Integrand size = 33, antiderivative size = 875 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {B^3 (b c-a d)^3 h^2 n^3 \log (c+d x)}{b^3 d^3}+\frac {B^2 (b c-a d)^2 h^2 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b^3 d^2}-\frac {2 B^2 (b c-a d)^2 h (3 b d g-2 b c h-a d h) n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b^3 d^3}-\frac {B (b c-a d) h (3 b d g-2 b c h-a d h) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b^3 d^2}-\frac {B (b c-a d) h^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b d^3}+\frac {B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{b^3 d^3}-\frac {(b g-a h)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 b^3 h}+\frac {(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 h}-\frac {B^2 (b c-a d)^3 h^2 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 d^3}-\frac {2 B^3 (b c-a d)^2 h (3 b d g-2 b c h-a d h) n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3}+\frac {B^3 (b c-a d)^3 h^2 n^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 d^3}-\frac {2 B^3 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b^3 d^3} \]
-B^3*(-a*d+b*c)^3*h^2*n^3*ln(d*x+c)/b^3/d^3+B^2*(-a*d+b*c)^2*h^2*n^2*(b*x+ a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b^3/d^2-2*B^2*(-a*d+b*c)^2*h*(-a*d*h- 2*b*c*h+3*b*d*g)*n^2*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c) ^n)))/b^3/d^3-B*(-a*d+b*c)*h*(-a*d*h-2*b*c*h+3*b*d*g)*n*(b*x+a)*(A+B*ln(e* (b*x+a)^n/((d*x+c)^n)))^2/b^3/d^2-1/2*B*(-a*d+b*c)*h^2*n*(d*x+c)^2*(A+B*ln (e*(b*x+a)^n/((d*x+c)^n)))^2/b/d^3+B*(-a*d+b*c)*(a^2*d^2*h^2-a*b*d*h*(-c*h +3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n*ln((-a*d+b*c)/b/(d*x+c))*(A+B *ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b^3/d^3-1/3*(-a*h+b*g)^3*(A+B*ln(e*(b*x+a) ^n/((d*x+c)^n)))^3/b^3/h+1/3*(h*x+g)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3 /h-B^2*(-a*d+b*c)^3*h^2*n^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))*ln(1-b*(d*x+ c)/d/(b*x+a))/b^3/d^3-2*B^3*(-a*d+b*c)^2*h*(-a*d*h-2*b*c*h+3*b*d*g)*n^3*po lylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^3+2*B^2*(-a*d+b*c)*(a^2*d^2*h^2-a*b*d*h *(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n^2*(A+B*ln(e*(b*x+a)^n/( (d*x+c)^n)))*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^3+B^3*(-a*d+b*c)^3*h^2*n ^3*polylog(2,b*(d*x+c)/d/(b*x+a))/b^3/d^3-2*B^3*(-a*d+b*c)*(a^2*d^2*h^2-a* b*d*h*(-c*h+3*d*g)+b^2*(c^2*h^2-3*c*d*g*h+3*d^2*g^2))*n^3*polylog(3,d*(b*x +a)/b/(d*x+c))/b^3/d^3
Leaf count is larger than twice the leaf count of optimal. \(7279\) vs. \(2(875)=1750\).
Time = 2.67 (sec) , antiderivative size = 7279, normalized size of antiderivative = 8.32 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \]
Time = 1.70 (sec) , antiderivative size = 1020, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2953, 2798, 2804, 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.
| \(\displaystyle \int (g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3 \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int (g+h x)^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3dx\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle (b c-a d) \int \frac {\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2798 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \frac {(c+d x) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{h (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \int \left (\frac {(b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 h^3}{b d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {(b c-a d)^2 (3 b d g-2 b c h-a d h) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 h^2}{b^2 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {(b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 h}{b^3 d^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b g-a h)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 (a+b x)}\right )d\frac {a+b x}{c+d x}}{h (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d) \left (\frac {\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 (b c-a d) h \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {B n \left (\frac {(b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 h^3}{2 b d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B (b c-a d)^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) h^3}{b^3 d^2 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B^2 (b c-a d)^3 n^2 \log \left (b-\frac {d (a+b x)}{c+d x}\right ) h^3}{b^3 d^3}+\frac {B (b c-a d)^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) h^3}{b^3 d^3}-\frac {B^2 (b c-a d)^3 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right ) h^3}{b^3 d^3}+\frac {(b c-a d)^2 (3 b d g-2 b c h-a d h) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 h^2}{b^3 d^2 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {2 B (b c-a d)^2 (3 b d g-2 b c h-a d h) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) h^2}{b^3 d^3}+\frac {2 B^2 (b c-a d)^2 (3 b d g-2 b c h-a d h) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) h^2}{b^3 d^3}-\frac {(b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) h}{b^3 d^3}-\frac {2 B (b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) h}{b^3 d^3}+\frac {2 B^2 (b c-a d) \left (\left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-a d h (3 d g-c h) b+a^2 d^2 h^2\right ) n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right ) h}{b^3 d^3}+\frac {(b g-a h)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 b^3 B n}\right )}{(b c-a d) h}\right )\) |
(b*c - a*d)*(((b*g - a*h - ((d*g - c*h)*(a + b*x))/(c + d*x))^3*(A + B*Log [e*((a + b*x)/(c + d*x))^n])^3)/(3*(b*c - a*d)*h*(b - (d*(a + b*x))/(c + d *x))^3) - (B*n*(-((B*(b*c - a*d)^3*h^3*n*(a + b*x)*(A + B*Log[e*((a + b*x) /(c + d*x))^n]))/(b^3*d^2*(c + d*x)*(b - (d*(a + b*x))/(c + d*x)))) + ((b* c - a*d)^3*h^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*b*d^3*(b - (d* (a + b*x))/(c + d*x))^2) + ((b*c - a*d)^2*h^2*(3*b*d*g - 2*b*c*h - a*d*h)* (a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*d^2*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + ((b*g - a*h)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*b^3*B*n) - (B^2*(b*c - a*d)^3*h^3*n^2*Log[b - (d*(a + b*x) )/(c + d*x)])/(b^3*d^3) + (2*B*(b*c - a*d)^2*h^2*(3*b*d*g - 2*b*c*h - a*d* h)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3) - ((b*c - a*d)*h*(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h^2))*(A + B*Log[e*((a + b*x)/(c + d*x))^ n])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3) + (B*(b*c - a*d)^3*h ^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^3*d^3) + (2*B^2*(b*c - a*d)^2*h^2*(3*b*d*g - 2*b*c*h - a*d*h)*n ^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^3*d^3) - (2*B*(b*c - a*d)*h *(a^2*d^2*h^2 - a*b*d*h*(3*d*g - c*h) + b^2*(3*d^2*g^2 - 3*c*d*g*h + c^2*h ^2))*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b* (c + d*x))])/(b^3*d^3) - (B^2*(b*c - a*d)^3*h^3*n^2*PolyLog[2, (b*(c + ...
3.4.9.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*(( f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Simp[b*n*(p/((q + 1) *(e*f - d*g))) Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{ u = ExpandIntegrand[(a + b*Log[c*x^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] / ; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d) Sub st[Int[(b*f - a*g - (d*f - c*g)*x)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2 )), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n} , x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
\[\int \left (h x +g \right )^{2} {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}d x\]
\[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (h x + g\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
integral(A^3*h^2*x^2 + 2*A^3*g*h*x + A^3*g^2 + (B^3*h^2*x^2 + 2*B^3*g*h*x + B^3*g^2)*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*(A*B^2*h^2*x^2 + 2*A*B^2*g *h*x + A*B^2*g^2)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*(A^2*B*h^2*x^2 + 2* A^2*B*g*h*x + A^2*B*g^2)*log((b*x + a)^n*e/(d*x + c)^n), x)
Exception generated. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (h x + g\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
A^2*B*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^3*h^2*x^3 + 3*A^2*B*g *h*x^2*log((b*x + a)^n*e/(d*x + c)^n) + A^3*g*h*x^2 + 3*A^2*B*g^2*x*log((b *x + a)^n*e/(d*x + c)^n) + A^3*g^2*x + 3*(a*e*n*log(b*x + a)/b - c*e*n*log (d*x + c)/d)*A^2*B*g^2/e - 3*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*A^2*B*g*h/e + 1/2*(2*a^3*e*n*log(b* x + a)/b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 - 2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*A^2*B*h^2/e - 1/6*(2*(B^3*b ^3*d^3*h^2*x^3 + 3*B^3*b^3*d^3*g*h*x^2 + 3*B^3*b^3*d^3*g^2*x)*log((d*x + c )^n)^3 + 3*(2*(3*c*d^2*g^2*n - 3*c^2*d*g*h*n + c^3*h^2*n)*B^3*b^3*log(d*x + c) - 2*(3*a*b^2*d^3*g^2*n - 3*a^2*b*d^3*g*h*n + a^3*d^3*h^2*n)*B^3*log(b *x + a) - 2*(B^3*b^3*d^3*h^2*log(e) + A*B^2*b^3*d^3*h^2)*x^3 - (6*A*B^2*b^ 3*d^3*g*h + (a*b^2*d^3*h^2*n - (c*d^2*h^2*n - 6*d^3*g*h*log(e))*b^3)*B^3)* x^2 - 2*(3*A*B^2*b^3*d^3*g^2 + (3*a*b^2*d^3*g*h*n - a^2*b*d^3*h^2*n - (3*c *d^2*g*h*n - c^2*d*h^2*n - 3*d^3*g^2*log(e))*b^3)*B^3)*x - 2*(B^3*b^3*d^3* h^2*x^3 + 3*B^3*b^3*d^3*g*h*x^2 + 3*B^3*b^3*d^3*g^2*x)*log((b*x + a)^n))*l og((d*x + c)^n)^2)/(b^3*d^3) - integrate(-(B^3*b^3*c*d^2*g^2*log(e)^3 + 3* A*B^2*b^3*c*d^2*g^2*log(e)^2 + (B^3*b^3*d^3*h^2*log(e)^3 + 3*A*B^2*b^3*d^3 *h^2*log(e)^2)*x^3 + (B^3*b^3*d^3*h^2*x^3 + B^3*b^3*c*d^2*g^2 + (2*d^3*g*h + c*d^2*h^2)*B^3*b^3*x^2 + (d^3*g^2 + 2*c*d^2*g*h)*B^3*b^3*x)*log((b*x + a)^n)^3 + (3*(2*d^3*g*h*log(e)^2 + c*d^2*h^2*log(e)^2)*A*B^2*b^3 + (2*d...
Timed out. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Timed out} \]
Timed out. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int {\left (g+h\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3 \,d x \]