Integrand size = 33, antiderivative size = 184 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=-\frac {6 B^3 n^3 (c+d x)}{(b c-a d) (a+b x)}-\frac {6 B^2 n^2 (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{(b c-a d) (a+b x)}-\frac {3 B n (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(b c-a d) (a+b x)}-\frac {(c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(b c-a d) (a+b x)} \]
-6*B^3*n^3*(d*x+c)/(-a*d+b*c)/(b*x+a)-6*B^2*n^2*(d*x+c)*(A+B*ln(e*(b*x+a)^ n/((d*x+c)^n)))/(-a*d+b*c)/(b*x+a)-3*B*n*(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x +c)^n)))^2/(-a*d+b*c)/(b*x+a)-(d*x+c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^3/ (-a*d+b*c)/(b*x+a)
Leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(184)=368\).
Time = 0.50 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.85 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=\frac {-B^3 d n^3 (a+b x) \log ^3(a+b x)+B^3 d n^3 (a+b x) \log ^3(c+d x)+3 B^2 d n^2 (a+b x) \log ^2(c+d x) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 B^2 d n^2 (a+b x) \log ^2(a+b x) \left (A+B n+B n \log (c+d x)+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+3 B d n (a+b x) \log (c+d x) \left (A^2+2 A B n+2 B^2 n^2+2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-(b c-a d) \left (A^3+3 A^2 B n+6 A B^2 n^2+6 B^3 n^3+3 B \left (A^2+2 A B n+2 B^2 n^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+3 B^2 (A+B n) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+B^3 \log ^3\left (e (a+b x)^n (c+d x)^{-n}\right )\right )-3 B d n (a+b x) \log (a+b x) \left (A^2+2 A B n+2 B^2 n^2+B^2 n^2 \log ^2(c+d x)+2 B (A+B n) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )+2 B n \log (c+d x) \left (A+B n+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )}{b (b c-a d) (a+b x)} \]
(-(B^3*d*n^3*(a + b*x)*Log[a + b*x]^3) + B^3*d*n^3*(a + b*x)*Log[c + d*x]^ 3 + 3*B^2*d*n^2*(a + b*x)*Log[c + d*x]^2*(A + B*n + B*Log[(e*(a + b*x)^n)/ (c + d*x)^n]) + 3*B^2*d*n^2*(a + b*x)*Log[a + b*x]^2*(A + B*n + B*n*Log[c + d*x] + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]) + 3*B*d*n*(a + b*x)*Log[c + d *x]*(A^2 + 2*A*B*n + 2*B^2*n^2 + 2*B*(A + B*n)*Log[(e*(a + b*x)^n)/(c + d* x)^n] + B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2) - (b*c - a*d)*(A^3 + 3*A^2 *B*n + 6*A*B^2*n^2 + 6*B^3*n^3 + 3*B*(A^2 + 2*A*B*n + 2*B^2*n^2)*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 3*B^2*(A + B*n)*Log[(e*(a + b*x)^n)/(c + d*x)^n] ^2 + B^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^3) - 3*B*d*n*(a + b*x)*Log[a + b *x]*(A^2 + 2*A*B*n + 2*B^2*n^2 + B^2*n^2*Log[c + d*x]^2 + 2*B*(A + B*n)*Lo g[(e*(a + b*x)^n)/(c + d*x)^n] + B^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + 2*B*n*Log[c + d*x]*(A + B*n + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])))/(b*(b* c - a*d)*(a + b*x))
Time = 0.44 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.82, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {2973, 2949, 2742, 2742, 2741}
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.
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{(a+b x)^2}dx\) |
| \(\Big \downarrow \) 2949 |
| \(\displaystyle \frac {\int \frac {(c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{(a+b x)^2}d\frac {a+b x}{c+d x}}{b c-a d}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {3 B n \int \frac {(c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^2}d\frac {a+b x}{c+d x}-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}}{b c-a d}\) |
| \(\Big \downarrow \) 2742 |
| \(\displaystyle \frac {3 B n \left (2 B n \int \frac {(c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^2}d\frac {a+b x}{c+d x}-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}\right )-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}}{b c-a d}\) |
| \(\Big \downarrow \) 2741 |
| \(\displaystyle \frac {3 B n \left (2 B n \left (-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac {B n (c+d x)}{a+b x}\right )-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a+b x}\right )-\frac {(c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^3}{a+b x}}{b c-a d}\) |
(-(((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^3)/(a + b*x)) + 3*B*n *(-(((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a + b*x)) + 2*B* n*(-((B*n*(c + d*x))/(a + b*x)) - ((c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x))))/(b*c - a*d)
3.2.68.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbo l] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])^p/(d*(m + 1))), x] - Simp[b*n* (p/(m + 1)) Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b , c, d, m, n}, x] && NeQ[m, -1] && GtQ[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)^(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]
Leaf count of result is larger than twice the leaf count of optimal. \(542\) vs. \(2(184)=368\).
Time = 38.58 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.95
| method | result | size |
| parallelrisch | \(-\frac {-3 A \,B^{2} x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} b^{3} d^{2} n -6 A \,B^{2} x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} d^{2} n^{2}-3 A^{2} B x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} d^{2} n -3 A \,B^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} b^{3} c d n -6 A \,B^{2} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} c d \,n^{2}-3 A^{2} B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} c d n +6 A \,B^{2} a \,b^{2} d^{2} n^{3}-6 A \,B^{2} b^{3} c d \,n^{3}+3 A^{2} B a \,b^{2} d^{2} n^{2}-3 A^{2} B \,b^{3} c d \,n^{2}-B^{3} x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} b^{3} d^{2} n -3 B^{3} x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} b^{3} d^{2} n^{2}-6 B^{3} x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} d^{2} n^{3}-B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3} b^{3} c d n -3 B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} b^{3} c d \,n^{2}+6 B^{3} a \,b^{2} d^{2} n^{4}-6 B^{3} b^{3} c d \,n^{4}+A^{3} a \,b^{2} d^{2} n -A^{3} b^{3} c d n -6 B^{3} \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) b^{3} c d \,n^{3}}{\left (b x +a \right ) b^{3} d n \left (a d -c b \right )}\) | \(543\) |
| risch | \(\text {Expression too large to display}\) | \(69354\) |
-(-3*A*B^2*x*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^3*d^2*n-6*A*B^2*x*ln(e*(b*x+a )^n/((d*x+c)^n))*b^3*d^2*n^2-3*A^2*B*x*ln(e*(b*x+a)^n/((d*x+c)^n))*b^3*d^2 *n-3*A*B^2*ln(e*(b*x+a)^n/((d*x+c)^n))^2*b^3*c*d*n-6*A*B^2*ln(e*(b*x+a)^n/ ((d*x+c)^n))*b^3*c*d*n^2-3*A^2*B*ln(e*(b*x+a)^n/((d*x+c)^n))*b^3*c*d*n+6*A *B^2*a*b^2*d^2*n^3-6*A*B^2*b^3*c*d*n^3+3*A^2*B*a*b^2*d^2*n^2-3*A^2*B*b^3*c *d*n^2-B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))^3*b^3*d^2*n-3*B^3*x*ln(e*(b*x+a)^ n/((d*x+c)^n))^2*b^3*d^2*n^2-6*B^3*x*ln(e*(b*x+a)^n/((d*x+c)^n))*b^3*d^2*n ^3-B^3*ln(e*(b*x+a)^n/((d*x+c)^n))^3*b^3*c*d*n-3*B^3*ln(e*(b*x+a)^n/((d*x+ c)^n))^2*b^3*c*d*n^2+6*B^3*a*b^2*d^2*n^4-6*B^3*b^3*c*d*n^4+A^3*a*b^2*d^2*n -A^3*b^3*c*d*n-6*B^3*ln(e*(b*x+a)^n/((d*x+c)^n))*b^3*c*d*n^3)/(b*x+a)/b^3/ d/n/(a*d-b*c)
Leaf count of result is larger than twice the leaf count of optimal. 825 vs. \(2 (184) = 368\).
Time = 0.29 (sec) , antiderivative size = 825, normalized size of antiderivative = 4.48 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=-\frac {A^{3} b c - A^{3} a d + 6 \, {\left (B^{3} b c - B^{3} a d\right )} n^{3} + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )^{3} - {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (d x + c\right )^{3} + {\left (B^{3} b c - B^{3} a d\right )} \log \left (e\right )^{3} + 6 \, {\left (A B^{2} b c - A B^{2} a d\right )} n^{2} + 3 \, {\left (B^{3} b c n^{3} + A B^{2} b c n^{2} + {\left (B^{3} b d n^{3} + A B^{2} b d n^{2}\right )} x + {\left (B^{3} b d n^{2} x + B^{3} b c n^{2}\right )} \log \left (e\right )\right )} \log \left (b x + a\right )^{2} + 3 \, {\left (B^{3} b c n^{3} + A B^{2} b c n^{2} + {\left (B^{3} b d n^{3} + A B^{2} b d n^{2}\right )} x + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right ) + {\left (B^{3} b d n^{2} x + B^{3} b c n^{2}\right )} \log \left (e\right )\right )} \log \left (d x + c\right )^{2} + 3 \, {\left (A B^{2} b c - A B^{2} a d + {\left (B^{3} b c - B^{3} a d\right )} n\right )} \log \left (e\right )^{2} + 3 \, {\left (A^{2} B b c - A^{2} B a d\right )} n + 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, A B^{2} b c n^{2} + A^{2} B b c n + {\left (B^{3} b d n x + B^{3} b c n\right )} \log \left (e\right )^{2} + {\left (2 \, B^{3} b d n^{3} + 2 \, A B^{2} b d n^{2} + A^{2} B b d n\right )} x + 2 \, {\left (B^{3} b c n^{2} + A B^{2} b c n + {\left (B^{3} b d n^{2} + A B^{2} b d n\right )} x\right )} \log \left (e\right )\right )} \log \left (b x + a\right ) - 3 \, {\left (2 \, B^{3} b c n^{3} + 2 \, A B^{2} b c n^{2} + A^{2} B b c n + {\left (B^{3} b d n^{3} x + B^{3} b c n^{3}\right )} \log \left (b x + a\right )^{2} + {\left (B^{3} b d n x + B^{3} b c n\right )} \log \left (e\right )^{2} + {\left (2 \, B^{3} b d n^{3} + 2 \, A B^{2} b d n^{2} + A^{2} B b d n\right )} x + 2 \, {\left (B^{3} b c n^{3} + A B^{2} b c n^{2} + {\left (B^{3} b d n^{3} + A B^{2} b d n^{2}\right )} x + {\left (B^{3} b d n^{2} x + B^{3} b c n^{2}\right )} \log \left (e\right )\right )} \log \left (b x + a\right ) + 2 \, {\left (B^{3} b c n^{2} + A B^{2} b c n + {\left (B^{3} b d n^{2} + A B^{2} b d n\right )} x\right )} \log \left (e\right )\right )} \log \left (d x + c\right ) + 3 \, {\left (A^{2} B b c - A^{2} B a d + 2 \, {\left (B^{3} b c - B^{3} a d\right )} n^{2} + 2 \, {\left (A B^{2} b c - A B^{2} a d\right )} n\right )} \log \left (e\right )}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \]
-(A^3*b*c - A^3*a*d + 6*(B^3*b*c - B^3*a*d)*n^3 + (B^3*b*d*n^3*x + B^3*b*c *n^3)*log(b*x + a)^3 - (B^3*b*d*n^3*x + B^3*b*c*n^3)*log(d*x + c)^3 + (B^3 *b*c - B^3*a*d)*log(e)^3 + 6*(A*B^2*b*c - A*B^2*a*d)*n^2 + 3*(B^3*b*c*n^3 + A*B^2*b*c*n^2 + (B^3*b*d*n^3 + A*B^2*b*d*n^2)*x + (B^3*b*d*n^2*x + B^3*b *c*n^2)*log(e))*log(b*x + a)^2 + 3*(B^3*b*c*n^3 + A*B^2*b*c*n^2 + (B^3*b*d *n^3 + A*B^2*b*d*n^2)*x + (B^3*b*d*n^3*x + B^3*b*c*n^3)*log(b*x + a) + (B^ 3*b*d*n^2*x + B^3*b*c*n^2)*log(e))*log(d*x + c)^2 + 3*(A*B^2*b*c - A*B^2*a *d + (B^3*b*c - B^3*a*d)*n)*log(e)^2 + 3*(A^2*B*b*c - A^2*B*a*d)*n + 3*(2* B^3*b*c*n^3 + 2*A*B^2*b*c*n^2 + A^2*B*b*c*n + (B^3*b*d*n*x + B^3*b*c*n)*lo g(e)^2 + (2*B^3*b*d*n^3 + 2*A*B^2*b*d*n^2 + A^2*B*b*d*n)*x + 2*(B^3*b*c*n^ 2 + A*B^2*b*c*n + (B^3*b*d*n^2 + A*B^2*b*d*n)*x)*log(e))*log(b*x + a) - 3* (2*B^3*b*c*n^3 + 2*A*B^2*b*c*n^2 + A^2*B*b*c*n + (B^3*b*d*n^3*x + B^3*b*c* n^3)*log(b*x + a)^2 + (B^3*b*d*n*x + B^3*b*c*n)*log(e)^2 + (2*B^3*b*d*n^3 + 2*A*B^2*b*d*n^2 + A^2*B*b*d*n)*x + 2*(B^3*b*c*n^3 + A*B^2*b*c*n^2 + (B^3 *b*d*n^3 + A*B^2*b*d*n^2)*x + (B^3*b*d*n^2*x + B^3*b*c*n^2)*log(e))*log(b* x + a) + 2*(B^3*b*c*n^2 + A*B^2*b*c*n + (B^3*b*d*n^2 + A*B^2*b*d*n)*x)*log (e))*log(d*x + c) + 3*(A^2*B*b*c - A^2*B*a*d + 2*(B^3*b*c - B^3*a*d)*n^2 + 2*(A*B^2*b*c - A*B^2*a*d)*n)*log(e))/(a*b^2*c - a^2*b*d + (b^3*c - a*b^2* d)*x)
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1129 vs. \(2 (184) = 368\).
Time = 0.26 (sec) , antiderivative size = 1129, normalized size of antiderivative = 6.14 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=\text {Too large to display} \]
-B^3*log((b*x + a)^n*e/(d*x + c)^n)^3/(b^2*x + a*b) - (3*(d*e*n*log(b*x + a)/(b^2*c - a*b*d) - d*e*n*log(d*x + c)/(b^2*c - a*b*d) + e*n/(b^2*x + a*b ))*log((b*x + a)^n*e/(d*x + c)^n)^2/e + (3*(2*b*c*e^2*n^2 - 2*a*d*e^2*n^2 - (b*d*e^2*n^2*x + a*d*e^2*n^2)*log(b*x + a)^2 - (b*d*e^2*n^2*x + a*d*e^2* n^2)*log(d*x + c)^2 + 2*(b*d*e^2*n^2*x + a*d*e^2*n^2)*log(b*x + a) - 2*(b* d*e^2*n^2*x + a*d*e^2*n^2 - (b*d*e^2*n^2*x + a*d*e^2*n^2)*log(b*x + a))*lo g(d*x + c))*log((b*x + a)^n*e/(d*x + c)^n)/((a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x)*e) + (6*b*c*e^3*n^3 - 6*a*d*e^3*n^3 + (b*d*e^3*n^3*x + a*d*e^3 *n^3)*log(b*x + a)^3 - (b*d*e^3*n^3*x + a*d*e^3*n^3)*log(d*x + c)^3 - 3*(b *d*e^3*n^3*x + a*d*e^3*n^3)*log(b*x + a)^2 - 3*(b*d*e^3*n^3*x + a*d*e^3*n^ 3 - (b*d*e^3*n^3*x + a*d*e^3*n^3)*log(b*x + a))*log(d*x + c)^2 + 6*(b*d*e^ 3*n^3*x + a*d*e^3*n^3)*log(b*x + a) - 3*(2*b*d*e^3*n^3*x + 2*a*d*e^3*n^3 + (b*d*e^3*n^3*x + a*d*e^3*n^3)*log(b*x + a)^2 - 2*(b*d*e^3*n^3*x + a*d*e^3 *n^3)*log(b*x + a))*log(d*x + c))/((a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)* x)*e^2))/e)*B^3 - 3*A*B^2*(2*(d*e*n*log(b*x + a)/(b^2*c - a*b*d) - d*e*n*l og(d*x + c)/(b^2*c - a*b*d) + e*n/(b^2*x + a*b))*log((b*x + a)^n*e/(d*x + c)^n)/e + (2*b*c*e^2*n^2 - 2*a*d*e^2*n^2 - (b*d*e^2*n^2*x + a*d*e^2*n^2)*l og(b*x + a)^2 - (b*d*e^2*n^2*x + a*d*e^2*n^2)*log(d*x + c)^2 + 2*(b*d*e^2* n^2*x + a*d*e^2*n^2)*log(b*x + a) - 2*(b*d*e^2*n^2*x + a*d*e^2*n^2 - (b*d* e^2*n^2*x + a*d*e^2*n^2)*log(b*x + a))*log(d*x + c))/((a*b^2*c - a^2*b*...
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{3}}{{\left (b x + a\right )}^{2}} \,d x } \]
Time = 2.76 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.58 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{(a+b x)^2} \, dx=-\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\,\left (\frac {3\,B\,b\,d\,A^2\,x^2+3\,B\,\left (a\,d+b\,c\right )\,A^2\,x+3\,B\,a\,c\,A^2}{b\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}+\frac {6\,d\,\left (n\,B^3+A\,B^2\right )\,\left (b^2\,n\,x^2\,\left (a\,d-b\,c\right )+\frac {a\,b\,c\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b\,n\,x\,\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d}\right )}{b^2\,\left (a\,d-b\,c\right )\,{\left (a+b\,x\right )}^2\,\left (c+d\,x\right )}\right )-\frac {A^3+3\,A^2\,B\,n+6\,A\,B^2\,n^2+6\,B^3\,n^3}{x\,b^2+a\,b}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^2\,\left (\frac {3\,A\,B^2}{x\,b^2+a\,b}+\frac {3\,B^3\,n}{x\,b^2+a\,b}-\frac {3\,d\,\left (n\,B^3+A\,B^2\right )}{b\,\left (a\,d-b\,c\right )}\right )-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )}^3\,\left (\frac {B^3}{b\,\left (a+b\,x\right )}-\frac {B^3\,d}{b\,\left (a\,d-b\,c\right )}\right )-\frac {B\,d\,n\,\mathrm {atan}\left (\frac {B\,d\,n\,\left (\frac {c\,b^2+a\,d\,b}{b}+2\,b\,d\,x\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,3{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (3\,d\,A^2\,B\,n+6\,d\,A\,B^2\,n^2+6\,d\,B^3\,n^3\right )}\right )\,\left (A^2+2\,A\,B\,n+2\,B^2\,n^2\right )\,6{}\mathrm {i}}{b\,\left (a\,d-b\,c\right )} \]
- log((e*(a + b*x)^n)/(c + d*x)^n)*((3*A^2*B*a*c + 3*A^2*B*x*(a*d + b*c) + 3*A^2*B*b*d*x^2)/(b*(a + b*x)^2*(c + d*x)) + (6*d*(A*B^2 + B^3*n)*(b^2*n* x^2*(a*d - b*c) + (a*b*c*n*(a*d - b*c))/d + (b*n*x*(a*d + b*c)*(a*d - b*c) )/d))/(b^2*(a*d - b*c)*(a + b*x)^2*(c + d*x))) - (A^3 + 6*B^3*n^3 + 6*A*B^ 2*n^2 + 3*A^2*B*n)/(a*b + b^2*x) - log((e*(a + b*x)^n)/(c + d*x)^n)^2*((3* A*B^2)/(a*b + b^2*x) + (3*B^3*n)/(a*b + b^2*x) - (3*d*(A*B^2 + B^3*n))/(b* (a*d - b*c))) - log((e*(a + b*x)^n)/(c + d*x)^n)^3*(B^3/(b*(a + b*x)) - (B ^3*d)/(b*(a*d - b*c))) - (B*d*n*atan((B*d*n*((b^2*c + a*b*d)/b + 2*b*d*x)* (A^2 + 2*B^2*n^2 + 2*A*B*n)*3i)/((a*d - b*c)*(6*B^3*d*n^3 + 3*A^2*B*d*n + 6*A*B^2*d*n^2)))*(A^2 + 2*B^2*n^2 + 2*A*B*n)*6i)/(b*(a*d - b*c))