Integrand size = 24, antiderivative size = 365 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {2 b^2 (e f-d g)^3 n^2 x}{e^3}+\frac {3 b^2 g (e f-d g)^2 n^2 (d+e x)^2}{4 e^4}+\frac {2 b^2 g^2 (e f-d g) n^2 (d+e x)^3}{9 e^4}+\frac {b^2 g^3 n^2 (d+e x)^4}{32 e^4}+\frac {b^2 (e f-d g)^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac {2 b (e f-d g)^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^4}-\frac {3 b g (e f-d g)^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4}-\frac {2 b g^2 (e f-d g) n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4}-\frac {b g^3 n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4}-\frac {b (e f-d g)^4 n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4 g}+\frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g} \]
2*b^2*(-d*g+e*f)^3*n^2*x/e^3+3/4*b^2*g*(-d*g+e*f)^2*n^2*(e*x+d)^2/e^4+2/9* b^2*g^2*(-d*g+e*f)*n^2*(e*x+d)^3/e^4+1/32*b^2*g^3*n^2*(e*x+d)^4/e^4+1/4*b^ 2*(-d*g+e*f)^4*n^2*ln(e*x+d)^2/e^4/g-2*b*(-d*g+e*f)^3*n*(e*x+d)*(a+b*ln(c* (e*x+d)^n))/e^4-3/2*b*g*(-d*g+e*f)^2*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^4 -2/3*b*g^2*(-d*g+e*f)*n*(e*x+d)^3*(a+b*ln(c*(e*x+d)^n))/e^4-1/8*b*g^3*n*(e *x+d)^4*(a+b*ln(c*(e*x+d)^n))/e^4-1/2*b*(-d*g+e*f)^4*n*ln(e*x+d)*(a+b*ln(c *(e*x+d)^n))/e^4/g+1/4*(g*x+f)^4*(a+b*ln(c*(e*x+d)^n))^2/g
Time = 0.17 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.99 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {288 (e f-d g)^3 (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2+432 g (e f-d g)^2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+288 g^2 (e f-d g) (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2+72 g^3 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2-576 b (e f-d g)^3 n \left (e (a-b n) x+b (d+e x) \log \left (c (d+e x)^n\right )\right )+216 b g (e f-d g)^2 n \left (b e n x (2 d+e x)-2 (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+64 b g^2 (e f-d g) n \left (b e n x \left (3 d^2+3 d e x+e^2 x^2\right )-3 (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )+9 b g^3 n \left (b e n x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{288 e^4} \]
(288*(e*f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n])^2 + 432*g*(e*f - d *g)^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2 + 288*g^2*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])^2 + 72*g^3*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n])^2 - 576*b*(e*f - d*g)^3*n*(e*(a - b*n)*x + b*(d + e*x)*Log[c*(d + e*x)^n]) + 216*b*g*(e*f - d*g)^2*n*(b*e*n*x*(2*d + e*x) - 2*(d + e*x)^2* (a + b*Log[c*(d + e*x)^n])) + 64*b*g^2*(e*f - d*g)*n*(b*e*n*x*(3*d^2 + 3*d *e*x + e^2*x^2) - 3*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n])) + 9*b*g^3*n*(b *e*n*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 4*(d + e*x)^4*(a + b* Log[c*(d + e*x)^n])))/(288*e^4)
Time = 0.57 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {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.
| \(\displaystyle \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {b e n \int \frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}dx}{2 g}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {b n \int \frac {\left (e \left (f-\frac {d g}{e}\right )+g (d+e x)\right )^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^4 (d+e x)}d(d+e x)}{2 g}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {b n \int \frac {(e f-d g+g (d+e x))^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x}d(d+e x)}{2 e^4 g}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {b n \left (-b n \int \left (\frac {1}{4} (d+e x)^3 g^4+\frac {4}{3} (e f-d g) (d+e x)^2 g^3+3 (e f-d g)^2 (d+e x) g^2+4 (e f-d g)^3 g+\frac {(e f-d g)^4 \log (d+e x)}{d+e x}\right )d(d+e x)+\frac {4}{3} g^3 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )+3 g^2 (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+(e f-d g)^4 \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )+4 g (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1}{4} g^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )\right )}{2 e^4 g}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f+g x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {b n \left (\frac {4}{3} g^3 (d+e x)^3 (e f-d g) \left (a+b \log \left (c (d+e x)^n\right )\right )+3 g^2 (d+e x)^2 (e f-d g)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )+(e f-d g)^4 \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )+4 g (d+e x) (e f-d g)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+\frac {1}{4} g^4 (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \left (\frac {4}{9} g^3 (d+e x)^3 (e f-d g)+\frac {3}{2} g^2 (d+e x)^2 (e f-d g)^2+4 g (d+e x) (e f-d g)^3+\frac {1}{2} (e f-d g)^4 \log ^2(d+e x)+\frac {1}{16} g^4 (d+e x)^4\right )\right )}{2 e^4 g}\) |
((f + g*x)^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g) - (b*n*(-(b*n*(4*g*(e*f - d*g)^3*(d + e*x) + (3*g^2*(e*f - d*g)^2*(d + e*x)^2)/2 + (4*g^3*(e*f - d* g)*(d + e*x)^3)/9 + (g^4*(d + e*x)^4)/16 + ((e*f - d*g)^4*Log[d + e*x]^2)/ 2)) + 4*g*(e*f - d*g)^3*(d + e*x)*(a + b*Log[c*(d + e*x)^n]) + 3*g^2*(e*f - d*g)^2*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]) + (4*g^3*(e*f - d*g)*(d + e*x)^3*(a + b*Log[c*(d + e*x)^n]))/3 + (g^4*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/4 + (e*f - d*g)^4*Log[d + e*x]*(a + b*Log[c*(d + e*x)^n])))/(2*e ^4*g)
3.1.44.3.1 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1342\) vs. \(2(347)=694\).
Time = 2.20 (sec) , antiderivative size = 1343, normalized size of antiderivative = 3.68
| method | result | size |
| parallelrisch | \(\text {Expression too large to display}\) | \(1343\) |
| risch | \(\text {Expression too large to display}\) | \(6770\) |
-1/288*(192*a*b*e^4*f*g^2*n*x^3+300*b^2*d^3*e*g^3*n^2*x-288*a^2*e^4*f^3*x+ 288*a^2*d*e^3*f^3-288*a*b*d*e^3*f*g^2*n*x^2-288*a^2*e^4*f*g^2*x^3+576*b*n* a*e^4*f^3*x+36*a*b*e^4*g^3*n*x^4+28*b^2*d*e^3*g^3*n^2*x^3-64*b^2*e^4*f*g^2 *n^2*x^3-9*b^2*e^4*g^3*n^2*x^4+72*ln(c*(e*x+d)^n)^2*b^2*d^4*g^3-864*a*b*d* e^3*f^2*g*n*x-576*b^2*n^2*e^4*f^3*x-300*b^2*d^4*g^3*n^2-288*ln(c*(e*x+d)^n )^2*b^2*d*e^3*f^3+144*ln(c*(e*x+d)^n)*b^2*d^4*g^3*n-444*ln(e*x+d)*b^2*d^4* g^3*n^2-72*a^2*e^4*g^3*x^4-78*b^2*d^2*e^2*g^3*n^2*x^2-216*b^2*e^4*f^2*g*n^ 2*x^2-432*a^2*e^4*f^2*g*x^2+240*b^2*d*e^3*f*g^2*n^2*x^2+432*a*b*e^4*f^2*g* n*x^2-1056*b^2*d^2*e^2*f*g^2*n^2*x+1296*b^2*d*e^3*f^2*g*n^2*x+576*b^2*d*e^ 3*f^3*n^2+144*a*b*d^4*g^3*n-72*x^4*ln(c*(e*x+d)^n)^2*b^2*e^4*g^3-288*x*ln( c*(e*x+d)^n)^2*b^2*e^4*f^3+576*x*ln(c*(e*x+d)^n)*b^2*d^2*e^2*f*g^2*n-864*x *ln(c*(e*x+d)^n)*b^2*d*e^3*f^2*g*n+1056*b^2*d^3*e*f*g^2*n^2-1296*b^2*d^2*e ^2*f^2*g*n^2-576*a*b*d*e^3*f^3*n-576*ln(e*x+d)*a*b*d^3*e*f*g^2*n+864*ln(e* x+d)*a*b*d^2*e^2*f^2*g*n+72*a*b*d^2*e^2*g^3*n*x^2-48*a*b*d*e^3*g^3*n*x^3-1 44*a*b*d^3*e*g^3*n*x-576*ln(c*(e*x+d)^n)*b^2*d^3*e*f*g^2*n+864*ln(c*(e*x+d )^n)*b^2*d^2*e^2*f^2*g*n+36*x^4*ln(c*(e*x+d)^n)*b^2*e^4*g^3*n-144*x^4*ln(c *(e*x+d)^n)*a*b*e^4*g^3-288*x^3*ln(c*(e*x+d)^n)^2*b^2*e^4*f*g^2-432*x^2*ln (c*(e*x+d)^n)^2*b^2*e^4*f^2*g+576*x*ln(c*(e*x+d)^n)*b^2*e^4*f^3*n-576*x*ln (c*(e*x+d)^n)*a*b*e^4*f^3-288*ln(c*(e*x+d)^n)^2*b^2*d^3*e*f*g^2+432*ln(c*( e*x+d)^n)^2*b^2*d^2*e^2*f^2*g-576*ln(c*(e*x+d)^n)*b^2*d*e^3*f^3*n+576*l...
Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (347) = 694\).
Time = 0.30 (sec) , antiderivative size = 1190, normalized size of antiderivative = 3.26 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Too large to display} \]
1/288*(9*(b^2*e^4*g^3*n^2 - 4*a*b*e^4*g^3*n + 8*a^2*e^4*g^3)*x^4 + 4*(72*a ^2*e^4*f*g^2 + (16*b^2*e^4*f*g^2 - 7*b^2*d*e^3*g^3)*n^2 - 12*(4*a*b*e^4*f* g^2 - a*b*d*e^3*g^3)*n)*x^3 + 6*(72*a^2*e^4*f^2*g + (36*b^2*e^4*f^2*g - 40 *b^2*d*e^3*f*g^2 + 13*b^2*d^2*e^2*g^3)*n^2 - 12*(6*a*b*e^4*f^2*g - 4*a*b*d *e^3*f*g^2 + a*b*d^2*e^2*g^3)*n)*x^2 + 72*(b^2*e^4*g^3*n^2*x^4 + 4*b^2*e^4 *f*g^2*n^2*x^3 + 6*b^2*e^4*f^2*g*n^2*x^2 + 4*b^2*e^4*f^3*n^2*x + (4*b^2*d* e^3*f^3 - 6*b^2*d^2*e^2*f^2*g + 4*b^2*d^3*e*f*g^2 - b^2*d^4*g^3)*n^2)*log( e*x + d)^2 + 72*(b^2*e^4*g^3*x^4 + 4*b^2*e^4*f*g^2*x^3 + 6*b^2*e^4*f^2*g*x ^2 + 4*b^2*e^4*f^3*x)*log(c)^2 + 12*(24*a^2*e^4*f^3 + (48*b^2*e^4*f^3 - 10 8*b^2*d*e^3*f^2*g + 88*b^2*d^2*e^2*f*g^2 - 25*b^2*d^3*e*g^3)*n^2 - 12*(4*a *b*e^4*f^3 - 6*a*b*d*e^3*f^2*g + 4*a*b*d^2*e^2*f*g^2 - a*b*d^3*e*g^3)*n)*x - 12*(3*(b^2*e^4*g^3*n^2 - 4*a*b*e^4*g^3*n)*x^4 - 4*(12*a*b*e^4*f*g^2*n - (4*b^2*e^4*f*g^2 - b^2*d*e^3*g^3)*n^2)*x^3 + (48*b^2*d*e^3*f^3 - 108*b^2* d^2*e^2*f^2*g + 88*b^2*d^3*e*f*g^2 - 25*b^2*d^4*g^3)*n^2 - 6*(12*a*b*e^4*f ^2*g*n - (6*b^2*e^4*f^2*g - 4*b^2*d*e^3*f*g^2 + b^2*d^2*e^2*g^3)*n^2)*x^2 - 12*(4*a*b*d*e^3*f^3 - 6*a*b*d^2*e^2*f^2*g + 4*a*b*d^3*e*f*g^2 - a*b*d^4* g^3)*n - 12*(4*a*b*e^4*f^3*n - (4*b^2*e^4*f^3 - 6*b^2*d*e^3*f^2*g + 4*b^2* d^2*e^2*f*g^2 - b^2*d^3*e*g^3)*n^2)*x - 12*(b^2*e^4*g^3*n*x^4 + 4*b^2*e^4* f*g^2*n*x^3 + 6*b^2*e^4*f^2*g*n*x^2 + 4*b^2*e^4*f^3*n*x + (4*b^2*d*e^3*f^3 - 6*b^2*d^2*e^2*f^2*g + 4*b^2*d^3*e*f*g^2 - b^2*d^4*g^3)*n)*log(c))*lo...
Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (348) = 696\).
Time = 2.35 (sec) , antiderivative size = 1241, normalized size of antiderivative = 3.40 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Too large to display} \]
Piecewise((a**2*f**3*x + 3*a**2*f**2*g*x**2/2 + a**2*f*g**2*x**3 + a**2*g* *3*x**4/4 - a*b*d**4*g**3*log(c*(d + e*x)**n)/(2*e**4) + 2*a*b*d**3*f*g**2 *log(c*(d + e*x)**n)/e**3 + a*b*d**3*g**3*n*x/(2*e**3) - 3*a*b*d**2*f**2*g *log(c*(d + e*x)**n)/e**2 - 2*a*b*d**2*f*g**2*n*x/e**2 - a*b*d**2*g**3*n*x **2/(4*e**2) + 2*a*b*d*f**3*log(c*(d + e*x)**n)/e + 3*a*b*d*f**2*g*n*x/e + a*b*d*f*g**2*n*x**2/e + a*b*d*g**3*n*x**3/(6*e) - 2*a*b*f**3*n*x + 2*a*b* f**3*x*log(c*(d + e*x)**n) - 3*a*b*f**2*g*n*x**2/2 + 3*a*b*f**2*g*x**2*log (c*(d + e*x)**n) - 2*a*b*f*g**2*n*x**3/3 + 2*a*b*f*g**2*x**3*log(c*(d + e* x)**n) - a*b*g**3*n*x**4/8 + a*b*g**3*x**4*log(c*(d + e*x)**n)/2 + 25*b**2 *d**4*g**3*n*log(c*(d + e*x)**n)/(24*e**4) - b**2*d**4*g**3*log(c*(d + e*x )**n)**2/(4*e**4) - 11*b**2*d**3*f*g**2*n*log(c*(d + e*x)**n)/(3*e**3) + b **2*d**3*f*g**2*log(c*(d + e*x)**n)**2/e**3 - 25*b**2*d**3*g**3*n**2*x/(24 *e**3) + b**2*d**3*g**3*n*x*log(c*(d + e*x)**n)/(2*e**3) + 9*b**2*d**2*f** 2*g*n*log(c*(d + e*x)**n)/(2*e**2) - 3*b**2*d**2*f**2*g*log(c*(d + e*x)**n )**2/(2*e**2) + 11*b**2*d**2*f*g**2*n**2*x/(3*e**2) - 2*b**2*d**2*f*g**2*n *x*log(c*(d + e*x)**n)/e**2 + 13*b**2*d**2*g**3*n**2*x**2/(48*e**2) - b**2 *d**2*g**3*n*x**2*log(c*(d + e*x)**n)/(4*e**2) - 2*b**2*d*f**3*n*log(c*(d + e*x)**n)/e + b**2*d*f**3*log(c*(d + e*x)**n)**2/e - 9*b**2*d*f**2*g*n**2 *x/(2*e) + 3*b**2*d*f**2*g*n*x*log(c*(d + e*x)**n)/e - 5*b**2*d*f*g**2*n** 2*x**2/(6*e) + b**2*d*f*g**2*n*x**2*log(c*(d + e*x)**n)/e - 7*b**2*d*g*...
Leaf count of result is larger than twice the leaf count of optimal. 827 vs. \(2 (347) = 694\).
Time = 0.21 (sec) , antiderivative size = 827, normalized size of antiderivative = 2.27 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\frac {1}{4} \, b^{2} g^{3} x^{4} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + \frac {1}{2} \, a b g^{3} x^{4} \log \left ({\left (e x + d\right )}^{n} c\right ) + b^{2} f g^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + \frac {1}{4} \, a^{2} g^{3} x^{4} + 2 \, a b f g^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {3}{2} \, b^{2} f^{2} g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + a^{2} f g^{2} x^{3} - 2 \, a b e f^{3} n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} - \frac {1}{24} \, a b e g^{3} n {\left (\frac {12 \, d^{4} \log \left (e x + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{4} - 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} - 12 \, d^{3} x}{e^{4}}\right )} + \frac {1}{3} \, a b e f g^{2} n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} - \frac {3}{2} \, a b e f^{2} g n {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} + 3 \, a b f^{2} g x^{2} \log \left ({\left (e x + d\right )}^{n} c\right ) + b^{2} f^{3} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + \frac {3}{2} \, a^{2} f^{2} g x^{2} + 2 \, a b f^{3} x \log \left ({\left (e x + d\right )}^{n} c\right ) - {\left (2 \, e n {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (d \log \left (e x + d\right )^{2} - 2 \, e x + 2 \, d \log \left (e x + d\right )\right )} n^{2}}{e}\right )} b^{2} f^{3} - \frac {3}{4} \, {\left (2 \, e n {\left (\frac {2 \, d^{2} \log \left (e x + d\right )}{e^{3}} + \frac {e x^{2} - 2 \, d x}{e^{2}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) - \frac {{\left (e^{2} x^{2} + 2 \, d^{2} \log \left (e x + d\right )^{2} - 6 \, d e x + 6 \, d^{2} \log \left (e x + d\right )\right )} n^{2}}{e^{2}}\right )} b^{2} f^{2} g + \frac {1}{18} \, {\left (6 \, e n {\left (\frac {6 \, d^{3} \log \left (e x + d\right )}{e^{4}} - \frac {2 \, e^{2} x^{3} - 3 \, d e x^{2} + 6 \, d^{2} x}{e^{3}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) + \frac {{\left (4 \, e^{3} x^{3} - 15 \, d e^{2} x^{2} - 18 \, d^{3} \log \left (e x + d\right )^{2} + 66 \, d^{2} e x - 66 \, d^{3} \log \left (e x + d\right )\right )} n^{2}}{e^{3}}\right )} b^{2} f g^{2} - \frac {1}{288} \, {\left (12 \, e n {\left (\frac {12 \, d^{4} \log \left (e x + d\right )}{e^{5}} + \frac {3 \, e^{3} x^{4} - 4 \, d e^{2} x^{3} + 6 \, d^{2} e x^{2} - 12 \, d^{3} x}{e^{4}}\right )} \log \left ({\left (e x + d\right )}^{n} c\right ) - \frac {{\left (9 \, e^{4} x^{4} - 28 \, d e^{3} x^{3} + 78 \, d^{2} e^{2} x^{2} + 72 \, d^{4} \log \left (e x + d\right )^{2} - 300 \, d^{3} e x + 300 \, d^{4} \log \left (e x + d\right )\right )} n^{2}}{e^{4}}\right )} b^{2} g^{3} + a^{2} f^{3} x \]
1/4*b^2*g^3*x^4*log((e*x + d)^n*c)^2 + 1/2*a*b*g^3*x^4*log((e*x + d)^n*c) + b^2*f*g^2*x^3*log((e*x + d)^n*c)^2 + 1/4*a^2*g^3*x^4 + 2*a*b*f*g^2*x^3*l og((e*x + d)^n*c) + 3/2*b^2*f^2*g*x^2*log((e*x + d)^n*c)^2 + a^2*f*g^2*x^3 - 2*a*b*e*f^3*n*(x/e - d*log(e*x + d)/e^2) - 1/24*a*b*e*g^3*n*(12*d^4*log (e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^4) + 1/3*a*b*e*f*g^2*n*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2 *x)/e^3) - 3/2*a*b*e*f^2*g*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x)/e^2 ) + 3*a*b*f^2*g*x^2*log((e*x + d)^n*c) + b^2*f^3*x*log((e*x + d)^n*c)^2 + 3/2*a^2*f^2*g*x^2 + 2*a*b*f^3*x*log((e*x + d)^n*c) - (2*e*n*(x/e - d*log(e *x + d)/e^2)*log((e*x + d)^n*c) + (d*log(e*x + d)^2 - 2*e*x + 2*d*log(e*x + d))*n^2/e)*b^2*f^3 - 3/4*(2*e*n*(2*d^2*log(e*x + d)/e^3 + (e*x^2 - 2*d*x )/e^2)*log((e*x + d)^n*c) - (e^2*x^2 + 2*d^2*log(e*x + d)^2 - 6*d*e*x + 6* d^2*log(e*x + d))*n^2/e^2)*b^2*f^2*g + 1/18*(6*e*n*(6*d^3*log(e*x + d)/e^4 - (2*e^2*x^3 - 3*d*e*x^2 + 6*d^2*x)/e^3)*log((e*x + d)^n*c) + (4*e^3*x^3 - 15*d*e^2*x^2 - 18*d^3*log(e*x + d)^2 + 66*d^2*e*x - 66*d^3*log(e*x + d)) *n^2/e^3)*b^2*f*g^2 - 1/288*(12*e*n*(12*d^4*log(e*x + d)/e^5 + (3*e^3*x^4 - 4*d*e^2*x^3 + 6*d^2*e*x^2 - 12*d^3*x)/e^4)*log((e*x + d)^n*c) - (9*e^4*x ^4 - 28*d*e^3*x^3 + 78*d^2*e^2*x^2 + 72*d^4*log(e*x + d)^2 - 300*d^3*e*x + 300*d^4*log(e*x + d))*n^2/e^4)*b^2*g^3 + a^2*f^3*x
Leaf count of result is larger than twice the leaf count of optimal. 2345 vs. \(2 (347) = 694\).
Time = 0.34 (sec) , antiderivative size = 2345, normalized size of antiderivative = 6.42 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=\text {Too large to display} \]
(e*x + d)*b^2*f^3*n^2*log(e*x + d)^2/e + 3/2*(e*x + d)^2*b^2*f^2*g*n^2*log (e*x + d)^2/e^2 - 3*(e*x + d)*b^2*d*f^2*g*n^2*log(e*x + d)^2/e^2 + (e*x + d)^3*b^2*f*g^2*n^2*log(e*x + d)^2/e^3 - 3*(e*x + d)^2*b^2*d*f*g^2*n^2*log( e*x + d)^2/e^3 + 3*(e*x + d)*b^2*d^2*f*g^2*n^2*log(e*x + d)^2/e^3 + 1/4*(e *x + d)^4*b^2*g^3*n^2*log(e*x + d)^2/e^4 - (e*x + d)^3*b^2*d*g^3*n^2*log(e *x + d)^2/e^4 + 3/2*(e*x + d)^2*b^2*d^2*g^3*n^2*log(e*x + d)^2/e^4 - (e*x + d)*b^2*d^3*g^3*n^2*log(e*x + d)^2/e^4 - 2*(e*x + d)*b^2*f^3*n^2*log(e*x + d)/e - 3/2*(e*x + d)^2*b^2*f^2*g*n^2*log(e*x + d)/e^2 + 6*(e*x + d)*b^2* d*f^2*g*n^2*log(e*x + d)/e^2 - 2/3*(e*x + d)^3*b^2*f*g^2*n^2*log(e*x + d)/ e^3 + 3*(e*x + d)^2*b^2*d*f*g^2*n^2*log(e*x + d)/e^3 - 6*(e*x + d)*b^2*d^2 *f*g^2*n^2*log(e*x + d)/e^3 - 1/8*(e*x + d)^4*b^2*g^3*n^2*log(e*x + d)/e^4 + 2/3*(e*x + d)^3*b^2*d*g^3*n^2*log(e*x + d)/e^4 - 3/2*(e*x + d)^2*b^2*d^ 2*g^3*n^2*log(e*x + d)/e^4 + 2*(e*x + d)*b^2*d^3*g^3*n^2*log(e*x + d)/e^4 + 2*(e*x + d)*b^2*f^3*n*log(e*x + d)*log(c)/e + 3*(e*x + d)^2*b^2*f^2*g*n* log(e*x + d)*log(c)/e^2 - 6*(e*x + d)*b^2*d*f^2*g*n*log(e*x + d)*log(c)/e^ 2 + 2*(e*x + d)^3*b^2*f*g^2*n*log(e*x + d)*log(c)/e^3 - 6*(e*x + d)^2*b^2* d*f*g^2*n*log(e*x + d)*log(c)/e^3 + 6*(e*x + d)*b^2*d^2*f*g^2*n*log(e*x + d)*log(c)/e^3 + 1/2*(e*x + d)^4*b^2*g^3*n*log(e*x + d)*log(c)/e^4 - 2*(e*x + d)^3*b^2*d*g^3*n*log(e*x + d)*log(c)/e^4 + 3*(e*x + d)^2*b^2*d^2*g^3*n* log(e*x + d)*log(c)/e^4 - 2*(e*x + d)*b^2*d^3*g^3*n*log(e*x + d)*log(c)...
Time = 1.12 (sec) , antiderivative size = 1051, normalized size of antiderivative = 2.88 \[ \int (f+g x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx=x\,\left (\frac {72\,a^2\,d\,e^2\,f^2\,g+24\,a^2\,e^3\,f^3-48\,a\,b\,e^3\,f^3\,n-12\,b^2\,d^3\,g^3\,n^2+48\,b^2\,d^2\,e\,f\,g^2\,n^2-72\,b^2\,d\,e^2\,f^2\,g\,n^2+48\,b^2\,e^3\,f^3\,n^2}{24\,e^3}+\frac {d\,\left (\frac {d\,\left (\frac {g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )}{6\,e}-\frac {d\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{8\,e}\right )}{e}-\frac {g\,\left (12\,a^2\,d\,e\,f\,g+12\,a^2\,e^2\,f^2-12\,a\,b\,e^2\,f^2\,n+b^2\,d^2\,g^2\,n^2-4\,b^2\,d\,e\,f\,g\,n^2+6\,b^2\,e^2\,f^2\,n^2\right )}{4\,e^2}\right )}{e}\right )-x^2\,\left (\frac {d\,\left (\frac {g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )}{6\,e}-\frac {d\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{8\,e}\right )}{2\,e}-\frac {g\,\left (12\,a^2\,d\,e\,f\,g+12\,a^2\,e^2\,f^2-12\,a\,b\,e^2\,f^2\,n+b^2\,d^2\,g^2\,n^2-4\,b^2\,d\,e\,f\,g\,n^2+6\,b^2\,e^2\,f^2\,n^2\right )}{8\,e^2}\right )+{\ln \left (c\,{\left (d+e\,x\right )}^n\right )}^2\,\left (b^2\,f^3\,x-\frac {d\,\left (b^2\,d^3\,g^3-4\,b^2\,d^2\,e\,f\,g^2+6\,b^2\,d\,e^2\,f^2\,g-4\,b^2\,e^3\,f^3\right )}{4\,e^4}+\frac {b^2\,g^3\,x^4}{4}+\frac {3\,b^2\,f^2\,g\,x^2}{2}+b^2\,f\,g^2\,x^3\right )+x^3\,\left (\frac {g^2\,\left (6\,a^2\,d\,g+18\,a^2\,e\,f-b^2\,d\,g\,n^2+4\,b^2\,e\,f\,n^2-12\,a\,b\,e\,f\,n\right )}{18\,e}-\frac {d\,g^3\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{24\,e}\right )+\ln \left (c\,{\left (d+e\,x\right )}^n\right )\,\left (\frac {x\,\left (\frac {d\,\left (\frac {d\,\left (\frac {8\,b\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {2\,b\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{e}-\frac {12\,b\,f\,g\,\left (2\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2\,e}+\frac {4\,b\,f^2\,\left (3\,a\,d\,g+a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2}+\frac {x^3\,\left (\frac {4\,b\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{3\,e}-\frac {b\,d\,g^3\,\left (4\,a-b\,n\right )}{3\,e}\right )}{2}-\frac {x^2\,\left (\frac {d\,\left (\frac {8\,b\,g^2\,\left (a\,d\,g+3\,a\,e\,f-b\,e\,f\,n\right )}{e}-\frac {2\,b\,d\,g^3\,\left (4\,a-b\,n\right )}{e}\right )}{4\,e}-\frac {3\,b\,f\,g\,\left (2\,a\,d\,g+2\,a\,e\,f-b\,e\,f\,n\right )}{e}\right )}{2}+\frac {b\,g^3\,x^4\,\left (4\,a-b\,n\right )}{8}\right )+\frac {\ln \left (d+e\,x\right )\,\left (25\,b^2\,d^4\,g^3\,n^2-88\,b^2\,d^3\,e\,f\,g^2\,n^2+108\,b^2\,d^2\,e^2\,f^2\,g\,n^2-48\,b^2\,d\,e^3\,f^3\,n^2-12\,a\,b\,d^4\,g^3\,n+48\,a\,b\,d^3\,e\,f\,g^2\,n-72\,a\,b\,d^2\,e^2\,f^2\,g\,n+48\,a\,b\,d\,e^3\,f^3\,n\right )}{24\,e^4}+\frac {g^3\,x^4\,\left (8\,a^2-4\,a\,b\,n+b^2\,n^2\right )}{32} \]
x*((24*a^2*e^3*f^3 - 12*b^2*d^3*g^3*n^2 + 48*b^2*e^3*f^3*n^2 - 48*a*b*e^3* f^3*n + 72*a^2*d*e^2*f^2*g - 72*b^2*d*e^2*f^2*g*n^2 + 48*b^2*d^2*e*f*g^2*n ^2)/(24*e^3) + (d*((d*((g^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2* e*f*n^2 - 12*a*b*e*f*n))/(6*e) - (d*g^3*(8*a^2 + b^2*n^2 - 4*a*b*n))/(8*e) ))/e - (g*(12*a^2*e^2*f^2 + b^2*d^2*g^2*n^2 + 6*b^2*e^2*f^2*n^2 - 12*a*b*e ^2*f^2*n + 12*a^2*d*e*f*g - 4*b^2*d*e*f*g*n^2))/(4*e^2)))/e) - x^2*((d*((g ^2*(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*n^2 - 12*a*b*e*f*n))/ (6*e) - (d*g^3*(8*a^2 + b^2*n^2 - 4*a*b*n))/(8*e)))/(2*e) - (g*(12*a^2*e^2 *f^2 + b^2*d^2*g^2*n^2 + 6*b^2*e^2*f^2*n^2 - 12*a*b*e^2*f^2*n + 12*a^2*d*e *f*g - 4*b^2*d*e*f*g*n^2))/(8*e^2)) + log(c*(d + e*x)^n)^2*(b^2*f^3*x - (d *(b^2*d^3*g^3 - 4*b^2*e^3*f^3 + 6*b^2*d*e^2*f^2*g - 4*b^2*d^2*e*f*g^2))/(4 *e^4) + (b^2*g^3*x^4)/4 + (3*b^2*f^2*g*x^2)/2 + b^2*f*g^2*x^3) + x^3*((g^2 *(6*a^2*d*g + 18*a^2*e*f - b^2*d*g*n^2 + 4*b^2*e*f*n^2 - 12*a*b*e*f*n))/(1 8*e) - (d*g^3*(8*a^2 + b^2*n^2 - 4*a*b*n))/(24*e)) + log(c*(d + e*x)^n)*(( x*((d*((d*((8*b*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (2*b*d*g^3*(4*a - b*n ))/e))/e - (12*b*f*g*(2*a*d*g + 2*a*e*f - b*e*f*n))/e))/(2*e) + (4*b*f^2*( 3*a*d*g + a*e*f - b*e*f*n))/e))/2 + (x^3*((4*b*g^2*(a*d*g + 3*a*e*f - b*e* f*n))/(3*e) - (b*d*g^3*(4*a - b*n))/(3*e)))/2 - (x^2*((d*((8*b*g^2*(a*d*g + 3*a*e*f - b*e*f*n))/e - (2*b*d*g^3*(4*a - b*n))/e))/(4*e) - (3*b*f*g*(2* a*d*g + 2*a*e*f - b*e*f*n))/e))/2 + (b*g^3*x^4*(4*a - b*n))/8) + (log(d...