Integrand size = 31, antiderivative size = 466 \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {3 B^2 (b c-a d)^2 h n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{b^2 d^2}-\frac {3 B (b c-a d) h n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b^2 d}+\frac {3 B (b c-a d) (2 b d g-b c h-a d h) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{2 b^2 d^2}-\frac {(b g-a h)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 b^2 h}+\frac {(g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{2 h}-\frac {3 B^3 (b c-a d)^2 h n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac {3 B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}-\frac {3 B^3 (b c-a d) (2 b d g-b c h-a d h) n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2} \]
-3*B^2*(-a*d+b*c)^2*h*n^2*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d *x+c)^n)))/b^2/d^2-3/2*B*(-a*d+b*c)*h*n*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+ c)^n)))^2/b^2/d+3/2*B*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*n*ln((-a*d+b*c)/b/ (d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b^2/d^2-1/2*(-a*h+b*g)^2*(A+B *ln(e*(b*x+a)^n/((d*x+c)^n)))^3/b^2/h+1/2*(h*x+g)^2*(A+B*ln(e*(b*x+a)^n/(( d*x+c)^n)))^3/h-3*B^3*(-a*d+b*c)^2*h*n^3*polylog(2,d*(b*x+a)/b/(d*x+c))/b^ 2/d^2+3*B^2*(-a*d+b*c)*(-a*d*h-b*c*h+2*b*d*g)*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^2/d^2-3*B^3*(-a*d+b*c)*(-a*d*h- b*c*h+2*b*d*g)*n^3*polylog(3,d*(b*x+a)/b/(d*x+c))/b^2/d^2
Leaf count is larger than twice the leaf count of optimal. \(3890\) vs. \(2(466)=932\).
Time = 0.94 (sec) , antiderivative size = 3890, normalized size of antiderivative = 8.35 \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \]
(-12*A*b^2*B^2*c*d*g*n^2 - 12*a*A*b*B^2*d^2*g*n^2 + 12*a*A*b*B^2*c*d*h*n^2 + 6*a*b*B^3*c*d*h*n^3 - 6*a^2*B^3*d^2*h*n^3 + 2*A^3*b^2*d^2*g*x - 3*A^2*b ^2*B*c*d*h*n*x + 3*a*A^2*b*B*d^2*h*n*x + A^3*b^2*d^2*h*x^2 + 6*a*A^2*b*B*d ^2*g*n*Log[a + b*x] - 3*a^2*A^2*B*d^2*h*n*Log[a + b*x] - 6*a*A*b*B^2*c*d*h *n^2*Log[a + b*x] + 6*a^2*A*B^2*d^2*h*n^2*Log[a + b*x] + 12*b^2*B^3*c*d*g* n^3*Log[a + b*x] + 12*a*b*B^3*d^2*g*n^3*Log[a + b*x] - 12*a*b*B^3*c*d*h*n^ 3*Log[a + b*x] - 6*a*A*b*B^2*d^2*g*n^2*Log[a + b*x]^2 + 3*a^2*A*B^2*d^2*h* n^2*Log[a + b*x]^2 + 3*a*b*B^3*c*d*h*n^3*Log[a + b*x]^2 - 3*a^2*B^3*d^2*h* n^3*Log[a + b*x]^2 + 2*a*b*B^3*d^2*g*n^3*Log[a + b*x]^3 - a^2*B^3*d^2*h*n^ 3*Log[a + b*x]^3 - 6*A^2*b^2*B*c*d*g*n*Log[c + d*x] + 3*A^2*b^2*B*c^2*h*n* Log[c + d*x] + 6*A*b^2*B^2*c^2*h*n^2*Log[c + d*x] - 6*a*A*b*B^2*c*d*h*n^2* Log[c + d*x] - 12*b^2*B^3*c*d*g*n^3*Log[c + d*x] - 12*a*b*B^3*d^2*g*n^3*Lo g[c + d*x] + 12*a*b*B^3*c*d*h*n^3*Log[c + d*x] + 12*A*b^2*B^2*c*d*g*n^2*Lo g[a + b*x]*Log[c + d*x] + 12*a*A*b*B^2*d^2*g*n^2*Log[a + b*x]*Log[c + d*x] - 6*A*b^2*B^2*c^2*h*n^2*Log[a + b*x]*Log[c + d*x] - 6*a^2*A*B^2*d^2*h*n^2 *Log[a + b*x]*Log[c + d*x] - 6*b^2*B^3*c^2*h*n^3*Log[a + b*x]*Log[c + d*x] + 6*a*b*B^3*c*d*h*n^3*Log[a + b*x]*Log[c + d*x] - 6*b^2*B^3*c*d*g*n^3*Log [a + b*x]^2*Log[c + d*x] - 12*a*b*B^3*d^2*g*n^3*Log[a + b*x]^2*Log[c + d*x ] + 3*b^2*B^3*c^2*h*n^3*Log[a + b*x]^2*Log[c + d*x] + 6*a^2*B^3*d^2*h*n^3* Log[a + b*x]^2*Log[c + d*x] - 12*a*A*b*B^2*d^2*g*n^2*Log[(d*(a + b*x))/...
Time = 1.09 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.22, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, 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.
| \(\displaystyle \int (g+h x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3 \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int (g+h x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3dx\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle (b c-a d) \int \frac {\left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2798 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {3 B n \int \frac {(c+d x) \left (b g-a h-\frac {(d g-c h) (a+b x)}{c+d x}\right )^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 h (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {3 B n \int \left (\frac {(b g-a h)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 (a+b x)}+\frac {(b c-a d) h (2 b d g-b c h-a d h) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 d \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(b c-a d)^2 h^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )d\frac {a+b x}{c+d x}}{2 h (b c-a d)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d) \left (\frac {\left (-\frac {(a+b x) (d g-c h)}{c+d x}-a h+b g\right )^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{2 h (b c-a d) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {3 B n \left (-\frac {2 B h n (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d^2}-\frac {h (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 d^2}+\frac {2 B h^2 n (b c-a d)^2 \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 d^2}+\frac {(b g-a h)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{3 b^2 B n}+\frac {h^2 (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b^2 d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {2 B^2 h n^2 (b c-a d) (-a d h-b c h+2 b d g) \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}+\frac {2 B^2 h^2 n^2 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^2 d^2}\right )}{2 h (b c-a d)}\right )\) |
(b*c - a*d)*(((b*g - a*h - ((d*g - c*h)*(a + b*x))/(c + d*x))^2*(A + B*Log [e*((a + b*x)/(c + d*x))^n])^3)/(2*(b*c - a*d)*h*(b - (d*(a + b*x))/(c + d *x))^2) - (3*B*n*(((b*c - a*d)^2*h^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^2*d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + ((b*g - a *h)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(3*b^2*B*n) + (2*B*(b*c - a*d)^2*h^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d*(a + b*x))/ (b*(c + d*x))])/(b^2*d^2) - ((b*c - a*d)*h*(2*b*d*g - b*c*h - a*d*h)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/ (b^2*d^2) + (2*B^2*(b*c - a*d)^2*h^2*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2) - (2*B*(b*c - a*d)*h*(2*b*d*g - b*c*h - a*d*h)*n*(A + B* Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/( b^2*d^2) + (2*B^2*(b*c - a*d)*h*(2*b*d*g - b*c*h - a*d*h)*n^2*PolyLog[3, ( d*(a + b*x))/(b*(c + d*x))])/(b^2*d^2)))/(2*(b*c - a*d)*h))
3.4.10.3.1 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]
\[\int \left (h x +g \right ) {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}d x\]
\[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (h x + g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
integral(A^3*h*x + A^3*g + (B^3*h*x + B^3*g)*log((b*x + a)^n*e/(d*x + c)^n )^3 + 3*(A*B^2*h*x + A*B^2*g)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*(A^2*B* h*x + A^2*B*g)*log((b*x + a)^n*e/(d*x + c)^n), x)
Exception generated. \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (h x + g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
3/2*A^2*B*h*x^2*log((b*x + a)^n*e/(d*x + c)^n) + 1/2*A^3*h*x^2 + 3*A^2*B*g *x*log((b*x + a)^n*e/(d*x + c)^n) + A^3*g*x + 3*(a*e*n*log(b*x + a)/b - c* e*n*log(d*x + c)/d)*A^2*B*g/e - 3/2*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*lo g(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))*A^2*B*h/e - 1/2*((B^3*b^2*d^ 2*h*x^2 + 2*B^3*b^2*d^2*g*x)*log((d*x + c)^n)^3 + 3*((2*c*d*g*n - c^2*h*n) *B^3*b^2*log(d*x + c) - (2*a*b*d^2*g*n - a^2*d^2*h*n)*B^3*log(b*x + a) - ( B^3*b^2*d^2*h*log(e) + A*B^2*b^2*d^2*h)*x^2 - (2*A*B^2*b^2*d^2*g + (a*b*d^ 2*h*n - (c*d*h*n - 2*d^2*g*log(e))*b^2)*B^3)*x - (B^3*b^2*d^2*h*x^2 + 2*B^ 3*b^2*d^2*g*x)*log((b*x + a)^n))*log((d*x + c)^n)^2)/(b^2*d^2) - integrate (-(B^3*b^2*c*d*g*log(e)^3 + 3*A*B^2*b^2*c*d*g*log(e)^2 + (B^3*b^2*d^2*h*x^ 2 + B^3*b^2*c*d*g + (d^2*g + c*d*h)*B^3*b^2*x)*log((b*x + a)^n)^3 + (B^3*b ^2*d^2*h*log(e)^3 + 3*A*B^2*b^2*d^2*h*log(e)^2)*x^2 + 3*(B^3*b^2*c*d*g*log (e) + A*B^2*b^2*c*d*g + (B^3*b^2*d^2*h*log(e) + A*B^2*b^2*d^2*h)*x^2 + ((d ^2*g + c*d*h)*A*B^2*b^2 + (d^2*g*log(e) + c*d*h*log(e))*B^3*b^2)*x)*log((b *x + a)^n)^2 + (3*(d^2*g*log(e)^2 + c*d*h*log(e)^2)*A*B^2*b^2 + (d^2*g*log (e)^3 + c*d*h*log(e)^3)*B^3*b^2)*x + 3*(B^3*b^2*c*d*g*log(e)^2 + 2*A*B^2*b ^2*c*d*g*log(e) + (B^3*b^2*d^2*h*log(e)^2 + 2*A*B^2*b^2*d^2*h*log(e))*x^2 + (2*(d^2*g*log(e) + c*d*h*log(e))*A*B^2*b^2 + (d^2*g*log(e)^2 + c*d*h*log (e)^2)*B^3*b^2)*x)*log((b*x + a)^n) - 3*(B^3*b^2*c*d*g*log(e)^2 + 2*A*B^2* b^2*c*d*g*log(e) - (2*c*d*g*n^2 - c^2*h*n^2)*B^3*b^2*log(d*x + c) + (2*...
\[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (h x + g\right )} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
Timed out. \[ \int (g+h x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int \left (g+h\,x\right )\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3 \,d x \]