Integrand size = 33, antiderivative size = 809 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=-\frac {B^3 (b c-a d)^3 n^3 x}{4 d^3}-\frac {B^3 (b c-a d)^4 n^3 \log \left (\frac {a+b x}{c+d x}\right )}{4 b d^4}+\frac {3 B^3 (b c-a d)^4 n^3 \log (c+d x)}{2 b d^4}-\frac {7 B^2 (b c-a d)^3 n^2 (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 b d^3}+\frac {b B^2 (b c-a d)^2 n^2 (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{4 d^4}-\frac {9 B^2 (b c-a d)^4 n^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{2 b d^4}-\frac {9 B (b c-a d)^3 n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 b d^3}+\frac {9 b B (b c-a d)^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{8 d^4}-\frac {b^2 B (b c-a d) n (c+d x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 d^4}-\frac {3 B (b c-a d)^4 n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{4 b d^4}+\frac {(a+b x)^4 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{4 b}+\frac {7 B^2 (b c-a d)^4 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{4 b d^4}-\frac {9 B^3 (b c-a d)^4 n^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac {3 B^2 (b c-a d)^4 n^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4}-\frac {7 B^3 (b c-a d)^4 n^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{4 b d^4}+\frac {3 B^3 (b c-a d)^4 n^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b d^4} \]
-1/4*B^3*(-a*d+b*c)^3*n^3*x/d^3-1/4*B^3*(-a*d+b*c)^4*n^3*ln((b*x+a)/(d*x+c ))/b/d^4+3/2*B^3*(-a*d+b*c)^4*n^3*ln(d*x+c)/b/d^4-7/4*B^2*(-a*d+b*c)^3*n^2 *(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^3+1/4*b*B^2*(-a*d+b*c)^2*n^ 2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/d^4-9/2*B^2*(-a*d+b*c)^4*n^2 *ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/b/d^4-9/4*B*(- a*d+b*c)^3*n*(b*x+a)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b/d^3+9/8*b*B*(-a *d+b*c)^2*n*(d*x+c)^2*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/d^4-1/4*b^2*B*(- a*d+b*c)*n*(d*x+c)^3*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/d^4-3/4*B*(-a*d+b *c)^4*n*ln((-a*d+b*c)/b/(d*x+c))*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/b/d^4 +1/4*(b*x+a)^4*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/b+7/4*B^2*(-a*d+b*c)^4* 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/d^4-9/2* B^3*(-a*d+b*c)^4*n^3*polylog(2,d*(b*x+a)/b/(d*x+c))/b/d^4-3/2*B^2*(-a*d+b* c)^4*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/d^4-7/4*B^3*(-a*d+b*c)^4*n^3*polylog(2,b*(d*x+c)/d/(b*x+a))/b/d^4+3/2*B^ 3*(-a*d+b*c)^4*n^3*polylog(3,d*(b*x+a)/b/(d*x+c))/b/d^4
Leaf count is larger than twice the leaf count of optimal. \(6885\) vs. \(2(809)=1618\).
Time = 2.32 (sec) , antiderivative size = 6885, normalized size of antiderivative = 8.51 \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Result too large to show} \]
Time = 1.21 (sec) , antiderivative size = 747, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2949, 2781, 2795, 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.
| \(\displaystyle \int (a+b x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3 \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int (a+b x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3dx\) |
| \(\Big \downarrow \) 2949 |
| \(\displaystyle (b c-a d)^4 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 B n \int \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^3}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{4 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d)^4 \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 B n \left (\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {b^2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {2 B n \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 )}{d^4}+\frac {\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}{d^4}+\frac {6 B n \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 )}{d^4}-\frac {7 B n \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^4}+\frac {3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {7 B n (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {6 B^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4}+\frac {7 B^2 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 d^4}-\frac {2 B^2 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4}+\frac {b B^2 n^2}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 d^4}+\frac {2 B^2 n^2 \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )\) |
(b*c - a*d)^4*(((a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(4*b *(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^4) - (3*B*n*((b*B^2*n^2)/(3*d^4 *(b - (d*(a + b*x))/(c + d*x))) - (b^2*B*n*(A + B*Log[e*((a + b*x)/(c + d* x))^n]))/(3*d^4*(b - (d*(a + b*x))/(c + d*x))^2) + (7*B*n*(a + b*x)*(A + B *Log[e*((a + b*x)/(c + d*x))^n]))/(3*d^3*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + (b^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*d^4*(b - (d*(a + b*x))/(c + d*x))^3) - (3*b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/ (2*d^4*(b - (d*(a + b*x))/(c + d*x))^2) + (3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(d^3*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + (B^ 2*n^2*Log[(a + b*x)/(c + d*x)])/(3*d^4) + (2*B^2*n^2*Log[b - (d*(a + b*x)) /(c + d*x)])/d^4 + (6*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - ( d*(a + b*x))/(b*(c + d*x))])/d^4 + ((A + B*Log[e*((a + b*x)/(c + d*x))^n]) ^2*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/d^4 - (7*B*n*(A + B*Log[e*((a + b *x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(3*d^4) + (6*B^2* n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^4 + (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/d^4 + (7*B ^2*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(3*d^4) - (2*B^2*n^2*PolyL og[3, (d*(a + b*x))/(b*(c + d*x))])/d^4))/(4*b))
3.2.64.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x) ^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^(m + 1)*(g/b)^m Subst[Int[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] && Ne Q[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[b*f - a*g, 0] && (GtQ[p, 0] || Lt Q[m, -1])
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]
\[\int \left (b x +a \right )^{3} {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{3}d x\]
\[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (b x + a\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
integral(A^3*b^3*x^3 + 3*A^3*a*b^2*x^2 + 3*A^3*a^2*b*x + A^3*a^3 + (B^3*b^ 3*x^3 + 3*B^3*a*b^2*x^2 + 3*B^3*a^2*b*x + B^3*a^3)*log((b*x + a)^n*e/(d*x + c)^n)^3 + 3*(A*B^2*b^3*x^3 + 3*A*B^2*a*b^2*x^2 + 3*A*B^2*a^2*b*x + A*B^2 *a^3)*log((b*x + a)^n*e/(d*x + c)^n)^2 + 3*(A^2*B*b^3*x^3 + 3*A^2*B*a*b^2* x^2 + 3*A^2*B*a^2*b*x + A^2*B*a^3)*log((b*x + a)^n*e/(d*x + c)^n), x)
Exception generated. \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (b x + a\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
3/4*A^2*B*b^3*x^4*log((b*x + a)^n*e/(d*x + c)^n) + 1/4*A^3*b^3*x^4 + 3*A^2 *B*a*b^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + A^3*a*b^2*x^3 + 9/2*A^2*B*a^ 2*b*x^2*log((b*x + a)^n*e/(d*x + c)^n) + 3/2*A^3*a^2*b*x^2 + 3*A^2*B*a^3*x *log((b*x + a)^n*e/(d*x + c)^n) + A^3*a^3*x + 3*(a*e*n*log(b*x + a)/b - c* e*n*log(d*x + c)/d)*A^2*B*a^3/e - 9/2*(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*a^2*b/e + 3/2*(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*a*b^2/e - 1/8*(6*a^4*e*n*log(b*x + a)/b^4 - 6*c^4*e*n*log(d*x + c)/d^4 + (2*(b^3*c*d ^2*e*n - a*b^2*d^3*e*n)*x^3 - 3*(b^3*c^2*d*e*n - a^2*b*d^3*e*n)*x^2 + 6*(b ^3*c^3*e*n - a^3*d^3*e*n)*x)/(b^3*d^3))*A^2*B*b^3/e - 1/8*(2*(B^3*b^4*d^4* x^4 + 4*B^3*a*b^3*d^4*x^3 + 6*B^3*a^2*b^2*d^4*x^2 + 4*B^3*a^3*b*d^4*x)*log ((d*x + c)^n)^3 - (6*B^3*a^4*d^4*n*log(b*x + a) + 6*(B^3*b^4*d^4*log(e) + A*B^2*b^4*d^4)*x^4 + 6*(b^4*c^4*n - 4*a*b^3*c^3*d*n + 6*a^2*b^2*c^2*d^2*n - 4*a^3*b*c*d^3*n)*B^3*log(d*x + c) + 2*(12*A*B^2*a*b^3*d^4 + (a*b^3*d^4*( n + 12*log(e)) - b^4*c*d^3*n)*B^3)*x^3 + 3*(12*A*B^2*a^2*b^2*d^4 + (3*a^2* b^2*d^4*(n + 4*log(e)) + b^4*c^2*d^2*n - 4*a*b^3*c*d^3*n)*B^3)*x^2 + 6*(4* A*B^2*a^3*b*d^4 + (a^3*b*d^4*(3*n + 4*log(e)) - b^4*c^3*d*n + 4*a*b^3*c^2* d^2*n - 6*a^2*b^2*c*d^3*n)*B^3)*x + 6*(B^3*b^4*d^4*x^4 + 4*B^3*a*b^3*d^4*x ^3 + 6*B^3*a^2*b^2*d^4*x^2 + 4*B^3*a^3*b*d^4*x)*log((b*x + a)^n))*log((...
\[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int { {\left (b x + a\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3} \,d x } \]
Timed out. \[ \int (a+b x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3 \, dx=\int {\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^3\,{\left (a+b\,x\right )}^3 \,d x \]