\(\int (g+h x)^2 (a+b \log (c (d (e+f x)^p)^q))^3 \, dx\) [435]

Optimal result

Integrand size = 28, antiderivative size = 492 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\frac {6 a b^2 (f g-e h)^2 p^2 q^2 x}{f^2}-\frac {6 b^3 (f g-e h)^2 p^3 q^3 x}{f^2}-\frac {3 b^3 h (f g-e h) p^3 q^3 (e+f x)^2}{4 f^3}-\frac {2 b^3 h^2 p^3 q^3 (e+f x)^3}{27 f^3}+\frac {6 b^3 (f g-e h)^2 p^2 q^2 (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^3}+\frac {3 b^2 h (f g-e h) p^2 q^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^3}+\frac {2 b^2 h^2 p^2 q^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}-\frac {3 b (f g-e h)^2 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}-\frac {3 b h (f g-e h) p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac {b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac {(f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac {h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2448, 2436, 2333, 2332, 2437, 2342, 2341, 2495} \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\frac {3 b^2 h p^2 q^2 (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^3}+\frac {2 b^2 h^2 p^2 q^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{9 f^3}+\frac {6 a b^2 p^2 q^2 x (f g-e h)^2}{f^2}-\frac {3 b h p q (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{2 f^3}-\frac {3 b p q (e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{f^3}+\frac {h (e+f x)^2 (f g-e h) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}+\frac {(e+f x) (f g-e h)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{f^3}-\frac {b h^2 p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{3 f^3}+\frac {h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3}{3 f^3}+\frac {6 b^3 p^2 q^2 (e+f x) (f g-e h)^2 \log \left (c \left (d (e+f x)^p\right )^q\right )}{f^3}-\frac {3 b^3 h p^3 q^3 (e+f x)^2 (f g-e h)}{4 f^3}-\frac {2 b^3 h^2 p^3 q^3 (e+f x)^3}{27 f^3}-\frac {6 b^3 p^3 q^3 x (f g-e h)^2}{f^2} \]

[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 2332

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

Rule 2333

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 2341

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 2342

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 2436

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 2437

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 2448

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 2495

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*} \text {integral}& = \text {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 ) \\ & = \text {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 ) \\ & = \text {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 )+\text {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 )+\text {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 ) \\ & = \text {Subst}\left (\frac {h^2 \text {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 )+\text {Subst}\left (\frac {(2 h (f g-e h)) \text {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 )+\text {Subst}\left (\frac {(f g-e h)^2 \text {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}-\text {Subst}\left (\frac {\left (b h^2 p q\right ) \text {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 )-\text {Subst}\left (\frac {(3 b h (f g-e h) p q) \text {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 )-\text {Subst}\left (\frac {\left (3 b (f g-e h)^2 p q\right ) \text {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}+\text {Subst}\left (\frac {\left (2 b^2 h^2 p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (3 b^2 h (f g-e h) p^2 q^2\right ) \text {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 )+\text {Subst}\left (\frac {\left (6 b^2 (f g-e h)^2 p^2 q^2\right ) \text {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}+\text {Subst}\left (\frac {\left (6 b^3 (f g-e h)^2 p^2 q^2\right ) \text {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] (verified)

Time = 0.19 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.77 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\frac {108 (f g-e h)^2 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+108 h (f g-e h) (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3+36 h^2 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3-324 b (f g-e h)^2 p q \left ((e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-2 b p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )-81 b h (f g-e h) p q \left (2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+b p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )-4 b h^2 p q \left (9 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+2 b p q \left (b f p q x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )\right )}{108 f^3} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2033\) vs. \(2(478)=956\).

Time = 16.19 (sec) , antiderivative size = 2034, normalized size of antiderivative = 4.13

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2034\)

[In]

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

[Out]

1/108*(648*x*ln(c*(d*(f*x+e)^p)^q)*a*b^2*e*f^2*g*h*p*q-972*x*ln(c*(d*(f*x+e)^p)^q)*b^3*e*f^2*g*h*p^2*q^2+108*x
^2*ln(c*(d*(f*x+e)^p)^q)*a*b^2*e*f^2*h^2*p*q-2106*ln(f*x+e)*b^3*e^2*f*g*h*p^3*q^3-324*x^2*ln(c*(d*(f*x+e)^p)^q
)*a*b^2*f^3*g*h*p*q+324*x*ln(c*(d*(f*x+e)^p)^q)^2*b^3*e*f^2*g*h*p*q-972*x*a*b^2*e*f^2*g*h*p^2*q^2-216*x*ln(c*(
d*(f*x+e)^p)^q)*a*b^2*e^2*f*h^2*p*q+324*x*a^2*b*e*f^2*g*h*p*q-648*ln(c*(d*(f*x+e)^p)^q)*a*b^2*e^2*f*g*h*p*q+16
20*ln(f*x+e)*a*b^2*e^2*f*g*h*p^2*q^2-324*ln(f*x+e)*a^2*b*e^2*f*g*h*p*q+972*a*b^2*e^2*f*g*h*p^2*q^2-324*a^2*b*e
^2*f*g*h*p*q-108*x*a^2*b*e^2*f*h^2*p*q+648*ln(c*(d*(f*x+e)^p)^q)*a*b^2*e*f^2*g^2*p*q-90*x^2*ln(c*(d*(f*x+e)^p)
^q)*b^3*e*f^2*h^2*p^2*q^2+162*x^2*ln(c*(d*(f*x+e)^p)^q)*b^3*f^3*g*h*p^2*q^2+1134*x*b^3*e*f^2*g*h*p^3*q^3-72*x^
3*ln(c*(d*(f*x+e)^p)^q)*a*b^2*f^3*h^2*p*q+54*x^2*ln(c*(d*(f*x+e)^p)^q)^2*b^3*e*f^2*h^2*p*q-1296*ln(f*x+e)*a*b^
2*e*f^2*g^2*p^2*q^2+648*ln(f*x+e)*a^2*b*e*f^2*g^2*p*q-162*x^2*ln(c*(d*(f*x+e)^p)^q)^2*b^3*f^3*g*h*p*q-90*x^2*a
*b^2*e*f^2*h^2*p^2*q^2+162*x^2*a*b^2*f^3*g*h*p^2*q^2+396*x*ln(c*(d*(f*x+e)^p)^q)*b^3*e^2*f*h^2*p^2*q^2-108*x*l
n(c*(d*(f*x+e)^p)^q)^2*b^3*e^2*f*h^2*p*q+396*x*a*b^2*e^2*f*h^2*p^2*q^2+972*ln(c*(d*(f*x+e)^p)^q)*b^3*e^2*f*g*h
*p^2*q^2+54*x^2*a^2*b*e*f^2*h^2*p*q-162*x^2*a^2*b*f^3*g*h*p*q-648*x*ln(c*(d*(f*x+e)^p)^q)*a*b^2*f^3*g^2*p*q+48
6*ln(c*(d*(f*x+e)^p)^q)^2*b^3*e^2*f*g*h*p*q+324*a^2*b*e*f^2*g^2*p*q-1134*b^3*e^2*f*g*h*p^3*q^3-648*a*b^2*e*f^2
*g^2*p^2*q^2-81*x^2*b^3*f^3*g*h*p^3*q^3-36*x^3*ln(c*(d*(f*x+e)^p)^q)^2*b^3*f^3*h^2*p*q+24*x^3*a*b^2*f^3*h^2*p^
2*q^2-510*x*b^3*e^2*f*h^2*p^3*q^3+648*x*ln(c*(d*(f*x+e)^p)^q)*b^3*f^3*g^2*p^2*q^2-36*x^3*a^2*b*f^3*h^2*p*q-324
*x*ln(c*(d*(f*x+e)^p)^q)^2*b^3*f^3*g^2*p*q+648*x*a*b^2*f^3*g^2*p^2*q^2-648*ln(c*(d*(f*x+e)^p)^q)*b^3*e*f^2*g^2
*p^2*q^2+324*x^2*ln(c*(d*(f*x+e)^p)^q)^2*a*b^2*f^3*g*h-324*ln(c*(d*(f*x+e)^p)^q)^2*b^3*e*f^2*g^2*p*q+324*x^2*l
n(c*(d*(f*x+e)^p)^q)*a^2*b*f^3*g*h-324*x*a^2*b*f^3*g^2*p*q+216*ln(c*(d*(f*x+e)^p)^q)*a*b^2*e^3*h^2*p*q-324*ln(
c*(d*(f*x+e)^p)^q)^2*a*b^2*e^2*f*g*h+1296*ln(f*x+e)*b^3*e*f^2*g^2*p^3*q^3-612*ln(f*x+e)*a*b^2*e^3*h^2*p^2*q^2+
108*ln(f*x+e)*a^2*b*e^3*h^2*p*q+57*x^2*b^3*e*f^2*h^2*p^3*q^3+24*x^3*ln(c*(d*(f*x+e)^p)^q)*b^3*f^3*h^2*p^2*q^2+
510*b^3*e^3*h^2*p^3*q^3-108*a^3*e*f^2*g^2+36*x^3*ln(c*(d*(f*x+e)^p)^q)^3*b^3*f^3*h^2+108*x*ln(c*(d*(f*x+e)^p)^
q)^3*b^3*f^3*g^2+108*ln(c*(d*(f*x+e)^p)^q)^3*b^3*e*f^2*g^2+108*x^2*a^3*f^3*g*h+108*ln(c*(d*(f*x+e)^p)^q)^2*a*b
^2*e^3*h^2+648*b^3*e*f^2*g^2*p^3*q^3-324*ln(c*(d*(f*x+e)^p)^q)*a^2*b*e*f^2*g^2+906*ln(f*x+e)*b^3*e^3*h^2*p^3*q
^3-8*x^3*b^3*f^3*h^2*p^3*q^3-648*x*b^3*f^3*g^2*p^3*q^3+108*x^3*ln(c*(d*(f*x+e)^p)^q)^2*a*b^2*f^3*h^2+108*x^2*l
n(c*(d*(f*x+e)^p)^q)^3*b^3*f^3*g*h-396*ln(c*(d*(f*x+e)^p)^q)*b^3*e^3*h^2*p^2*q^2+108*x^3*ln(c*(d*(f*x+e)^p)^q)
*a^2*b*f^3*h^2-198*ln(c*(d*(f*x+e)^p)^q)^2*b^3*e^3*h^2*p*q+324*x*ln(c*(d*(f*x+e)^p)^q)^2*a*b^2*f^3*g^2-108*ln(
c*(d*(f*x+e)^p)^q)^3*b^3*e^2*f*g*h+324*x*ln(c*(d*(f*x+e)^p)^q)*a^2*b*f^3*g^2+324*ln(c*(d*(f*x+e)^p)^q)^2*a*b^2
*e*f^2*g^2-396*a*b^2*e^3*h^2*p^2*q^2+108*a^2*b*e^3*h^2*p*q+36*x^3*a^3*f^3*h^2+36*ln(c*(d*(f*x+e)^p)^q)^3*b^3*e
^3*h^2+108*x*a^3*f^3*g^2)/f^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3121 vs. \(2 (478) = 956\).

Time = 0.38 (sec) , antiderivative size = 3121, normalized size of antiderivative = 6.34 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\text {Too large to display} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1846 vs. \(2 (481) = 962\).

Time = 5.38 (sec) , antiderivative size = 1846, normalized size of antiderivative = 3.75 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\text {Too large to display} \]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1245 vs. \(2 (478) = 956\).

Time = 0.25 (sec) , antiderivative size = 1245, normalized size of antiderivative = 2.53 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\text {Too large to display} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5146 vs. \(2 (478) = 956\).

Time = 0.42 (sec) , antiderivative size = 5146, normalized size of antiderivative = 10.46 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((h*x+g)^2*(a+b*log(c*(d*(f*x+e)^p)^q))^3,x, algorithm="giac")

[Out]

1/108*(108*(f*x + e)*b^3*f^2*g^2*p^3*q^3*log(f*x + e)^3 + 108*(f*x + e)^2*b^3*f*g*h*p^3*q^3*log(f*x + e)^3 - 2
16*(f*x + e)*b^3*e*f*g*h*p^3*q^3*log(f*x + e)^3 + 36*(f*x + e)^3*b^3*h^2*p^3*q^3*log(f*x + e)^3 - 108*(f*x + e
)^2*b^3*e*h^2*p^3*q^3*log(f*x + e)^3 + 108*(f*x + e)*b^3*e^2*h^2*p^3*q^3*log(f*x + e)^3 - 324*(f*x + e)*b^3*f^
2*g^2*p^3*q^3*log(f*x + e)^2 - 162*(f*x + e)^2*b^3*f*g*h*p^3*q^3*log(f*x + e)^2 + 648*(f*x + e)*b^3*e*f*g*h*p^
3*q^3*log(f*x + e)^2 - 36*(f*x + e)^3*b^3*h^2*p^3*q^3*log(f*x + e)^2 + 162*(f*x + e)^2*b^3*e*h^2*p^3*q^3*log(f
*x + e)^2 - 324*(f*x + e)*b^3*e^2*h^2*p^3*q^3*log(f*x + e)^2 + 324*(f*x + e)*b^3*f^2*g^2*p^2*q^3*log(f*x + e)^
2*log(d) + 324*(f*x + e)^2*b^3*f*g*h*p^2*q^3*log(f*x + e)^2*log(d) - 648*(f*x + e)*b^3*e*f*g*h*p^2*q^3*log(f*x
+ e)^2*log(d) + 108*(f*x + e)^3*b^3*h^2*p^2*q^3*log(f*x + e)^2*log(d) - 324*(f*x + e)^2*b^3*e*h^2*p^2*q^3*log
(f*x + e)^2*log(d) + 324*(f*x + e)*b^3*e^2*h^2*p^2*q^3*log(f*x + e)^2*log(d) + 648*(f*x + e)*b^3*f^2*g^2*p^3*q
^3*log(f*x + e) + 162*(f*x + e)^2*b^3*f*g*h*p^3*q^3*log(f*x + e) - 1296*(f*x + e)*b^3*e*f*g*h*p^3*q^3*log(f*x
+ e) + 24*(f*x + e)^3*b^3*h^2*p^3*q^3*log(f*x + e) - 162*(f*x + e)^2*b^3*e*h^2*p^3*q^3*log(f*x + e) + 648*(f*x
+ e)*b^3*e^2*h^2*p^3*q^3*log(f*x + e) + 324*(f*x + e)*b^3*f^2*g^2*p^2*q^2*log(f*x + e)^2*log(c) + 324*(f*x +
e)^2*b^3*f*g*h*p^2*q^2*log(f*x + e)^2*log(c) - 648*(f*x + e)*b^3*e*f*g*h*p^2*q^2*log(f*x + e)^2*log(c) + 108*(
f*x + e)^3*b^3*h^2*p^2*q^2*log(f*x + e)^2*log(c) - 324*(f*x + e)^2*b^3*e*h^2*p^2*q^2*log(f*x + e)^2*log(c) + 3
24*(f*x + e)*b^3*e^2*h^2*p^2*q^2*log(f*x + e)^2*log(c) - 648*(f*x + e)*b^3*f^2*g^2*p^2*q^3*log(f*x + e)*log(d)
- 324*(f*x + e)^2*b^3*f*g*h*p^2*q^3*log(f*x + e)*log(d) + 1296*(f*x + e)*b^3*e*f*g*h*p^2*q^3*log(f*x + e)*log
(d) - 72*(f*x + e)^3*b^3*h^2*p^2*q^3*log(f*x + e)*log(d) + 324*(f*x + e)^2*b^3*e*h^2*p^2*q^3*log(f*x + e)*log(
d) - 648*(f*x + e)*b^3*e^2*h^2*p^2*q^3*log(f*x + e)*log(d) + 324*(f*x + e)*b^3*f^2*g^2*p*q^3*log(f*x + e)*log(
d)^2 + 324*(f*x + e)^2*b^3*f*g*h*p*q^3*log(f*x + e)*log(d)^2 - 648*(f*x + e)*b^3*e*f*g*h*p*q^3*log(f*x + e)*lo
g(d)^2 + 108*(f*x + e)^3*b^3*h^2*p*q^3*log(f*x + e)*log(d)^2 - 324*(f*x + e)^2*b^3*e*h^2*p*q^3*log(f*x + e)*lo
g(d)^2 + 324*(f*x + e)*b^3*e^2*h^2*p*q^3*log(f*x + e)*log(d)^2 - 648*(f*x + e)*b^3*f^2*g^2*p^3*q^3 - 81*(f*x +
e)^2*b^3*f*g*h*p^3*q^3 + 1296*(f*x + e)*b^3*e*f*g*h*p^3*q^3 - 8*(f*x + e)^3*b^3*h^2*p^3*q^3 + 81*(f*x + e)^2*
b^3*e*h^2*p^3*q^3 - 648*(f*x + e)*b^3*e^2*h^2*p^3*q^3 + 324*(f*x + e)*a*b^2*f^2*g^2*p^2*q^2*log(f*x + e)^2 + 3
24*(f*x + e)^2*a*b^2*f*g*h*p^2*q^2*log(f*x + e)^2 - 648*(f*x + e)*a*b^2*e*f*g*h*p^2*q^2*log(f*x + e)^2 + 108*(
f*x + e)^3*a*b^2*h^2*p^2*q^2*log(f*x + e)^2 - 324*(f*x + e)^2*a*b^2*e*h^2*p^2*q^2*log(f*x + e)^2 + 324*(f*x +
e)*a*b^2*e^2*h^2*p^2*q^2*log(f*x + e)^2 - 648*(f*x + e)*b^3*f^2*g^2*p^2*q^2*log(f*x + e)*log(c) - 324*(f*x + e
)^2*b^3*f*g*h*p^2*q^2*log(f*x + e)*log(c) + 1296*(f*x + e)*b^3*e*f*g*h*p^2*q^2*log(f*x + e)*log(c) - 72*(f*x +
e)^3*b^3*h^2*p^2*q^2*log(f*x + e)*log(c) + 324*(f*x + e)^2*b^3*e*h^2*p^2*q^2*log(f*x + e)*log(c) - 648*(f*x +
e)*b^3*e^2*h^2*p^2*q^2*log(f*x + e)*log(c) + 648*(f*x + e)*b^3*f^2*g^2*p^2*q^3*log(d) + 162*(f*x + e)^2*b^3*f
*g*h*p^2*q^3*log(d) - 1296*(f*x + e)*b^3*e*f*g*h*p^2*q^3*log(d) + 24*(f*x + e)^3*b^3*h^2*p^2*q^3*log(d) - 162*
(f*x + e)^2*b^3*e*h^2*p^2*q^3*log(d) + 648*(f*x + e)*b^3*e^2*h^2*p^2*q^3*log(d) + 648*(f*x + e)*b^3*f^2*g^2*p*
q^2*log(f*x + e)*log(c)*log(d) + 648*(f*x + e)^2*b^3*f*g*h*p*q^2*log(f*x + e)*log(c)*log(d) - 1296*(f*x + e)*b
^3*e*f*g*h*p*q^2*log(f*x + e)*log(c)*log(d) + 216*(f*x + e)^3*b^3*h^2*p*q^2*log(f*x + e)*log(c)*log(d) - 648*(
f*x + e)^2*b^3*e*h^2*p*q^2*log(f*x + e)*log(c)*log(d) + 648*(f*x + e)*b^3*e^2*h^2*p*q^2*log(f*x + e)*log(c)*lo
g(d) - 324*(f*x + e)*b^3*f^2*g^2*p*q^3*log(d)^2 - 162*(f*x + e)^2*b^3*f*g*h*p*q^3*log(d)^2 + 648*(f*x + e)*b^3
*e*f*g*h*p*q^3*log(d)^2 - 36*(f*x + e)^3*b^3*h^2*p*q^3*log(d)^2 + 162*(f*x + e)^2*b^3*e*h^2*p*q^3*log(d)^2 - 3
24*(f*x + e)*b^3*e^2*h^2*p*q^3*log(d)^2 + 108*(f*x + e)*b^3*f^2*g^2*q^3*log(d)^3 + 108*(f*x + e)^2*b^3*f*g*h*q
^3*log(d)^3 - 216*(f*x + e)*b^3*e*f*g*h*q^3*log(d)^3 + 36*(f*x + e)^3*b^3*h^2*q^3*log(d)^3 - 108*(f*x + e)^2*b
^3*e*h^2*q^3*log(d)^3 + 108*(f*x + e)*b^3*e^2*h^2*q^3*log(d)^3 - 648*(f*x + e)*a*b^2*f^2*g^2*p^2*q^2*log(f*x +
e) - 324*(f*x + e)^2*a*b^2*f*g*h*p^2*q^2*log(f*x + e) + 1296*(f*x + e)*a*b^2*e*f*g*h*p^2*q^2*log(f*x + e) - 7
2*(f*x + e)^3*a*b^2*h^2*p^2*q^2*log(f*x + e) + 324*(f*x + e)^2*a*b^2*e*h^2*p^2*q^2*log(f*x + e) - 648*(f*x + e
)*a*b^2*e^2*h^2*p^2*q^2*log(f*x + e) + 648*(f*x + e)*b^3*f^2*g^2*p^2*q^2*log(c) + 162*(f*x + e)^2*b^3*f*g*h*p^
2*q^2*log(c) - 1296*(f*x + e)*b^3*e*f*g*h*p^2*q^2*log(c) + 24*(f*x + e)^3*b^3*h^2*p^2*q^2*log(c) - 162*(f*x +
e)^2*b^3*e*h^2*p^2*q^2*log(c) + 648*(f*x + e)*b^3*e^2*h^2*p^2*q^2*log(c) + 324*(f*x + e)*b^3*f^2*g^2*p*q*log(f
*x + e)*log(c)^2 + 324*(f*x + e)^2*b^3*f*g*h*p*q*log(f*x + e)*log(c)^2 - 648*(f*x + e)*b^3*e*f*g*h*p*q*log(f*x
+ e)*log(c)^2 + 108*(f*x + e)^3*b^3*h^2*p*q*log(f*x + e)*log(c)^2 - 324*(f*x + e)^2*b^3*e*h^2*p*q*log(f*x + e
)*log(c)^2 + 324*(f*x + e)*b^3*e^2*h^2*p*q*log(f*x + e)*log(c)^2 + 648*(f*x + e)*a*b^2*f^2*g^2*p*q^2*log(f*x +
e)*log(d) + 648*(f*x + e)^2*a*b^2*f*g*h*p*q^2*log(f*x + e)*log(d) - 1296*(f*x + e)*a*b^2*e*f*g*h*p*q^2*log(f*
x + e)*log(d) + 216*(f*x + e)^3*a*b^2*h^2*p*q^2*log(f*x + e)*log(d) - 648*(f*x + e)^2*a*b^2*e*h^2*p*q^2*log(f*
x + e)*log(d) + 648*(f*x + e)*a*b^2*e^2*h^2*p*q^2*log(f*x + e)*log(d) - 648*(f*x + e)*b^3*f^2*g^2*p*q^2*log(c)
*log(d) - 324*(f*x + e)^2*b^3*f*g*h*p*q^2*log(c)*log(d) + 1296*(f*x + e)*b^3*e*f*g*h*p*q^2*log(c)*log(d) - 72*
(f*x + e)^3*b^3*h^2*p*q^2*log(c)*log(d) + 324*(f*x + e)^2*b^3*e*h^2*p*q^2*log(c)*log(d) - 648*(f*x + e)*b^3*e^
2*h^2*p*q^2*log(c)*log(d) + 324*(f*x + e)*b^3*f^2*g^2*q^2*log(c)*log(d)^2 + 324*(f*x + e)^2*b^3*f*g*h*q^2*log(
c)*log(d)^2 - 648*(f*x + e)*b^3*e*f*g*h*q^2*log(c)*log(d)^2 + 108*(f*x + e)^3*b^3*h^2*q^2*log(c)*log(d)^2 - 32
4*(f*x + e)^2*b^3*e*h^2*q^2*log(c)*log(d)^2 + 324*(f*x + e)*b^3*e^2*h^2*q^2*log(c)*log(d)^2 + 648*(f*x + e)*a*
b^2*f^2*g^2*p^2*q^2 + 162*(f*x + e)^2*a*b^2*f*g*h*p^2*q^2 - 1296*(f*x + e)*a*b^2*e*f*g*h*p^2*q^2 + 24*(f*x + e
)^3*a*b^2*h^2*p^2*q^2 - 162*(f*x + e)^2*a*b^2*e*h^2*p^2*q^2 + 648*(f*x + e)*a*b^2*e^2*h^2*p^2*q^2 + 648*(f*x +
e)*a*b^2*f^2*g^2*p*q*log(f*x + e)*log(c) + 648*(f*x + e)^2*a*b^2*f*g*h*p*q*log(f*x + e)*log(c) - 1296*(f*x +
e)*a*b^2*e*f*g*h*p*q*log(f*x + e)*log(c) + 216*(f*x + e)^3*a*b^2*h^2*p*q*log(f*x + e)*log(c) - 648*(f*x + e)^2
*a*b^2*e*h^2*p*q*log(f*x + e)*log(c) + 648*(f*x + e)*a*b^2*e^2*h^2*p*q*log(f*x + e)*log(c) - 324*(f*x + e)*b^3
*f^2*g^2*p*q*log(c)^2 - 162*(f*x + e)^2*b^3*f*g*h*p*q*log(c)^2 + 648*(f*x + e)*b^3*e*f*g*h*p*q*log(c)^2 - 36*(
f*x + e)^3*b^3*h^2*p*q*log(c)^2 + 162*(f*x + e)^2*b^3*e*h^2*p*q*log(c)^2 - 324*(f*x + e)*b^3*e^2*h^2*p*q*log(c
)^2 - 648*(f*x + e)*a*b^2*f^2*g^2*p*q^2*log(d) - 324*(f*x + e)^2*a*b^2*f*g*h*p*q^2*log(d) + 1296*(f*x + e)*a*b
^2*e*f*g*h*p*q^2*log(d) - 72*(f*x + e)^3*a*b^2*h^2*p*q^2*log(d) + 324*(f*x + e)^2*a*b^2*e*h^2*p*q^2*log(d) - 6
48*(f*x + e)*a*b^2*e^2*h^2*p*q^2*log(d) + 324*(f*x + e)*b^3*f^2*g^2*q*log(c)^2*log(d) + 324*(f*x + e)^2*b^3*f*
g*h*q*log(c)^2*log(d) - 648*(f*x + e)*b^3*e*f*g*h*q*log(c)^2*log(d) + 108*(f*x + e)^3*b^3*h^2*q*log(c)^2*log(d
) - 324*(f*x + e)^2*b^3*e*h^2*q*log(c)^2*log(d) + 324*(f*x + e)*b^3*e^2*h^2*q*log(c)^2*log(d) + 324*(f*x + e)*
a*b^2*f^2*g^2*q^2*log(d)^2 + 324*(f*x + e)^2*a*b^2*f*g*h*q^2*log(d)^2 - 648*(f*x + e)*a*b^2*e*f*g*h*q^2*log(d)
^2 + 108*(f*x + e)^3*a*b^2*h^2*q^2*log(d)^2 - 324*(f*x + e)^2*a*b^2*e*h^2*q^2*log(d)^2 + 324*(f*x + e)*a*b^2*e
^2*h^2*q^2*log(d)^2 + 324*(f*x + e)*a^2*b*f^2*g^2*p*q*log(f*x + e) + 324*(f*x + e)^2*a^2*b*f*g*h*p*q*log(f*x +
e) - 648*(f*x + e)*a^2*b*e*f*g*h*p*q*log(f*x + e) + 108*(f*x + e)^3*a^2*b*h^2*p*q*log(f*x + e) - 324*(f*x + e
)^2*a^2*b*e*h^2*p*q*log(f*x + e) + 324*(f*x + e)*a^2*b*e^2*h^2*p*q*log(f*x + e) - 648*(f*x + e)*a*b^2*f^2*g^2*
p*q*log(c) - 324*(f*x + e)^2*a*b^2*f*g*h*p*q*log(c) + 1296*(f*x + e)*a*b^2*e*f*g*h*p*q*log(c) - 72*(f*x + e)^3
*a*b^2*h^2*p*q*log(c) + 324*(f*x + e)^2*a*b^2*e*h^2*p*q*log(c) - 648*(f*x + e)*a*b^2*e^2*h^2*p*q*log(c) + 108*
(f*x + e)*b^3*f^2*g^2*log(c)^3 + 108*(f*x + e)^2*b^3*f*g*h*log(c)^3 - 216*(f*x + e)*b^3*e*f*g*h*log(c)^3 + 36*
(f*x + e)^3*b^3*h^2*log(c)^3 - 108*(f*x + e)^2*b^3*e*h^2*log(c)^3 + 108*(f*x + e)*b^3*e^2*h^2*log(c)^3 + 648*(
f*x + e)*a*b^2*f^2*g^2*q*log(c)*log(d) + 648*(f*x + e)^2*a*b^2*f*g*h*q*log(c)*log(d) - 1296*(f*x + e)*a*b^2*e*
f*g*h*q*log(c)*log(d) + 216*(f*x + e)^3*a*b^2*h^2*q*log(c)*log(d) - 648*(f*x + e)^2*a*b^2*e*h^2*q*log(c)*log(d
) + 648*(f*x + e)*a*b^2*e^2*h^2*q*log(c)*log(d) - 324*(f*x + e)*a^2*b*f^2*g^2*p*q - 162*(f*x + e)^2*a^2*b*f*g*
h*p*q + 648*(f*x + e)*a^2*b*e*f*g*h*p*q - 36*(f*x + e)^3*a^2*b*h^2*p*q + 162*(f*x + e)^2*a^2*b*e*h^2*p*q - 324
*(f*x + e)*a^2*b*e^2*h^2*p*q + 324*(f*x + e)*a*b^2*f^2*g^2*log(c)^2 + 324*(f*x + e)^2*a*b^2*f*g*h*log(c)^2 - 6
48*(f*x + e)*a*b^2*e*f*g*h*log(c)^2 + 108*(f*x + e)^3*a*b^2*h^2*log(c)^2 - 324*(f*x + e)^2*a*b^2*e*h^2*log(c)^
2 + 324*(f*x + e)*a*b^2*e^2*h^2*log(c)^2 + 324*(f*x + e)*a^2*b*f^2*g^2*q*log(d) + 324*(f*x + e)^2*a^2*b*f*g*h*
q*log(d) - 648*(f*x + e)*a^2*b*e*f*g*h*q*log(d) + 108*(f*x + e)^3*a^2*b*h^2*q*log(d) - 324*(f*x + e)^2*a^2*b*e
*h^2*q*log(d) + 324*(f*x + e)*a^2*b*e^2*h^2*q*log(d) + 324*(f*x + e)*a^2*b*f^2*g^2*log(c) + 324*(f*x + e)^2*a^
2*b*f*g*h*log(c) - 648*(f*x + e)*a^2*b*e*f*g*h*log(c) + 108*(f*x + e)^3*a^2*b*h^2*log(c) - 324*(f*x + e)^2*a^2
*b*e*h^2*log(c) + 324*(f*x + e)*a^2*b*e^2*h^2*log(c) + 108*(f*x + e)*a^3*f^2*g^2 + 108*(f*x + e)^2*a^3*f*g*h -
216*(f*x + e)*a^3*e*f*g*h + 36*(f*x + e)^3*a^3*h^2 - 108*(f*x + e)^2*a^3*e*h^2 + 108*(f*x + e)*a^3*e^2*h^2)/f
^3

Mupad [B] (verification not implemented)

Time = 2.34 (sec) , antiderivative size = 1400, normalized size of antiderivative = 2.85 \[ \int (g+h x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^3 \, dx=\text {Too large to display} \]

[In]

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

[Out]

x*((18*a^3*f^2*g^2 - 66*b^3*e^2*h^2*p^3*q^3 - 108*b^3*f^2*g^2*p^3*q^3 + 36*a^3*e*f*g*h + 36*a*b^2*e^2*h^2*p^2*
q^2 + 108*a*b^2*f^2*g^2*p^2*q^2 - 54*a^2*b*f^2*g^2*p*q + 162*b^3*e*f*g*h*p^3*q^3 - 108*a*b^2*e*f*g*h*p^2*q^2)/
(18*f^2) - (e*((h*(6*a^3*e*h + 12*a^3*f*g + 5*b^3*e*h*p^3*q^3 - 9*b^3*f*g*p^3*q^3 - 18*a^2*b*f*g*p*q - 6*a*b^2
*e*h*p^2*q^2 + 18*a*b^2*f*g*p^2*q^2))/(6*f) - (e*h^2*(9*a^3 - 2*b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 9*a^2*b*p*q))/
(9*f)))/f) + log(c*(d*(e + f*x)^p)^q)^2*(x^2*((3*b^2*h*(a*e*h + 2*a*f*g - b*f*g*p*q))/(2*f) - (b^2*e*h^2*(3*a
- b*p*q))/(2*f)) - x*((e*((3*b^2*h*(a*e*h + 2*a*f*g - b*f*g*p*q))/f - (b^2*e*h^2*(3*a - b*p*q))/f))/f - (3*b^2
*g*(2*a*e*h + a*f*g - b*f*g*p*q))/f) + (e*(6*a*b^2*e^2*h^2 + 18*a*b^2*f^2*g^2 - 11*b^3*e^2*h^2*p*q - 18*b^3*f^
2*g^2*p*q - 18*a*b^2*e*f*g*h + 27*b^3*e*f*g*h*p*q))/(6*f^3) + (b^2*h^2*x^3*(3*a - b*p*q))/3) + log(c*(d*(e + f
*x)^p)^q)^3*(b^3*g^2*x + (b^3*h^2*x^3)/3 + (e*(b^3*e^2*h^2 + 3*b^3*f^2*g^2 - 3*b^3*e*f*g*h))/(3*f^3) + b^3*g*h
*x^2) + x^2*((h*(6*a^3*e*h + 12*a^3*f*g + 5*b^3*e*h*p^3*q^3 - 9*b^3*f*g*p^3*q^3 - 18*a^2*b*f*g*p*q - 6*a*b^2*e
*h*p^2*q^2 + 18*a*b^2*f*g*p^2*q^2))/(12*f) - (e*h^2*(9*a^3 - 2*b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 9*a^2*b*p*q))/(
18*f)) + (log(e + f*x)*(85*b^3*e^3*h^2*p^3*q^3 - 66*a*b^2*e^3*h^2*p^2*q^2 + 108*b^3*e*f^2*g^2*p^3*q^3 + 18*a^2
*b*e^3*h^2*p*q - 108*a*b^2*e*f^2*g^2*p^2*q^2 + 54*a^2*b*e*f^2*g^2*p*q - 189*b^3*e^2*f*g*h*p^3*q^3 + 162*a*b^2*
e^2*f*g*h*p^2*q^2 - 54*a^2*b*e^2*f*g*h*p*q))/(18*f^3) + (h^2*x^3*(9*a^3 - 2*b^3*p^3*q^3 + 6*a*b^2*p^2*q^2 - 9*
a^2*b*p*q))/27 + (log(c*(d*(e + f*x)^p)^q)*(x^3*(f*(9*a^2*b*f*g*h - (5*b^3*e*h^2*p^2*q^2)/2 + 3*a*b^2*e*h^2*p*
q + (9*b^3*f*g*h*p^2*q^2)/2 - 9*a*b^2*f*g*h*p*q) + (b*e*f*h^2*(9*a^2 + 2*b^2*p^2*q^2 - 6*a*b*p*q))/3) + x^2*(e
*(9*a^2*b*f*g*h - (5*b^3*e*h^2*p^2*q^2)/2 + 3*a*b^2*e*h^2*p*q + (9*b^3*f*g*h*p^2*q^2)/2 - 9*a*b^2*f*g*h*p*q) +
9*a^2*b*f^2*g^2 + 11*b^3*e^2*h^2*p^2*q^2 + 18*b^3*f^2*g^2*p^2*q^2 - 6*a*b^2*e^2*h^2*p*q - 18*a*b^2*f^2*g^2*p*
q - 27*b^3*e*f*g*h*p^2*q^2 + 18*a*b^2*e*f*g*h*p*q) + (e*x*(9*a^2*b*f^2*g^2 + 11*b^3*e^2*h^2*p^2*q^2 + 18*b^3*f
^2*g^2*p^2*q^2 - 6*a*b^2*e^2*h^2*p*q - 18*a*b^2*f^2*g^2*p*q - 27*b^3*e*f*g*h*p^2*q^2 + 18*a*b^2*e*f*g*h*p*q))/
f + (b*f^2*h^2*x^4*(9*a^2 + 2*b^2*p^2*q^2 - 6*a*b*p*q))/3))/(3*f*(e + f*x))