Integrand size = 28, antiderivative size = 409 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {2 b^2 (f g-e h)^3 p^2 q^2 x}{f^3}+\frac {3 b^2 h (f g-e h)^2 p^2 q^2 (e+f x)^2}{4 f^4}+\frac {2 b^2 h^2 (f g-e h) p^2 q^2 (e+f x)^3}{9 f^4}+\frac {b^2 h^3 p^2 q^2 (e+f x)^4}{32 f^4}+\frac {b^2 (f g-e h)^4 p^2 q^2 \log ^2(e+f x)}{4 f^4 h}-\frac {2 b (f g-e h)^3 p q (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{f^4}-\frac {3 b h (f g-e h)^2 p q (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4}-\frac {2 b h^2 (f g-e h) p q (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{3 f^4}-\frac {b h^3 p q (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{8 f^4}-\frac {b (f g-e h)^4 p q \log (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{2 f^4 h}+\frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h} \]
2*b^2*(-e*h+f*g)^3*p^2*q^2*x/f^3+3/4*b^2*h*(-e*h+f*g)^2*p^2*q^2*(f*x+e)^2/ f^4+2/9*b^2*h^2*(-e*h+f*g)*p^2*q^2*(f*x+e)^3/f^4+1/32*b^2*h^3*p^2*q^2*(f*x +e)^4/f^4+1/4*b^2*(-e*h+f*g)^4*p^2*q^2*ln(f*x+e)^2/f^4/h-2*b*(-e*h+f*g)^3* p*q*(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^4-3/2*b*h*(-e*h+f*g)^2*p*q*(f*x+ e)^2*(a+b*ln(c*(d*(f*x+e)^p)^q))/f^4-2/3*b*h^2*(-e*h+f*g)*p*q*(f*x+e)^3*(a +b*ln(c*(d*(f*x+e)^p)^q))/f^4-1/8*b*h^3*p*q*(f*x+e)^4*(a+b*ln(c*(d*(f*x+e) ^p)^q))/f^4-1/2*b*(-e*h+f*g)^4*p*q*ln(f*x+e)*(a+b*ln(c*(d*(f*x+e)^p)^q))/f ^4/h+1/4*(h*x+g)^4*(a+b*ln(c*(d*(f*x+e)^p)^q))^2/h
Time = 0.20 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.98 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\frac {288 (f g-e h)^3 (e+f x) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+432 h (f g-e h)^2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+288 h^2 (f g-e h) (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2+72 h^3 (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2-576 b (f g-e h)^3 p q \left (f (a-b p q) x+b (e+f x) \log \left (c \left (d (e+f x)^p\right )^q\right )\right )+216 b h (f g-e h)^2 p q \left (b f p q x (2 e+f x)-2 (e+f x)^2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+64 b h^2 (f g-e h) p q \left (b f p q x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )+9 b h^3 p q \left (b f p q x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 (e+f x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )\right )}{288 f^4} \]
(288*(f*g - e*h)^3*(e + f*x)*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 432*h*(f *g - e*h)^2*(e + f*x)^2*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 288*h^2*(f*g - e*h)*(e + f*x)^3*(a + b*Log[c*(d*(e + f*x)^p)^q])^2 + 72*h^3*(e + f*x)^4 *(a + b*Log[c*(d*(e + f*x)^p)^q])^2 - 576*b*(f*g - e*h)^3*p*q*(f*(a - b*p* q)*x + b*(e + f*x)*Log[c*(d*(e + f*x)^p)^q]) + 216*b*h*(f*g - e*h)^2*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])) + 64*b*h^2*(f*g - e*h)*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])) + 9*b*h^3*p*q*(b*f*p*q*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3) - 4*(e + f*x)^4*(a + b*Log[c*(d*(e + f* x)^p)^q])))/(288*f^4)
Time = 1.14 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2895, 2845, 2858, 27, 2772, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2895 |
| \(\displaystyle \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\frac {b f p q \int \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{e+f x}dx}{2 h}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\frac {b p q \int \frac {\left (f \left (g-\frac {e h}{f}\right )+h (e+f x)\right )^4 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{f^4 (e+f x)}d(e+f x)}{2 h}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\frac {b p q \int \frac {(f g-e h+h (e+f x))^4 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )}{e+f x}d(e+f x)}{2 f^4 h}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\frac {b p q \left (-b p q \int \left (\frac {1}{4} (e+f x)^3 h^4+\frac {4}{3} (f g-e h) (e+f x)^2 h^3+3 (f g-e h)^2 (e+f x) h^2+4 (f g-e h)^3 h+\frac {(f g-e h)^4 \log (e+f x)}{e+f x}\right )d(e+f x)+\frac {4}{3} h^3 (e+f x)^3 (f g-e h) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+3 h^2 (e+f x)^2 (f g-e h)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+(f g-e h)^4 \log (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+4 h (e+f x) (f g-e h)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+\frac {1}{4} h^4 (e+f x)^4 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )\right )}{2 f^4 h}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(g+h x)^4 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2}{4 h}-\frac {b p q \left (\frac {4}{3} h^3 (e+f x)^3 (f g-e h) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+3 h^2 (e+f x)^2 (f g-e h)^2 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+(f g-e h)^4 \log (e+f x) \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+4 h (e+f x) (f g-e h)^3 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )+\frac {1}{4} h^4 (e+f x)^4 \left (a+b \log \left (c d^q (e+f x)^{p q}\right )\right )-b p q \left (\frac {4}{9} h^3 (e+f x)^3 (f g-e h)+\frac {3}{2} h^2 (e+f x)^2 (f g-e h)^2+4 h (e+f x) (f g-e h)^3+\frac {1}{2} (f g-e h)^4 \log ^2(e+f x)+\frac {1}{16} h^4 (e+f x)^4\right )\right )}{2 f^4 h}\) |
-1/2*(b*p*q*(-(b*p*q*(4*h*(f*g - e*h)^3*(e + f*x) + (3*h^2*(f*g - e*h)^2*( e + f*x)^2)/2 + (4*h^3*(f*g - e*h)*(e + f*x)^3)/9 + (h^4*(e + f*x)^4)/16 + ((f*g - e*h)^4*Log[e + f*x]^2)/2)) + 4*h*(f*g - e*h)^3*(e + f*x)*(a + b*L og[c*d^q*(e + f*x)^(p*q)]) + 3*h^2*(f*g - e*h)^2*(e + f*x)^2*(a + b*Log[c* d^q*(e + f*x)^(p*q)]) + (4*h^3*(f*g - e*h)*(e + f*x)^3*(a + b*Log[c*d^q*(e + f*x)^(p*q)]))/3 + (h^4*(e + f*x)^4*(a + b*Log[c*d^q*(e + f*x)^(p*q)]))/ 4 + (f*g - e*h)^4*Log[e + f*x]*(a + b*Log[c*d^q*(e + f*x)^(p*q)])))/(f^4*h ) + ((g + h*x)^4*(a + b*Log[c*(d*(e + f*x)^p)^q])^2)/(4*h)
3.5.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
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]]
Leaf count of result is larger than twice the leaf count of optimal. \(1536\) vs. \(2(391)=782\).
Time = 16.24 (sec) , antiderivative size = 1537, normalized size of antiderivative = 3.76
-1/288*(1632*ln(f*x+e)*b^2*e^3*f*g*h^2*p^2*q^2-864*x*ln(c*(d*(f*x+e)^p)^q) *b^2*e*f^3*g^2*h*p*q+576*x*a*b*e^2*f^2*g*h^2*p*q-864*x*a*b*e*f^3*g^2*h*p*q -576*ln(f*x+e)*a*b*e^3*f*g*h^2*p*q+864*ln(f*x+e)*a*b*e^2*f^2*g^2*h*p*q-288 *x^2*ln(c*(d*(f*x+e)^p)^q)*b^2*e*f^3*g*h^2*p*q-288*x^2*a*b*e*f^3*g*h^2*p*q +576*x*ln(c*(d*(f*x+e)^p)^q)*b^2*e^2*f^2*g*h^2*p*q-2160*ln(f*x+e)*b^2*e^2* f^2*g^2*h*p^2*q^2-1152*ln(f*x+e)*a*b*e*f^3*g^3*p*q-48*x^3*ln(c*(d*(f*x+e)^ p)^q)*b^2*e*f^3*h^3*p*q+192*x^3*ln(c*(d*(f*x+e)^p)^q)*b^2*f^4*g*h^2*p*q+24 0*x^2*b^2*e*f^3*g*h^2*p^2*q^2-48*x^3*a*b*e*f^3*h^3*p*q+192*x^3*a*b*f^4*g*h ^2*p*q+72*x^2*ln(c*(d*(f*x+e)^p)^q)*b^2*e^2*f^2*h^3*p*q+432*x^2*ln(c*(d*(f *x+e)^p)^q)*b^2*f^4*g^2*h*p*q-1056*x*b^2*e^2*f^2*g*h^2*p^2*q^2+864*a*b*e^2 *f^2*g^2*h*p*q-576*a*b*e^3*f*g*h^2*p*q+1296*x*b^2*e*f^3*g^2*h*p^2*q^2+72*x ^2*a*b*e^2*f^2*h^3*p*q+432*x^2*a*b*f^4*g^2*h*p*q-144*x*ln(c*(d*(f*x+e)^p)^ q)*b^2*e^3*f*h^3*p*q-144*x*a*b*e^3*f*h^3*p*q-576*ln(c*(d*(f*x+e)^p)^q)*b^2 *e^3*f*g*h^2*p*q+864*ln(c*(d*(f*x+e)^p)^q)*b^2*e^2*f^2*g^2*h*p*q+36*x^4*ln (c*(d*(f*x+e)^p)^q)*b^2*f^4*h^3*p*q+28*x^3*b^2*e*f^3*h^3*p^2*q^2-64*x^3*b^ 2*f^4*g*h^2*p^2*q^2-576*a*b*e*f^3*g^3*p*q+1056*b^2*e^3*f*g*h^2*p^2*q^2-129 6*b^2*e^2*f^2*g^2*h*p^2*q^2+36*x^4*a*b*f^4*h^3*p*q-78*x^2*b^2*e^2*f^2*h^3* p^2*q^2-216*x^2*b^2*f^4*g^2*h*p^2*q^2+300*x*b^2*e^3*f*h^3*p^2*q^2-576*x^3* ln(c*(d*(f*x+e)^p)^q)*a*b*f^4*g*h^2+576*x*ln(c*(d*(f*x+e)^p)^q)*b^2*f^4*g^ 3*p*q-864*x^2*ln(c*(d*(f*x+e)^p)^q)*a*b*f^4*g^2*h+576*x*a*b*f^4*g^3*p*q...
Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (391) = 782\).
Time = 0.35 (sec) , antiderivative size = 1742, normalized size of antiderivative = 4.26 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]
1/288*(9*(b^2*f^4*h^3*p^2*q^2 - 4*a*b*f^4*h^3*p*q + 8*a^2*f^4*h^3)*x^4 + 4 *(72*a^2*f^4*g*h^2 + (16*b^2*f^4*g*h^2 - 7*b^2*e*f^3*h^3)*p^2*q^2 - 12*(4* a*b*f^4*g*h^2 - a*b*e*f^3*h^3)*p*q)*x^3 + 6*(72*a^2*f^4*g^2*h + (36*b^2*f^ 4*g^2*h - 40*b^2*e*f^3*g*h^2 + 13*b^2*e^2*f^2*h^3)*p^2*q^2 - 12*(6*a*b*f^4 *g^2*h - 4*a*b*e*f^3*g*h^2 + a*b*e^2*f^2*h^3)*p*q)*x^2 + 72*(b^2*f^4*h^3*p ^2*q^2*x^4 + 4*b^2*f^4*g*h^2*p^2*q^2*x^3 + 6*b^2*f^4*g^2*h*p^2*q^2*x^2 + 4 *b^2*f^4*g^3*p^2*q^2*x + (4*b^2*e*f^3*g^3 - 6*b^2*e^2*f^2*g^2*h + 4*b^2*e^ 3*f*g*h^2 - b^2*e^4*h^3)*p^2*q^2)*log(f*x + e)^2 + 72*(b^2*f^4*h^3*x^4 + 4 *b^2*f^4*g*h^2*x^3 + 6*b^2*f^4*g^2*h*x^2 + 4*b^2*f^4*g^3*x)*log(c)^2 + 72* (b^2*f^4*h^3*q^2*x^4 + 4*b^2*f^4*g*h^2*q^2*x^3 + 6*b^2*f^4*g^2*h*q^2*x^2 + 4*b^2*f^4*g^3*q^2*x)*log(d)^2 + 12*(24*a^2*f^4*g^3 + (48*b^2*f^4*g^3 - 10 8*b^2*e*f^3*g^2*h + 88*b^2*e^2*f^2*g*h^2 - 25*b^2*e^3*f*h^3)*p^2*q^2 - 12* (4*a*b*f^4*g^3 - 6*a*b*e*f^3*g^2*h + 4*a*b*e^2*f^2*g*h^2 - a*b*e^3*f*h^3)* p*q)*x - 12*((48*b^2*e*f^3*g^3 - 108*b^2*e^2*f^2*g^2*h + 88*b^2*e^3*f*g*h^ 2 - 25*b^2*e^4*h^3)*p^2*q^2 + 3*(b^2*f^4*h^3*p^2*q^2 - 4*a*b*f^4*h^3*p*q)* x^4 - 4*(12*a*b*f^4*g*h^2*p*q - (4*b^2*f^4*g*h^2 - b^2*e*f^3*h^3)*p^2*q^2) *x^3 - 12*(4*a*b*e*f^3*g^3 - 6*a*b*e^2*f^2*g^2*h + 4*a*b*e^3*f*g*h^2 - a*b *e^4*h^3)*p*q - 6*(12*a*b*f^4*g^2*h*p*q - (6*b^2*f^4*g^2*h - 4*b^2*e*f^3*g *h^2 + b^2*e^2*f^2*h^3)*p^2*q^2)*x^2 - 12*(4*a*b*f^4*g^3*p*q - (4*b^2*f^4* g^3 - 6*b^2*e*f^3*g^2*h + 4*b^2*e^2*f^2*g*h^2 - b^2*e^3*f*h^3)*p^2*q^2)...
Leaf count of result is larger than twice the leaf count of optimal. 1421 vs. \(2 (394) = 788\).
Time = 5.39 (sec) , antiderivative size = 1421, normalized size of antiderivative = 3.47 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]
Piecewise((a**2*g**3*x + 3*a**2*g**2*h*x**2/2 + a**2*g*h**2*x**3 + a**2*h* *3*x**4/4 - a*b*e**4*h**3*log(c*(d*(e + f*x)**p)**q)/(2*f**4) + 2*a*b*e**3 *g*h**2*log(c*(d*(e + f*x)**p)**q)/f**3 + a*b*e**3*h**3*p*q*x/(2*f**3) - 3 *a*b*e**2*g**2*h*log(c*(d*(e + f*x)**p)**q)/f**2 - 2*a*b*e**2*g*h**2*p*q*x /f**2 - a*b*e**2*h**3*p*q*x**2/(4*f**2) + 2*a*b*e*g**3*log(c*(d*(e + f*x)* *p)**q)/f + 3*a*b*e*g**2*h*p*q*x/f + a*b*e*g*h**2*p*q*x**2/f + a*b*e*h**3* p*q*x**3/(6*f) - 2*a*b*g**3*p*q*x + 2*a*b*g**3*x*log(c*(d*(e + f*x)**p)**q ) - 3*a*b*g**2*h*p*q*x**2/2 + 3*a*b*g**2*h*x**2*log(c*(d*(e + f*x)**p)**q) - 2*a*b*g*h**2*p*q*x**3/3 + 2*a*b*g*h**2*x**3*log(c*(d*(e + f*x)**p)**q) - a*b*h**3*p*q*x**4/8 + a*b*h**3*x**4*log(c*(d*(e + f*x)**p)**q)/2 + 25*b* *2*e**4*h**3*p*q*log(c*(d*(e + f*x)**p)**q)/(24*f**4) - b**2*e**4*h**3*log (c*(d*(e + f*x)**p)**q)**2/(4*f**4) - 11*b**2*e**3*g*h**2*p*q*log(c*(d*(e + f*x)**p)**q)/(3*f**3) + b**2*e**3*g*h**2*log(c*(d*(e + f*x)**p)**q)**2/f **3 - 25*b**2*e**3*h**3*p**2*q**2*x/(24*f**3) + b**2*e**3*h**3*p*q*x*log(c *(d*(e + f*x)**p)**q)/(2*f**3) + 9*b**2*e**2*g**2*h*p*q*log(c*(d*(e + f*x) **p)**q)/(2*f**2) - 3*b**2*e**2*g**2*h*log(c*(d*(e + f*x)**p)**q)**2/(2*f* *2) + 11*b**2*e**2*g*h**2*p**2*q**2*x/(3*f**2) - 2*b**2*e**2*g*h**2*p*q*x* log(c*(d*(e + f*x)**p)**q)/f**2 + 13*b**2*e**2*h**3*p**2*q**2*x**2/(48*f** 2) - b**2*e**2*h**3*p*q*x**2*log(c*(d*(e + f*x)**p)**q)/(4*f**2) - 2*b**2* e*g**3*p*q*log(c*(d*(e + f*x)**p)**q)/f + b**2*e*g**3*log(c*(d*(e + f*x...
Leaf count of result is larger than twice the leaf count of optimal. 895 vs. \(2 (391) = 782\).
Time = 0.23 (sec) , antiderivative size = 895, normalized size of antiderivative = 2.19 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]
1/4*b^2*h^3*x^4*log(((f*x + e)^p*d)^q*c)^2 + 1/2*a*b*h^3*x^4*log(((f*x + e )^p*d)^q*c) + b^2*g*h^2*x^3*log(((f*x + e)^p*d)^q*c)^2 + 1/4*a^2*h^3*x^4 - 2*a*b*f*g^3*p*q*(x/f - e*log(f*x + e)/f^2) - 1/24*a*b*f*h^3*p*q*(12*e^4*l og(f*x + e)/f^5 + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/f^4) + 1/3*a*b*f*g*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*b*f*g^2*h*p*q*(2*e^2*log(f*x + e)/f^3 + (f*x^2 - 2*e* x)/f^2) + 2*a*b*g*h^2*x^3*log(((f*x + e)^p*d)^q*c) + 3/2*b^2*g^2*h*x^2*log (((f*x + e)^p*d)^q*c)^2 + a^2*g*h^2*x^3 + 3*a*b*g^2*h*x^2*log(((f*x + e)^p *d)^q*c) + b^2*g^3*x*log(((f*x + e)^p*d)^q*c)^2 + 3/2*a^2*g^2*h*x^2 + 2*a* b*g^3*x*log(((f*x + e)^p*d)^q*c) - (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)*b^2*g^3 - 3/4*(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)*b^2*g^2*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)*b^2*g*h^2 - 1/288*(12*f*p*q*(12*e^4*log(f*x + e)/f^5 + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/f^4)*log(((f*x + e)^p*d)^q*c) - (9*f^4*x^4 - 28*e*f^3*x^3 + 78*e^2*f^2*x^2 + 72*e^4*log( f*x + e)^2 - 300*e^3*f*x + 300*e^4*log(f*x + e))*p^2*q^2/f^4)*b^2*h^3 +...
Leaf count of result is larger than twice the leaf count of optimal. 3738 vs. \(2 (391) = 782\).
Time = 0.39 (sec) , antiderivative size = 3738, normalized size of antiderivative = 9.14 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=\text {Too large to display} \]
(f*x + e)*b^2*g^3*p^2*q^2*log(f*x + e)^2/f + 3/2*(f*x + e)^2*b^2*g^2*h*p^2 *q^2*log(f*x + e)^2/f^2 - 3*(f*x + e)*b^2*e*g^2*h*p^2*q^2*log(f*x + e)^2/f ^2 + (f*x + e)^3*b^2*g*h^2*p^2*q^2*log(f*x + e)^2/f^3 - 3*(f*x + e)^2*b^2* e*g*h^2*p^2*q^2*log(f*x + e)^2/f^3 + 3*(f*x + e)*b^2*e^2*g*h^2*p^2*q^2*log (f*x + e)^2/f^3 + 1/4*(f*x + e)^4*b^2*h^3*p^2*q^2*log(f*x + e)^2/f^4 - (f* x + e)^3*b^2*e*h^3*p^2*q^2*log(f*x + e)^2/f^4 + 3/2*(f*x + e)^2*b^2*e^2*h^ 3*p^2*q^2*log(f*x + e)^2/f^4 - (f*x + e)*b^2*e^3*h^3*p^2*q^2*log(f*x + e)^ 2/f^4 - 2*(f*x + e)*b^2*g^3*p^2*q^2*log(f*x + e)/f - 3/2*(f*x + e)^2*b^2*g ^2*h*p^2*q^2*log(f*x + e)/f^2 + 6*(f*x + e)*b^2*e*g^2*h*p^2*q^2*log(f*x + e)/f^2 - 2/3*(f*x + e)^3*b^2*g*h^2*p^2*q^2*log(f*x + e)/f^3 + 3*(f*x + e)^ 2*b^2*e*g*h^2*p^2*q^2*log(f*x + e)/f^3 - 6*(f*x + e)*b^2*e^2*g*h^2*p^2*q^2 *log(f*x + e)/f^3 - 1/8*(f*x + e)^4*b^2*h^3*p^2*q^2*log(f*x + e)/f^4 + 2/3 *(f*x + e)^3*b^2*e*h^3*p^2*q^2*log(f*x + e)/f^4 - 3/2*(f*x + e)^2*b^2*e^2* h^3*p^2*q^2*log(f*x + e)/f^4 + 2*(f*x + e)*b^2*e^3*h^3*p^2*q^2*log(f*x + e )/f^4 + 2*(f*x + e)*b^2*g^3*p*q^2*log(f*x + e)*log(d)/f + 3*(f*x + e)^2*b^ 2*g^2*h*p*q^2*log(f*x + e)*log(d)/f^2 - 6*(f*x + e)*b^2*e*g^2*h*p*q^2*log( f*x + e)*log(d)/f^2 + 2*(f*x + e)^3*b^2*g*h^2*p*q^2*log(f*x + e)*log(d)/f^ 3 - 6*(f*x + e)^2*b^2*e*g*h^2*p*q^2*log(f*x + e)*log(d)/f^3 + 6*(f*x + e)* b^2*e^2*g*h^2*p*q^2*log(f*x + e)*log(d)/f^3 + 1/2*(f*x + e)^4*b^2*h^3*p*q^ 2*log(f*x + e)*log(d)/f^4 - 2*(f*x + e)^3*b^2*e*h^3*p*q^2*log(f*x + e)*...
Time = 1.96 (sec) , antiderivative size = 1154, normalized size of antiderivative = 2.82 \[ \int (g+h x)^3 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^2 \, dx=x^3\,\left (\frac {h^2\,\left (6\,a^2\,e\,h+18\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-12\,a\,b\,f\,g\,p\,q\right )}{18\,f}-\frac {e\,h^3\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{24\,f}\right )+\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\,\left (\frac {x\,\left (\frac {e\,\left (\frac {e\,\left (\frac {4\,b\,h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {b\,e\,h^3\,\left (4\,a-b\,p\,q\right )}{f}\right )}{f}-\frac {6\,b\,g\,h\,\left (2\,a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{f}+\frac {4\,b\,g^2\,\left (3\,a\,e\,h+a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{2}+\frac {x^3\,\left (\frac {4\,b\,h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{3\,f}-\frac {b\,e\,h^3\,\left (4\,a-b\,p\,q\right )}{3\,f}\right )}{2}-\frac {x^2\,\left (\frac {e\,\left (\frac {4\,b\,h^2\,\left (a\,e\,h+3\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}-\frac {b\,e\,h^3\,\left (4\,a-b\,p\,q\right )}{f}\right )}{2\,f}-\frac {3\,b\,g\,h\,\left (2\,a\,e\,h+2\,a\,f\,g-b\,f\,g\,p\,q\right )}{f}\right )}{2}+\frac {b\,h^3\,x^4\,\left (4\,a-b\,p\,q\right )}{8}\right )+{\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}^2\,\left (b^2\,g^3\,x-\frac {e\,\left (b^2\,e^3\,h^3-4\,b^2\,e^2\,f\,g\,h^2+6\,b^2\,e\,f^2\,g^2\,h-4\,b^2\,f^3\,g^3\right )}{4\,f^4}+\frac {b^2\,h^3\,x^4}{4}+\frac {3\,b^2\,g^2\,h\,x^2}{2}+b^2\,g\,h^2\,x^3\right )+x\,\left (\frac {72\,a^2\,e\,f^2\,g^2\,h+24\,a^2\,f^3\,g^3-48\,a\,b\,f^3\,g^3\,p\,q-12\,b^2\,e^3\,h^3\,p^2\,q^2+48\,b^2\,e^2\,f\,g\,h^2\,p^2\,q^2-72\,b^2\,e\,f^2\,g^2\,h\,p^2\,q^2+48\,b^2\,f^3\,g^3\,p^2\,q^2}{24\,f^3}+\frac {e\,\left (\frac {e\,\left (\frac {h^2\,\left (6\,a^2\,e\,h+18\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-12\,a\,b\,f\,g\,p\,q\right )}{6\,f}-\frac {e\,h^3\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{8\,f}\right )}{f}-\frac {h\,\left (12\,a^2\,e\,f\,g\,h+12\,a^2\,f^2\,g^2-12\,a\,b\,f^2\,g^2\,p\,q+b^2\,e^2\,h^2\,p^2\,q^2-4\,b^2\,e\,f\,g\,h\,p^2\,q^2+6\,b^2\,f^2\,g^2\,p^2\,q^2\right )}{4\,f^2}\right )}{f}\right )-x^2\,\left (\frac {e\,\left (\frac {h^2\,\left (6\,a^2\,e\,h+18\,a^2\,f\,g-b^2\,e\,h\,p^2\,q^2+4\,b^2\,f\,g\,p^2\,q^2-12\,a\,b\,f\,g\,p\,q\right )}{6\,f}-\frac {e\,h^3\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{8\,f}\right )}{2\,f}-\frac {h\,\left (12\,a^2\,e\,f\,g\,h+12\,a^2\,f^2\,g^2-12\,a\,b\,f^2\,g^2\,p\,q+b^2\,e^2\,h^2\,p^2\,q^2-4\,b^2\,e\,f\,g\,h\,p^2\,q^2+6\,b^2\,f^2\,g^2\,p^2\,q^2\right )}{8\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (25\,b^2\,e^4\,h^3\,p^2\,q^2-88\,b^2\,e^3\,f\,g\,h^2\,p^2\,q^2+108\,b^2\,e^2\,f^2\,g^2\,h\,p^2\,q^2-48\,b^2\,e\,f^3\,g^3\,p^2\,q^2-12\,a\,b\,e^4\,h^3\,p\,q+48\,a\,b\,e^3\,f\,g\,h^2\,p\,q-72\,a\,b\,e^2\,f^2\,g^2\,h\,p\,q+48\,a\,b\,e\,f^3\,g^3\,p\,q\right )}{24\,f^4}+\frac {h^3\,x^4\,\left (8\,a^2-4\,a\,b\,p\,q+b^2\,p^2\,q^2\right )}{32} \]
x^3*((h^2*(6*a^2*e*h + 18*a^2*f*g - b^2*e*h*p^2*q^2 + 4*b^2*f*g*p^2*q^2 - 12*a*b*f*g*p*q))/(18*f) - (e*h^3*(8*a^2 + b^2*p^2*q^2 - 4*a*b*p*q))/(24*f) ) + log(c*(d*(e + f*x)^p)^q)*((x*((e*((e*((4*b*h^2*(a*e*h + 3*a*f*g - b*f* g*p*q))/f - (b*e*h^3*(4*a - b*p*q))/f))/f - (6*b*g*h*(2*a*e*h + 2*a*f*g - b*f*g*p*q))/f))/f + (4*b*g^2*(3*a*e*h + a*f*g - b*f*g*p*q))/f))/2 + (x^3*( (4*b*h^2*(a*e*h + 3*a*f*g - b*f*g*p*q))/(3*f) - (b*e*h^3*(4*a - b*p*q))/(3 *f)))/2 - (x^2*((e*((4*b*h^2*(a*e*h + 3*a*f*g - b*f*g*p*q))/f - (b*e*h^3*( 4*a - b*p*q))/f))/(2*f) - (3*b*g*h*(2*a*e*h + 2*a*f*g - b*f*g*p*q))/f))/2 + (b*h^3*x^4*(4*a - b*p*q))/8) + log(c*(d*(e + f*x)^p)^q)^2*(b^2*g^3*x - ( e*(b^2*e^3*h^3 - 4*b^2*f^3*g^3 + 6*b^2*e*f^2*g^2*h - 4*b^2*e^2*f*g*h^2))/( 4*f^4) + (b^2*h^3*x^4)/4 + (3*b^2*g^2*h*x^2)/2 + b^2*g*h^2*x^3) + x*((24*a ^2*f^3*g^3 - 12*b^2*e^3*h^3*p^2*q^2 + 48*b^2*f^3*g^3*p^2*q^2 + 72*a^2*e*f^ 2*g^2*h - 48*a*b*f^3*g^3*p*q - 72*b^2*e*f^2*g^2*h*p^2*q^2 + 48*b^2*e^2*f*g *h^2*p^2*q^2)/(24*f^3) + (e*((e*((h^2*(6*a^2*e*h + 18*a^2*f*g - b^2*e*h*p^ 2*q^2 + 4*b^2*f*g*p^2*q^2 - 12*a*b*f*g*p*q))/(6*f) - (e*h^3*(8*a^2 + b^2*p ^2*q^2 - 4*a*b*p*q))/(8*f)))/f - (h*(12*a^2*f^2*g^2 + b^2*e^2*h^2*p^2*q^2 + 6*b^2*f^2*g^2*p^2*q^2 + 12*a^2*e*f*g*h - 12*a*b*f^2*g^2*p*q - 4*b^2*e*f* g*h*p^2*q^2))/(4*f^2)))/f) - x^2*((e*((h^2*(6*a^2*e*h + 18*a^2*f*g - b^2*e *h*p^2*q^2 + 4*b^2*f*g*p^2*q^2 - 12*a*b*f*g*p*q))/(6*f) - (e*h^3*(8*a^2 + b^2*p^2*q^2 - 4*a*b*p*q))/(8*f)))/(2*f) - (h*(12*a^2*f^2*g^2 + b^2*e^2*...