^2/f)*a*b^2*g^2 - (3*f*p*q*(x/f - e*lo
g(f*x + e)/f^2)*log(((f*x + e)^p*d)^q*c)^2 - ((e*log(f*x + e)^3 + 3*e*log(f*x + e)^2 - 6*f*x + 6*e*log(f*x + e
))*p^2*q^2/f^2 - 3*(e*log(f*x + e)^2 - 2*f*x + 2*e*log(f*x + e))*p*q*log(((f*x + e)^p*d)^q*c)/f^2)*f*p*q)*b^3*
g^2 - 3/2*(2*f*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^2 - 2*e*x)/f^2)*log(((f*x + e)^p*d)^q*c) - (f^2*x^2 + 2*e^2*
log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*p^2*q^2/f^2)*a*b^2*g*h - 1/4*(6*f*p*q*(2*e^2*log(f*x + e)/f^3 +
(f*x^2 - 2*e*x)/f^2)*log(((f*x + e)^p*d)^q*c)^2 + ((4*e^2*log(f*x + e)^3 + 3*f^2*x^2 + 18*e^2*log(f*x + e)^2
- 42*e*f*x + 42*e^2*log(f*x + e))*p^2*q^2/f^3 - 6*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x +
e))*p*q*log(((f*x + e)^p*d)^q*c)/f^3)*f*p*q)*b^3*g*h + 1/18*(6*f*p*q*(6*e^3*log(f*x + e)/f^4 - (2*f^2*x^3 - 3*
e*f*x^2 + 6*e^2*x)/f^3)*log(((f*x + e)^p*d)^q*c) + (4*f^3*x^3 - 15*e*f^2*x^2 - 18*e^3*log(f*x + e)^2 + 66*e^2*
f*x - 66*e^3*log(f*x + e))*p^2*q^2/f^3)*a*b^2*h^2 + 1/108*(18*f*p*q*(6*e^3*log(f*x + e)/f^4 - (2*f^2*x^3 - 3*e
*f*x^2 + 6*e^2*x)/f^3)*log(((f*x + e)^p*d)^q*c)^2 - f*p*q*((8*f^3*x^3 - 36*e^3*log(f*x + e)^3 - 57*e*f^2*x^2 -
198*e^3*log(f*x + e)^2 + 510*e^2*f*x - 510*e^3*log(f*x + e))*p^2*q^2/f^4 - 6*(4*f^3*x^3 - 15*e*f^2*x^2 - 18*e
^3*log(f*x + e)^2 + 66*e^2*f*x - 66*e^3*log(f*x + e))*p*q*log(((f*x + e)^p*d)^q*c)/f^4))*b^3*h^2 + a^3*g^2*x

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Fricas [B]  time = 3.12716, size = 6372, normalized size = 12.95 \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)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="fricas")

[Out]

-1/108*(4*(2*b^3*f^3*h^2*p^3*q^3 - 6*a*b^2*f^3*h^2*p^2*q^2 + 9*a^2*b*f^3*h^2*p*q - 9*a^3*f^3*h^2)*x^3 - 36*(b^
3*f^3*h^2*p^3*q^3*x^3 + 3*b^3*f^3*g*h*p^3*q^3*x^2 + 3*b^3*f^3*g^2*p^3*q^3*x + (3*b^3*e*f^2*g^2 - 3*b^3*e^2*f*g
*h + b^3*e^3*h^2)*p^3*q^3)*log(f*x + e)^3 - 36*(b^3*f^3*h^2*x^3 + 3*b^3*f^3*g*h*x^2 + 3*b^3*f^3*g^2*x)*log(c)^
3 - 36*(b^3*f^3*h^2*q^3*x^3 + 3*b^3*f^3*g*h*q^3*x^2 + 3*b^3*f^3*g^2*q^3*x)*log(d)^3 - 3*(36*a^3*f^3*g*h - (27*
b^3*f^3*g*h - 19*b^3*e*f^2*h^2)*p^3*q^3 + 6*(9*a*b^2*f^3*g*h - 5*a*b^2*e*f^2*h^2)*p^2*q^2 - 18*(3*a^2*b*f^3*g*
h - a^2*b*e*f^2*h^2)*p*q)*x^2 + 18*((18*b^3*e*f^2*g^2 - 27*b^3*e^2*f*g*h + 11*b^3*e^3*h^2)*p^3*q^3 - 6*(3*a*b^
2*e*f^2*g^2 - 3*a*b^2*e^2*f*g*h + a*b^2*e^3*h^2)*p^2*q^2 + 2*(b^3*f^3*h^2*p^3*q^3 - 3*a*b^2*f^3*h^2*p^2*q^2)*x
^3 - 3*(6*a*b^2*f^3*g*h*p^2*q^2 - (3*b^3*f^3*g*h - b^3*e*f^2*h^2)*p^3*q^3)*x^2 - 6*(3*a*b^2*f^3*g^2*p^2*q^2 -
(3*b^3*f^3*g^2 - 3*b^3*e*f^2*g*h + b^3*e^2*f*h^2)*p^3*q^3)*x - 6*(b^3*f^3*h^2*p^2*q^2*x^3 + 3*b^3*f^3*g*h*p^2*
q^2*x^2 + 3*b^3*f^3*g^2*p^2*q^2*x + (3*b^3*e*f^2*g^2 - 3*b^3*e^2*f*g*h + b^3*e^3*h^2)*p^2*q^2)*log(c) - 6*(b^3
*f^3*h^2*p^2*q^3*x^3 + 3*b^3*f^3*g*h*p^2*q^3*x^2 + 3*b^3*f^3*g^2*p^2*q^3*x + (3*b^3*e*f^2*g^2 - 3*b^3*e^2*f*g*
h + b^3*e^3*h^2)*p^2*q^3)*log(d))*log(f*x + e)^2 + 18*(2*(b^3*f^3*h^2*p*q - 3*a*b^2*f^3*h^2)*x^3 - 3*(6*a*b^2*
f^3*g*h - (3*b^3*f^3*g*h - b^3*e*f^2*h^2)*p*q)*x^2 - 6*(3*a*b^2*f^3*g^2 - (3*b^3*f^3*g^2 - 3*b^3*e*f^2*g*h + b
^3*e^2*f*h^2)*p*q)*x)*log(c)^2 + 18*(2*(b^3*f^3*h^2*p*q^3 - 3*a*b^2*f^3*h^2*q^2)*x^3 - 3*(6*a*b^2*f^3*g*h*q^2
- (3*b^3*f^3*g*h - b^3*e*f^2*h^2)*p*q^3)*x^2 - 6*(3*a*b^2*f^3*g^2*q^2 - (3*b^3*f^3*g^2 - 3*b^3*e*f^2*g*h + b^3
*e^2*f*h^2)*p*q^3)*x - 6*(b^3*f^3*h^2*q^2*x^3 + 3*b^3*f^3*g*h*q^2*x^2 + 3*b^3*f^3*g^2*q^2*x)*log(c))*log(d)^2
- 6*(18*a^3*f^3*g^2 - (108*b^3*f^3*g^2 - 189*b^3*e*f^2*g*h + 85*b^3*e^2*f*h^2)*p^3*q^3 + 6*(18*a*b^2*f^3*g^2 -
27*a*b^2*e*f^2*g*h + 11*a*b^2*e^2*f*h^2)*p^2*q^2 - 18*(3*a^2*b*f^3*g^2 - 3*a^2*b*e*f^2*g*h + a^2*b*e^2*f*h^2)
*p*q)*x - 6*((108*b^3*e*f^2*g^2 - 189*b^3*e^2*f*g*h + 85*b^3*e^3*h^2)*p^3*q^3 - 6*(18*a*b^2*e*f^2*g^2 - 27*a*b
^2*e^2*f*g*h + 11*a*b^2*e^3*h^2)*p^2*q^2 + 2*(2*b^3*f^3*h^2*p^3*q^3 - 6*a*b^2*f^3*h^2*p^2*q^2 + 9*a^2*b*f^3*h^
2*p*q)*x^3 + 18*(3*a^2*b*e*f^2*g^2 - 3*a^2*b*e^2*f*g*h + a^2*b*e^3*h^2)*p*q + 3*(18*a^2*b*f^3*g*h*p*q + (9*b^3
*f^3*g*h - 5*b^3*e*f^2*h^2)*p^3*q^3 - 6*(3*a*b^2*f^3*g*h - a*b^2*e*f^2*h^2)*p^2*q^2)*x^2 + 18*(b^3*f^3*h^2*p*q
*x^3 + 3*b^3*f^3*g*h*p*q*x^2 + 3*b^3*f^3*g^2*p*q*x + (3*b^3*e*f^2*g^2 - 3*b^3*e^2*f*g*h + b^3*e^3*h^2)*p*q)*lo
g(c)^2 + 18*(b^3*f^3*h^2*p*q^3*x^3 + 3*b^3*f^3*g*h*p*q^3*x^2 + 3*b^3*f^3*g^2*p*q^3*x + (3*b^3*e*f^2*g^2 - 3*b^
3*e^2*f*g*h + b^3*e^3*h^2)*p*q^3)*log(d)^2 + 6*(9*a^2*b*f^3*g^2*p*q + (18*b^3*f^3*g^2 - 27*b^3*e*f^2*g*h + 11*
b^3*e^2*f*h^2)*p^3*q^3 - 6*(3*a*b^2*f^3*g^2 - 3*a*b^2*e*f^2*g*h + a*b^2*e^2*f*h^2)*p^2*q^2)*x - 6*((18*b^3*e*f
^2*g^2 - 27*b^3*e^2*f*g*h + 11*b^3*e^3*h^2)*p^2*q^2 + 2*(b^3*f^3*h^2*p^2*q^2 - 3*a*b^2*f^3*h^2*p*q)*x^3 - 6*(3
*a*b^2*e*f^2*g^2 - 3*a*b^2*e^2*f*g*h + a*b^2*e^3*h^2)*p*q - 3*(6*a*b^2*f^3*g*h*p*q - (3*b^3*f^3*g*h - b^3*e*f^
2*h^2)*p^2*q^2)*x^2 - 6*(3*a*b^2*f^3*g^2*p*q - (3*b^3*f^3*g^2 - 3*b^3*e*f^2*g*h + b^3*e^2*f*h^2)*p^2*q^2)*x)*l
og(c) - 6*((18*b^3*e*f^2*g^2 - 27*b^3*e^2*f*g*h + 11*b^3*e^3*h^2)*p^2*q^3 - 6*(3*a*b^2*e*f^2*g^2 - 3*a*b^2*e^2
*f*g*h + a*b^2*e^3*h^2)*p*q^2 + 2*(b^3*f^3*h^2*p^2*q^3 - 3*a*b^2*f^3*h^2*p*q^2)*x^3 - 3*(6*a*b^2*f^3*g*h*p*q^2
- (3*b^3*f^3*g*h - b^3*e*f^2*h^2)*p^2*q^3)*x^2 - 6*(3*a*b^2*f^3*g^2*p*q^2 - (3*b^3*f^3*g^2 - 3*b^3*e*f^2*g*h
+ b^3*e^2*f*h^2)*p^2*q^3)*x - 6*(b^3*f^3*h^2*p*q^2*x^3 + 3*b^3*f^3*g*h*p*q^2*x^2 + 3*b^3*f^3*g^2*p*q^2*x + (3*
b^3*e*f^2*g^2 - 3*b^3*e^2*f*g*h + b^3*e^3*h^2)*p*q^2)*log(c))*log(d))*log(f*x + e) - 6*(2*(2*b^3*f^3*h^2*p^2*q
^2 - 6*a*b^2*f^3*h^2*p*q + 9*a^2*b*f^3*h^2)*x^3 + 3*(18*a^2*b*f^3*g*h + (9*b^3*f^3*g*h - 5*b^3*e*f^2*h^2)*p^2*
q^2 - 6*(3*a*b^2*f^3*g*h - a*b^2*e*f^2*h^2)*p*q)*x^2 + 6*(9*a^2*b*f^3*g^2 + (18*b^3*f^3*g^2 - 27*b^3*e*f^2*g*h
+ 11*b^3*e^2*f*h^2)*p^2*q^2 - 6*(3*a*b^2*f^3*g^2 - 3*a*b^2*e*f^2*g*h + a*b^2*e^2*f*h^2)*p*q)*x)*log(c) - 6*(2
*(2*b^3*f^3*h^2*p^2*q^3 - 6*a*b^2*f^3*h^2*p*q^2 + 9*a^2*b*f^3*h^2*q)*x^3 + 3*(18*a^2*b*f^3*g*h*q + (9*b^3*f^3*
g*h - 5*b^3*e*f^2*h^2)*p^2*q^3 - 6*(3*a*b^2*f^3*g*h - a*b^2*e*f^2*h^2)*p*q^2)*x^2 + 18*(b^3*f^3*h^2*q*x^3 + 3*
b^3*f^3*g*h*q*x^2 + 3*b^3*f^3*g^2*q*x)*log(c)^2 + 6*(9*a^2*b*f^3*g^2*q + (18*b^3*f^3*g^2 - 27*b^3*e*f^2*g*h +
11*b^3*e^2*f*h^2)*p^2*q^3 - 6*(3*a*b^2*f^3*g^2 - 3*a*b^2*e*f^2*g*h + a*b^2*e^2*f*h^2)*p*q^2)*x - 6*(2*(b^3*f^3
*h^2*p*q^2 - 3*a*b^2*f^3*h^2*q)*x^3 - 3*(6*a*b^2*f^3*g*h*q - (3*b^3*f^3*g*h - b^3*e*f^2*h^2)*p*q^2)*x^2 - 6*(3
*a*b^2*f^3*g^2*q - (3*b^3*f^3*g^2 - 3*b^3*e*f^2*g*h + b^3*e^2*f*h^2)*p*q^2)*x)*log(c))*log(d))/f^3

________________________________________________________________________________________

Sympy [A]  time = 124.17, size = 5008, normalized size = 10.18 \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)**2*(a+b*ln(c*(d*(f*x+e)**p)**q))**3,x)

[Out]

Piecewise((a**3*g**2*x + a**3*g*h*x**2 + a**3*h**2*x**3/3 + a**2*b*e**3*h**2*p*q*log(e + f*x)/f**3 - 3*a**2*b*
e**2*g*h*p*q*log(e + f*x)/f**2 - a**2*b*e**2*h**2*p*q*x/f**2 + 3*a**2*b*e*g**2*p*q*log(e + f*x)/f + 3*a**2*b*e
*g*h*p*q*x/f + a**2*b*e*h**2*p*q*x**2/(2*f) + 3*a**2*b*g**2*p*q*x*log(e + f*x) - 3*a**2*b*g**2*p*q*x + 3*a**2*
b*g**2*q*x*log(d) + 3*a**2*b*g**2*x*log(c) + 3*a**2*b*g*h*p*q*x**2*log(e + f*x) - 3*a**2*b*g*h*p*q*x**2/2 + 3*
a**2*b*g*h*q*x**2*log(d) + 3*a**2*b*g*h*x**2*log(c) + a**2*b*h**2*p*q*x**3*log(e + f*x) - a**2*b*h**2*p*q*x**3
/3 + a**2*b*h**2*q*x**3*log(d) + a**2*b*h**2*x**3*log(c) + a*b**2*e**3*h**2*p**2*q**2*log(e + f*x)**2/f**3 - 1
1*a*b**2*e**3*h**2*p**2*q**2*log(e + f*x)/(3*f**3) + 2*a*b**2*e**3*h**2*p*q**2*log(d)*log(e + f*x)/f**3 + 2*a*
b**2*e**3*h**2*p*q*log(c)*log(e + f*x)/f**3 - 3*a*b**2*e**2*g*h*p**2*q**2*log(e + f*x)**2/f**2 + 9*a*b**2*e**2
*g*h*p**2*q**2*log(e + f*x)/f**2 - 6*a*b**2*e**2*g*h*p*q**2*log(d)*log(e + f*x)/f**2 - 6*a*b**2*e**2*g*h*p*q*l
og(c)*log(e + f*x)/f**2 - 2*a*b**2*e**2*h**2*p**2*q**2*x*log(e + f*x)/f**2 + 11*a*b**2*e**2*h**2*p**2*q**2*x/(
3*f**2) - 2*a*b**2*e**2*h**2*p*q**2*x*log(d)/f**2 - 2*a*b**2*e**2*h**2*p*q*x*log(c)/f**2 + 3*a*b**2*e*g**2*p**
2*q**2*log(e + f*x)**2/f - 6*a*b**2*e*g**2*p**2*q**2*log(e + f*x)/f + 6*a*b**2*e*g**2*p*q**2*log(d)*log(e + f*
x)/f + 6*a*b**2*e*g**2*p*q*log(c)*log(e + f*x)/f + 6*a*b**2*e*g*h*p**2*q**2*x*log(e + f*x)/f - 9*a*b**2*e*g*h*
p**2*q**2*x/f + 6*a*b**2*e*g*h*p*q**2*x*log(d)/f + 6*a*b**2*e*g*h*p*q*x*log(c)/f + a*b**2*e*h**2*p**2*q**2*x**
2*log(e + f*x)/f - 5*a*b**2*e*h**2*p**2*q**2*x**2/(6*f) + a*b**2*e*h**2*p*q**2*x**2*log(d)/f + a*b**2*e*h**2*p
*q*x**2*log(c)/f + 3*a*b**2*g**2*p**2*q**2*x*log(e + f*x)**2 - 6*a*b**2*g**2*p**2*q**2*x*log(e + f*x) + 6*a*b*
*2*g**2*p**2*q**2*x + 6*a*b**2*g**2*p*q**2*x*log(d)*log(e + f*x) - 6*a*b**2*g**2*p*q**2*x*log(d) + 6*a*b**2*g*
*2*p*q*x*log(c)*log(e + f*x) - 6*a*b**2*g**2*p*q*x*log(c) + 3*a*b**2*g**2*q**2*x*log(d)**2 + 6*a*b**2*g**2*q*x
*log(c)*log(d) + 3*a*b**2*g**2*x*log(c)**2 + 3*a*b**2*g*h*p**2*q**2*x**2*log(e + f*x)**2 - 3*a*b**2*g*h*p**2*q
**2*x**2*log(e + f*x) + 3*a*b**2*g*h*p**2*q**2*x**2/2 + 6*a*b**2*g*h*p*q**2*x**2*log(d)*log(e + f*x) - 3*a*b**
2*g*h*p*q**2*x**2*log(d) + 6*a*b**2*g*h*p*q*x**2*log(c)*log(e + f*x) - 3*a*b**2*g*h*p*q*x**2*log(c) + 3*a*b**2
*g*h*q**2*x**2*log(d)**2 + 6*a*b**2*g*h*q*x**2*log(c)*log(d) + 3*a*b**2*g*h*x**2*log(c)**2 + a*b**2*h**2*p**2*
q**2*x**3*log(e + f*x)**2 - 2*a*b**2*h**2*p**2*q**2*x**3*log(e + f*x)/3 + 2*a*b**2*h**2*p**2*q**2*x**3/9 + 2*a
*b**2*h**2*p*q**2*x**3*log(d)*log(e + f*x) - 2*a*b**2*h**2*p*q**2*x**3*log(d)/3 + 2*a*b**2*h**2*p*q*x**3*log(c
)*log(e + f*x) - 2*a*b**2*h**2*p*q*x**3*log(c)/3 + a*b**2*h**2*q**2*x**3*log(d)**2 + 2*a*b**2*h**2*q*x**3*log(
c)*log(d) + a*b**2*h**2*x**3*log(c)**2 + b**3*e**3*h**2*p**3*q**3*log(e + f*x)**3/(3*f**3) - 11*b**3*e**3*h**2
*p**3*q**3*log(e + f*x)**2/(6*f**3) + 85*b**3*e**3*h**2*p**3*q**3*log(e + f*x)/(18*f**3) + b**3*e**3*h**2*p**2
*q**3*log(d)*log(e + f*x)**2/f**3 - 11*b**3*e**3*h**2*p**2*q**3*log(d)*log(e + f*x)/(3*f**3) + b**3*e**3*h**2*
p**2*q**2*log(c)*log(e + f*x)**2/f**3 - 11*b**3*e**3*h**2*p**2*q**2*log(c)*log(e + f*x)/(3*f**3) + b**3*e**3*h
**2*p*q**3*log(d)**2*log(e + f*x)/f**3 + 2*b**3*e**3*h**2*p*q**2*log(c)*log(d)*log(e + f*x)/f**3 + b**3*e**3*h
**2*p*q*log(c)**2*log(e + f*x)/f**3 - b**3*e**2*g*h*p**3*q**3*log(e + f*x)**3/f**2 + 9*b**3*e**2*g*h*p**3*q**3
*log(e + f*x)**2/(2*f**2) - 21*b**3*e**2*g*h*p**3*q**3*log(e + f*x)/(2*f**2) - 3*b**3*e**2*g*h*p**2*q**3*log(d
)*log(e + f*x)**2/f**2 + 9*b**3*e**2*g*h*p**2*q**3*log(d)*log(e + f*x)/f**2 - 3*b**3*e**2*g*h*p**2*q**2*log(c)
*log(e + f*x)**2/f**2 + 9*b**3*e**2*g*h*p**2*q**2*log(c)*log(e + f*x)/f**2 - 3*b**3*e**2*g*h*p*q**3*log(d)**2*
log(e + f*x)/f**2 - 6*b**3*e**2*g*h*p*q**2*log(c)*log(d)*log(e + f*x)/f**2 - 3*b**3*e**2*g*h*p*q*log(c)**2*log
(e + f*x)/f**2 - b**3*e**2*h**2*p**3*q**3*x*log(e + f*x)**2/f**2 + 11*b**3*e**2*h**2*p**3*q**3*x*log(e + f*x)/
(3*f**2) - 85*b**3*e**2*h**2*p**3*q**3*x/(18*f**2) - 2*b**3*e**2*h**2*p**2*q**3*x*log(d)*log(e + f*x)/f**2 + 1
1*b**3*e**2*h**2*p**2*q**3*x*log(d)/(3*f**2) - 2*b**3*e**2*h**2*p**2*q**2*x*log(c)*log(e + f*x)/f**2 + 11*b**3
*e**2*h**2*p**2*q**2*x*log(c)/(3*f**2) - b**3*e**2*h**2*p*q**3*x*log(d)**2/f**2 - 2*b**3*e**2*h**2*p*q**2*x*lo
g(c)*log(d)/f**2 - b**3*e**2*h**2*p*q*x*log(c)**2/f**2 + b**3*e*g**2*p**3*q**3*log(e + f*x)**3/f - 3*b**3*e*g*
*2*p**3*q**3*log(e + f*x)**2/f + 6*b**3*e*g**2*p**3*q**3*log(e + f*x)/f + 3*b**3*e*g**2*p**2*q**3*log(d)*log(e
+ f*x)**2/f - 6*b**3*e*g**2*p**2*q**3*log(d)*log(e + f*x)/f + 3*b**3*e*g**2*p**2*q**2*log(c)*log(e + f*x)**2/
f - 6*b**3*e*g**2*p**2*q**2*log(c)*log(e + f*x)/f + 3*b**3*e*g**2*p*q**3*log(d)**2*log(e + f*x)/f + 6*b**3*e*g
**2*p*q**2*log(c)*log(d)*log(e + f*x)/f + 3*b**3*e*g**2*p*q*log(c)**2*log(e + f*x)/f + 3*b**3*e*g*h*p**3*q**3*
x*log(e + f*x)**2/f - 9*b**3*e*g*h*p**3*q**3*x*log(e + f*x)/f + 21*b**3*e*g*h*p**3*q**3*x/(2*f) + 6*b**3*e*g*h
*p**2*q**3*x*log(d)*log(e + f*x)/f - 9*b**3*e*g*h*p**2*q**3*x*log(d)/f + 6*b**3*e*g*h*p**2*q**2*x*log(c)*log(e
+ f*x)/f - 9*b**3*e*g*h*p**2*q**2*x*log(c)/f + 3*b**3*e*g*h*p*q**3*x*log(d)**2/f + 6*b**3*e*g*h*p*q**2*x*log(
c)*log(d)/f + 3*b**3*e*g*h*p*q*x*log(c)**2/f + b**3*e*h**2*p**3*q**3*x**2*log(e + f*x)**2/(2*f) - 5*b**3*e*h**
2*p**3*q**3*x**2*log(e + f*x)/(6*f) + 19*b**3*e*h**2*p**3*q**3*x**2/(36*f) + b**3*e*h**2*p**2*q**3*x**2*log(d)
*log(e + f*x)/f - 5*b**3*e*h**2*p**2*q**3*x**2*log(d)/(6*f) + b**3*e*h**2*p**2*q**2*x**2*log(c)*log(e + f*x)/f
- 5*b**3*e*h**2*p**2*q**2*x**2*log(c)/(6*f) + b**3*e*h**2*p*q**3*x**2*log(d)**2/(2*f) + b**3*e*h**2*p*q**2*x*
*2*log(c)*log(d)/f + b**3*e*h**2*p*q*x**2*log(c)**2/(2*f) + b**3*g**2*p**3*q**3*x*log(e + f*x)**3 - 3*b**3*g**
2*p**3*q**3*x*log(e + f*x)**2 + 6*b**3*g**2*p**3*q**3*x*log(e + f*x) - 6*b**3*g**2*p**3*q**3*x + 3*b**3*g**2*p
**2*q**3*x*log(d)*log(e + f*x)**2 - 6*b**3*g**2*p**2*q**3*x*log(d)*log(e + f*x) + 6*b**3*g**2*p**2*q**3*x*log(
d) + 3*b**3*g**2*p**2*q**2*x*log(c)*log(e + f*x)**2 - 6*b**3*g**2*p**2*q**2*x*log(c)*log(e + f*x) + 6*b**3*g**
2*p**2*q**2*x*log(c) + 3*b**3*g**2*p*q**3*x*log(d)**2*log(e + f*x) - 3*b**3*g**2*p*q**3*x*log(d)**2 + 6*b**3*g
**2*p*q**2*x*log(c)*log(d)*log(e + f*x) - 6*b**3*g**2*p*q**2*x*log(c)*log(d) + 3*b**3*g**2*p*q*x*log(c)**2*log
(e + f*x) - 3*b**3*g**2*p*q*x*log(c)**2 + b**3*g**2*q**3*x*log(d)**3 + 3*b**3*g**2*q**2*x*log(c)*log(d)**2 + 3
*b**3*g**2*q*x*log(c)**2*log(d) + b**3*g**2*x*log(c)**3 + b**3*g*h*p**3*q**3*x**2*log(e + f*x)**3 - 3*b**3*g*h
*p**3*q**3*x**2*log(e + f*x)**2/2 + 3*b**3*g*h*p**3*q**3*x**2*log(e + f*x)/2 - 3*b**3*g*h*p**3*q**3*x**2/4 + 3
*b**3*g*h*p**2*q**3*x**2*log(d)*log(e + f*x)**2 - 3*b**3*g*h*p**2*q**3*x**2*log(d)*log(e + f*x) + 3*b**3*g*h*p
**2*q**3*x**2*log(d)/2 + 3*b**3*g*h*p**2*q**2*x**2*log(c)*log(e + f*x)**2 - 3*b**3*g*h*p**2*q**2*x**2*log(c)*l
og(e + f*x) + 3*b**3*g*h*p**2*q**2*x**2*log(c)/2 + 3*b**3*g*h*p*q**3*x**2*log(d)**2*log(e + f*x) - 3*b**3*g*h*
p*q**3*x**2*log(d)**2/2 + 6*b**3*g*h*p*q**2*x**2*log(c)*log(d)*log(e + f*x) - 3*b**3*g*h*p*q**2*x**2*log(c)*lo
g(d) + 3*b**3*g*h*p*q*x**2*log(c)**2*log(e + f*x) - 3*b**3*g*h*p*q*x**2*log(c)**2/2 + b**3*g*h*q**3*x**2*log(d
)**3 + 3*b**3*g*h*q**2*x**2*log(c)*log(d)**2 + 3*b**3*g*h*q*x**2*log(c)**2*log(d) + b**3*g*h*x**2*log(c)**3 +
b**3*h**2*p**3*q**3*x**3*log(e + f*x)**3/3 - b**3*h**2*p**3*q**3*x**3*log(e + f*x)**2/3 + 2*b**3*h**2*p**3*q**
3*x**3*log(e + f*x)/9 - 2*b**3*h**2*p**3*q**3*x**3/27 + b**3*h**2*p**2*q**3*x**3*log(d)*log(e + f*x)**2 - 2*b*
*3*h**2*p**2*q**3*x**3*log(d)*log(e + f*x)/3 + 2*b**3*h**2*p**2*q**3*x**3*log(d)/9 + b**3*h**2*p**2*q**2*x**3*
log(c)*log(e + f*x)**2 - 2*b**3*h**2*p**2*q**2*x**3*log(c)*log(e + f*x)/3 + 2*b**3*h**2*p**2*q**2*x**3*log(c)/
9 + b**3*h**2*p*q**3*x**3*log(d)**2*log(e + f*x) - b**3*h**2*p*q**3*x**3*log(d)**2/3 + 2*b**3*h**2*p*q**2*x**3
*log(c)*log(d)*log(e + f*x) - 2*b**3*h**2*p*q**2*x**3*log(c)*log(d)/3 + b**3*h**2*p*q*x**3*log(c)**2*log(e + f
*x) - b**3*h**2*p*q*x**3*log(c)**2/3 + b**3*h**2*q**3*x**3*log(d)**3/3 + b**3*h**2*q**2*x**3*log(c)*log(d)**2
+ b**3*h**2*q*x**3*log(c)**2*log(d) + b**3*h**2*x**3*log(c)**3/3, Ne(f, 0)), ((a + b*log(c*(d*e**p)**q))**3*(g
**2*x + g*h*x**2 + h**2*x**3/3), True))

________________________________________________________________________________________

Giac [B]  time = 1.67604, size = 7974, normalized size = 16.21 \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)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="giac")

[Out]

(f*x + e)*b^3*g^2*p^3*q^3*log(f*x + e)^3/f + (f*x + e)^2*b^3*g*h*p^3*q^3*log(f*x + e)^3/f^2 + 1/3*(f*x + e)^3*
b^3*h^2*p^3*q^3*log(f*x + e)^3/f^3 - 2*(f*x + e)*b^3*g*h*p^3*q^3*e*log(f*x + e)^3/f^2 - (f*x + e)^2*b^3*h^2*p^
3*q^3*e*log(f*x + e)^3/f^3 - 3*(f*x + e)*b^3*g^2*p^3*q^3*log(f*x + e)^2/f - 3/2*(f*x + e)^2*b^3*g*h*p^3*q^3*lo
g(f*x + e)^2/f^2 - 1/3*(f*x + e)^3*b^3*h^2*p^3*q^3*log(f*x + e)^2/f^3 + 6*(f*x + e)*b^3*g*h*p^3*q^3*e*log(f*x
+ e)^2/f^2 + 3/2*(f*x + e)^2*b^3*h^2*p^3*q^3*e*log(f*x + e)^2/f^3 + (f*x + e)*b^3*h^2*p^3*q^3*e^2*log(f*x + e)
^3/f^3 + 3*(f*x + e)*b^3*g^2*p^2*q^3*log(f*x + e)^2*log(d)/f + 3*(f*x + e)^2*b^3*g*h*p^2*q^3*log(f*x + e)^2*lo
g(d)/f^2 + (f*x + e)^3*b^3*h^2*p^2*q^3*log(f*x + e)^2*log(d)/f^3 - 6*(f*x + e)*b^3*g*h*p^2*q^3*e*log(f*x + e)^
2*log(d)/f^2 - 3*(f*x + e)^2*b^3*h^2*p^2*q^3*e*log(f*x + e)^2*log(d)/f^3 + 6*(f*x + e)*b^3*g^2*p^3*q^3*log(f*x
+ e)/f + 3/2*(f*x + e)^2*b^3*g*h*p^3*q^3*log(f*x + e)/f^2 + 2/9*(f*x + e)^3*b^3*h^2*p^3*q^3*log(f*x + e)/f^3
- 12*(f*x + e)*b^3*g*h*p^3*q^3*e*log(f*x + e)/f^2 - 3/2*(f*x + e)^2*b^3*h^2*p^3*q^3*e*log(f*x + e)/f^3 - 3*(f*
x + e)*b^3*h^2*p^3*q^3*e^2*log(f*x + e)^2/f^3 + 3*(f*x + e)*b^3*g^2*p^2*q^2*log(f*x + e)^2*log(c)/f + 3*(f*x +
e)^2*b^3*g*h*p^2*q^2*log(f*x + e)^2*log(c)/f^2 + (f*x + e)^3*b^3*h^2*p^2*q^2*log(f*x + e)^2*log(c)/f^3 - 6*(f
*x + e)*b^3*g*h*p^2*q^2*e*log(f*x + e)^2*log(c)/f^2 - 3*(f*x + e)^2*b^3*h^2*p^2*q^2*e*log(f*x + e)^2*log(c)/f^
3 - 6*(f*x + e)*b^3*g^2*p^2*q^3*log(f*x + e)*log(d)/f - 3*(f*x + e)^2*b^3*g*h*p^2*q^3*log(f*x + e)*log(d)/f^2
- 2/3*(f*x + e)^3*b^3*h^2*p^2*q^3*log(f*x + e)*log(d)/f^3 + 12*(f*x + e)*b^3*g*h*p^2*q^3*e*log(f*x + e)*log(d)
/f^2 + 3*(f*x + e)^2*b^3*h^2*p^2*q^3*e*log(f*x + e)*log(d)/f^3 + 3*(f*x + e)*b^3*h^2*p^2*q^3*e^2*log(f*x + e)^
2*log(d)/f^3 + 3*(f*x + e)*b^3*g^2*p*q^3*log(f*x + e)*log(d)^2/f + 3*(f*x + e)^2*b^3*g*h*p*q^3*log(f*x + e)*lo
g(d)^2/f^2 + (f*x + e)^3*b^3*h^2*p*q^3*log(f*x + e)*log(d)^2/f^3 - 6*(f*x + e)*b^3*g*h*p*q^3*e*log(f*x + e)*lo
g(d)^2/f^2 - 3*(f*x + e)^2*b^3*h^2*p*q^3*e*log(f*x + e)*log(d)^2/f^3 - 6*(f*x + e)*b^3*g^2*p^3*q^3/f - 3/4*(f*
x + e)^2*b^3*g*h*p^3*q^3/f^2 - 2/27*(f*x + e)^3*b^3*h^2*p^3*q^3/f^3 + 12*(f*x + e)*b^3*g*h*p^3*q^3*e/f^2 + 3/4
*(f*x + e)^2*b^3*h^2*p^3*q^3*e/f^3 + 6*(f*x + e)*b^3*h^2*p^3*q^3*e^2*log(f*x + e)/f^3 + 3*(f*x + e)*a*b^2*g^2*
p^2*q^2*log(f*x + e)^2/f + 3*(f*x + e)^2*a*b^2*g*h*p^2*q^2*log(f*x + e)^2/f^2 + (f*x + e)^3*a*b^2*h^2*p^2*q^2*
log(f*x + e)^2/f^3 - 6*(f*x + e)*a*b^2*g*h*p^2*q^2*e*log(f*x + e)^2/f^2 - 3*(f*x + e)^2*a*b^2*h^2*p^2*q^2*e*lo
g(f*x + e)^2/f^3 - 6*(f*x + e)*b^3*g^2*p^2*q^2*log(f*x + e)*log(c)/f - 3*(f*x + e)^2*b^3*g*h*p^2*q^2*log(f*x +
e)*log(c)/f^2 - 2/3*(f*x + e)^3*b^3*h^2*p^2*q^2*log(f*x + e)*log(c)/f^3 + 12*(f*x + e)*b^3*g*h*p^2*q^2*e*log(
f*x + e)*log(c)/f^2 + 3*(f*x + e)^2*b^3*h^2*p^2*q^2*e*log(f*x + e)*log(c)/f^3 + 3*(f*x + e)*b^3*h^2*p^2*q^2*e^
2*log(f*x + e)^2*log(c)/f^3 + 6*(f*x + e)*b^3*g^2*p^2*q^3*log(d)/f + 3/2*(f*x + e)^2*b^3*g*h*p^2*q^3*log(d)/f^
2 + 2/9*(f*x + e)^3*b^3*h^2*p^2*q^3*log(d)/f^3 - 12*(f*x + e)*b^3*g*h*p^2*q^3*e*log(d)/f^2 - 3/2*(f*x + e)^2*b
^3*h^2*p^2*q^3*e*log(d)/f^3 - 6*(f*x + e)*b^3*h^2*p^2*q^3*e^2*log(f*x + e)*log(d)/f^3 + 6*(f*x + e)*b^3*g^2*p*
q^2*log(f*x + e)*log(c)*log(d)/f + 6*(f*x + e)^2*b^3*g*h*p*q^2*log(f*x + e)*log(c)*log(d)/f^2 + 2*(f*x + e)^3*
b^3*h^2*p*q^2*log(f*x + e)*log(c)*log(d)/f^3 - 12*(f*x + e)*b^3*g*h*p*q^2*e*log(f*x + e)*log(c)*log(d)/f^2 - 6
*(f*x + e)^2*b^3*h^2*p*q^2*e*log(f*x + e)*log(c)*log(d)/f^3 - 3*(f*x + e)*b^3*g^2*p*q^3*log(d)^2/f - 3/2*(f*x
+ e)^2*b^3*g*h*p*q^3*log(d)^2/f^2 - 1/3*(f*x + e)^3*b^3*h^2*p*q^3*log(d)^2/f^3 + 6*(f*x + e)*b^3*g*h*p*q^3*e*l
og(d)^2/f^2 + 3/2*(f*x + e)^2*b^3*h^2*p*q^3*e*log(d)^2/f^3 + 3*(f*x + e)*b^3*h^2*p*q^3*e^2*log(f*x + e)*log(d)
^2/f^3 + (f*x + e)*b^3*g^2*q^3*log(d)^3/f + (f*x + e)^2*b^3*g*h*q^3*log(d)^3/f^2 + 1/3*(f*x + e)^3*b^3*h^2*q^3
*log(d)^3/f^3 - 2*(f*x + e)*b^3*g*h*q^3*e*log(d)^3/f^2 - (f*x + e)^2*b^3*h^2*q^3*e*log(d)^3/f^3 - 6*(f*x + e)*
b^3*h^2*p^3*q^3*e^2/f^3 - 6*(f*x + e)*a*b^2*g^2*p^2*q^2*log(f*x + e)/f - 3*(f*x + e)^2*a*b^2*g*h*p^2*q^2*log(f
*x + e)/f^2 - 2/3*(f*x + e)^3*a*b^2*h^2*p^2*q^2*log(f*x + e)/f^3 + 12*(f*x + e)*a*b^2*g*h*p^2*q^2*e*log(f*x +
e)/f^2 + 3*(f*x + e)^2*a*b^2*h^2*p^2*q^2*e*log(f*x + e)/f^3 + 3*(f*x + e)*a*b^2*h^2*p^2*q^2*e^2*log(f*x + e)^2
/f^3 + 6*(f*x + e)*b^3*g^2*p^2*q^2*log(c)/f + 3/2*(f*x + e)^2*b^3*g*h*p^2*q^2*log(c)/f^2 + 2/9*(f*x + e)^3*b^3
*h^2*p^2*q^2*log(c)/f^3 - 12*(f*x + e)*b^3*g*h*p^2*q^2*e*log(c)/f^2 - 3/2*(f*x + e)^2*b^3*h^2*p^2*q^2*e*log(c)
/f^3 - 6*(f*x + e)*b^3*h^2*p^2*q^2*e^2*log(f*x + e)*log(c)/f^3 + 3*(f*x + e)*b^3*g^2*p*q*log(f*x + e)*log(c)^2
/f + 3*(f*x + e)^2*b^3*g*h*p*q*log(f*x + e)*log(c)^2/f^2 + (f*x + e)^3*b^3*h^2*p*q*log(f*x + e)*log(c)^2/f^3 -
6*(f*x + e)*b^3*g*h*p*q*e*log(f*x + e)*log(c)^2/f^2 - 3*(f*x + e)^2*b^3*h^2*p*q*e*log(f*x + e)*log(c)^2/f^3 +
6*(f*x + e)*b^3*h^2*p^2*q^3*e^2*log(d)/f^3 + 6*(f*x + e)*a*b^2*g^2*p*q^2*log(f*x + e)*log(d)/f + 6*(f*x + e)^
2*a*b^2*g*h*p*q^2*log(f*x + e)*log(d)/f^2 + 2*(f*x + e)^3*a*b^2*h^2*p*q^2*log(f*x + e)*log(d)/f^3 - 12*(f*x +
e)*a*b^2*g*h*p*q^2*e*log(f*x + e)*log(d)/f^2 - 6*(f*x + e)^2*a*b^2*h^2*p*q^2*e*log(f*x + e)*log(d)/f^3 - 6*(f*
x + e)*b^3*g^2*p*q^2*log(c)*log(d)/f - 3*(f*x + e)^2*b^3*g*h*p*q^2*log(c)*log(d)/f^2 - 2/3*(f*x + e)^3*b^3*h^2
*p*q^2*log(c)*log(d)/f^3 + 12*(f*x + e)*b^3*g*h*p*q^2*e*log(c)*log(d)/f^2 + 3*(f*x + e)^2*b^3*h^2*p*q^2*e*log(
c)*log(d)/f^3 + 6*(f*x + e)*b^3*h^2*p*q^2*e^2*log(f*x + e)*log(c)*log(d)/f^3 - 3*(f*x + e)*b^3*h^2*p*q^3*e^2*l
og(d)^2/f^3 + 3*(f*x + e)*b^3*g^2*q^2*log(c)*log(d)^2/f + 3*(f*x + e)^2*b^3*g*h*q^2*log(c)*log(d)^2/f^2 + (f*x
+ e)^3*b^3*h^2*q^2*log(c)*log(d)^2/f^3 - 6*(f*x + e)*b^3*g*h*q^2*e*log(c)*log(d)^2/f^2 - 3*(f*x + e)^2*b^3*h^
2*q^2*e*log(c)*log(d)^2/f^3 + (f*x + e)*b^3*h^2*q^3*e^2*log(d)^3/f^3 + 6*(f*x + e)*a*b^2*g^2*p^2*q^2/f + 3/2*(
f*x + e)^2*a*b^2*g*h*p^2*q^2/f^2 + 2/9*(f*x + e)^3*a*b^2*h^2*p^2*q^2/f^3 - 12*(f*x + e)*a*b^2*g*h*p^2*q^2*e/f^
2 - 3/2*(f*x + e)^2*a*b^2*h^2*p^2*q^2*e/f^3 - 6*(f*x + e)*a*b^2*h^2*p^2*q^2*e^2*log(f*x + e)/f^3 + 6*(f*x + e)
*b^3*h^2*p^2*q^2*e^2*log(c)/f^3 + 6*(f*x + e)*a*b^2*g^2*p*q*log(f*x + e)*log(c)/f + 6*(f*x + e)^2*a*b^2*g*h*p*
q*log(f*x + e)*log(c)/f^2 + 2*(f*x + e)^3*a*b^2*h^2*p*q*log(f*x + e)*log(c)/f^3 - 12*(f*x + e)*a*b^2*g*h*p*q*e
*log(f*x + e)*log(c)/f^2 - 6*(f*x + e)^2*a*b^2*h^2*p*q*e*log(f*x + e)*log(c)/f^3 - 3*(f*x + e)*b^3*g^2*p*q*log
(c)^2/f - 3/2*(f*x + e)^2*b^3*g*h*p*q*log(c)^2/f^2 - 1/3*(f*x + e)^3*b^3*h^2*p*q*log(c)^2/f^3 + 6*(f*x + e)*b^
3*g*h*p*q*e*log(c)^2/f^2 + 3/2*(f*x + e)^2*b^3*h^2*p*q*e*log(c)^2/f^3 + 3*(f*x + e)*b^3*h^2*p*q*e^2*log(f*x +
e)*log(c)^2/f^3 - 6*(f*x + e)*a*b^2*g^2*p*q^2*log(d)/f - 3*(f*x + e)^2*a*b^2*g*h*p*q^2*log(d)/f^2 - 2/3*(f*x +
e)^3*a*b^2*h^2*p*q^2*log(d)/f^3 + 12*(f*x + e)*a*b^2*g*h*p*q^2*e*log(d)/f^2 + 3*(f*x + e)^2*a*b^2*h^2*p*q^2*e
*log(d)/f^3 + 6*(f*x + e)*a*b^2*h^2*p*q^2*e^2*log(f*x + e)*log(d)/f^3 - 6*(f*x + e)*b^3*h^2*p*q^2*e^2*log(c)*l
og(d)/f^3 + 3*(f*x + e)*b^3*g^2*q*log(c)^2*log(d)/f + 3*(f*x + e)^2*b^3*g*h*q*log(c)^2*log(d)/f^2 + (f*x + e)^
3*b^3*h^2*q*log(c)^2*log(d)/f^3 - 6*(f*x + e)*b^3*g*h*q*e*log(c)^2*log(d)/f^2 - 3*(f*x + e)^2*b^3*h^2*q*e*log(
c)^2*log(d)/f^3 + 3*(f*x + e)*a*b^2*g^2*q^2*log(d)^2/f + 3*(f*x + e)^2*a*b^2*g*h*q^2*log(d)^2/f^2 + (f*x + e)^
3*a*b^2*h^2*q^2*log(d)^2/f^3 - 6*(f*x + e)*a*b^2*g*h*q^2*e*log(d)^2/f^2 - 3*(f*x + e)^2*a*b^2*h^2*q^2*e*log(d)
^2/f^3 + 3*(f*x + e)*b^3*h^2*q^2*e^2*log(c)*log(d)^2/f^3 + 6*(f*x + e)*a*b^2*h^2*p^2*q^2*e^2/f^3 + 3*(f*x + e)
*a^2*b*g^2*p*q*log(f*x + e)/f + 3*(f*x + e)^2*a^2*b*g*h*p*q*log(f*x + e)/f^2 + (f*x + e)^3*a^2*b*h^2*p*q*log(f
*x + e)/f^3 - 6*(f*x + e)*a^2*b*g*h*p*q*e*log(f*x + e)/f^2 - 3*(f*x + e)^2*a^2*b*h^2*p*q*e*log(f*x + e)/f^3 -
6*(f*x + e)*a*b^2*g^2*p*q*log(c)/f - 3*(f*x + e)^2*a*b^2*g*h*p*q*log(c)/f^2 - 2/3*(f*x + e)^3*a*b^2*h^2*p*q*lo
g(c)/f^3 + 12*(f*x + e)*a*b^2*g*h*p*q*e*log(c)/f^2 + 3*(f*x + e)^2*a*b^2*h^2*p*q*e*log(c)/f^3 + 6*(f*x + e)*a*
b^2*h^2*p*q*e^2*log(f*x + e)*log(c)/f^3 - 3*(f*x + e)*b^3*h^2*p*q*e^2*log(c)^2/f^3 + (f*x + e)*b^3*g^2*log(c)^
3/f + (f*x + e)^2*b^3*g*h*log(c)^3/f^2 + 1/3*(f*x + e)^3*b^3*h^2*log(c)^3/f^3 - 2*(f*x + e)*b^3*g*h*e*log(c)^3
/f^2 - (f*x + e)^2*b^3*h^2*e*log(c)^3/f^3 - 6*(f*x + e)*a*b^2*h^2*p*q^2*e^2*log(d)/f^3 + 6*(f*x + e)*a*b^2*g^2
*q*log(c)*log(d)/f + 6*(f*x + e)^2*a*b^2*g*h*q*log(c)*log(d)/f^2 + 2*(f*x + e)^3*a*b^2*h^2*q*log(c)*log(d)/f^3
- 12*(f*x + e)*a*b^2*g*h*q*e*log(c)*log(d)/f^2 - 6*(f*x + e)^2*a*b^2*h^2*q*e*log(c)*log(d)/f^3 + 3*(f*x + e)*
b^3*h^2*q*e^2*log(c)^2*log(d)/f^3 + 3*(f*x + e)*a*b^2*h^2*q^2*e^2*log(d)^2/f^3 - 3*(f*x + e)*a^2*b*g^2*p*q/f -
3/2*(f*x + e)^2*a^2*b*g*h*p*q/f^2 - 1/3*(f*x + e)^3*a^2*b*h^2*p*q/f^3 + 6*(f*x + e)*a^2*b*g*h*p*q*e/f^2 + 3/2
*(f*x + e)^2*a^2*b*h^2*p*q*e/f^3 + 3*(f*x + e)*a^2*b*h^2*p*q*e^2*log(f*x + e)/f^3 - 6*(f*x + e)*a*b^2*h^2*p*q*
e^2*log(c)/f^3 + 3*(f*x + e)*a*b^2*g^2*log(c)^2/f + 3*(f*x + e)^2*a*b^2*g*h*log(c)^2/f^2 + (f*x + e)^3*a*b^2*h
^2*log(c)^2/f^3 - 6*(f*x + e)*a*b^2*g*h*e*log(c)^2/f^2 - 3*(f*x + e)^2*a*b^2*h^2*e*log(c)^2/f^3 + (f*x + e)*b^
3*h^2*e^2*log(c)^3/f^3 + 3*(f*x + e)*a^2*b*g^2*q*log(d)/f + 3*(f*x + e)^2*a^2*b*g*h*q*log(d)/f^2 + (f*x + e)^3
*a^2*b*h^2*q*log(d)/f^3 - 6*(f*x + e)*a^2*b*g*h*q*e*log(d)/f^2 - 3*(f*x + e)^2*a^2*b*h^2*q*e*log(d)/f^3 + 6*(f
*x + e)*a*b^2*h^2*q*e^2*log(c)*log(d)/f^3 - 3*(f*x + e)*a^2*b*h^2*p*q*e^2/f^3 + 3*(f*x + e)*a^2*b*g^2*log(c)/f
+ 3*(f*x + e)^2*a^2*b*g*h*log(c)/f^2 + (f*x + e)^3*a^2*b*h^2*log(c)/f^3 - 6*(f*x + e)*a^2*b*g*h*e*log(c)/f^2
- 3*(f*x + e)^2*a^2*b*h^2*e*log(c)/f^3 + 3*(f*x + e)*a*b^2*h^2*e^2*log(c)^2/f^3 + 3*(f*x + e)*a^2*b*h^2*q*e^2*
log(d)/f^3 + (f*x + e)*a^3*g^2/f + (f*x + e)^2*a^3*g*h/f^2 + 1/3*(f*x + e)^3*a^3*h^2/f^3 - 2*(f*x + e)*a^3*g*h
*e/f^2 - (f*x + e)^2*a^3*h^2*e/f^3 + 3*(f*x + e)*a^2*b*h^2*e^2*log(c)/f^3 + (f*x + e)*a^3*h^2*e^2/f^3