Integrand size = 29, antiderivative size = 193 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=-\frac {p r}{16 b (a+b x)^4}-\frac {d q r}{12 b (b c-a d) (a+b x)^3}+\frac {d^2 q r}{8 b (b c-a d)^2 (a+b x)^2}-\frac {d^3 q r}{4 b (b c-a d)^3 (a+b x)}-\frac {d^4 q r \log (a+b x)}{4 b (b c-a d)^4}+\frac {d^4 q r \log (c+d x)}{4 b (b c-a d)^4}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4} \]
-1/16*p*r/b/(b*x+a)^4-1/12*d*q*r/b/(-a*d+b*c)/(b*x+a)^3+1/8*d^2*q*r/b/(-a* d+b*c)^2/(b*x+a)^2-1/4*d^3*q*r/b/(-a*d+b*c)^3/(b*x+a)-1/4*d^4*q*r*ln(b*x+a )/b/(-a*d+b*c)^4+1/4*d^4*q*r*ln(d*x+c)/b/(-a*d+b*c)^4-1/4*ln(e*(f*(b*x+a)^ p*(d*x+c)^q)^r)/b/(b*x+a)^4
Time = 0.21 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.85 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\frac {r \left (\frac {-3 p+\frac {4 d q (a+b x)}{-b c+a d}+\frac {6 d^2 q (a+b x)^2}{(b c-a d)^2}-\frac {12 d^3 q (a+b x)^3}{(b c-a d)^3}}{12 (a+b x)^4}-\frac {d^4 q \log (a+b x)}{(b c-a d)^4}+\frac {d^4 q \log (c+d x)}{(b c-a d)^4}\right )-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^4}}{4 b} \]
(r*((-3*p + (4*d*q*(a + b*x))/(-(b*c) + a*d) + (6*d^2*q*(a + b*x)^2)/(b*c - a*d)^2 - (12*d^3*q*(a + b*x)^3)/(b*c - a*d)^3)/(12*(a + b*x)^4) - (d^4*q *Log[a + b*x])/(b*c - a*d)^4 + (d^4*q*Log[c + d*x])/(b*c - a*d)^4) - Log[e *(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^4)/(4*b)
Time = 0.34 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.88, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2981, 17, 54, 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.
| \(\displaystyle \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx\) |
| \(\Big \downarrow \) 2981 |
| \(\displaystyle \frac {d q r \int \frac {1}{(a+b x)^4 (c+d x)}dx}{4 b}+\frac {1}{4} p r \int \frac {1}{(a+b x)^5}dx-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle \frac {d q r \int \frac {1}{(a+b x)^4 (c+d x)}dx}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {d q r \int \left (\frac {d^4}{(b c-a d)^4 (c+d x)}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b}{(b c-a d) (a+b x)^4}\right )dx}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d q r \left (-\frac {d^3 \log (a+b x)}{(b c-a d)^4}+\frac {d^3 \log (c+d x)}{(b c-a d)^4}-\frac {d^2}{(a+b x) (b c-a d)^3}+\frac {d}{2 (a+b x)^2 (b c-a d)^2}-\frac {1}{3 (a+b x)^3 (b c-a d)}\right )}{4 b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{4 b (a+b x)^4}-\frac {p r}{16 b (a+b x)^4}\) |
-1/16*(p*r)/(b*(a + b*x)^4) + (d*q*r*(-1/3*1/((b*c - a*d)*(a + b*x)^3) + d /(2*(b*c - a*d)^2*(a + b*x)^2) - d^2/((b*c - a*d)^3*(a + b*x)) - (d^3*Log[ a + b*x])/(b*c - a*d)^4 + (d^3*Log[c + d*x])/(b*c - a*d)^4))/(4*b) - Log[e *(f*(a + b*x)^p*(c + d*x)^q)^r]/(4*b*(a + b*x)^4)
3.1.15.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)*(Lo g[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1))), x] + (-Simp[b*p*(r/(h*(m + 1))) Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Simp[d*q*(r/(h*(m + 1))) Int[(g + h*x)^(m + 1)/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h , m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
\[\int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (b x +a \right )^{5}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (179) = 358\).
Time = 0.35 (sec) , antiderivative size = 861, normalized size of antiderivative = 4.46 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=-\frac {12 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )} q r x^{3} - 6 \, {\left (b^{4} c^{2} d^{2} - 8 \, a b^{3} c d^{3} + 7 \, a^{2} b^{2} d^{4}\right )} q r x^{2} + 4 \, {\left (b^{4} c^{3} d - 6 \, a b^{3} c^{2} d^{2} + 18 \, a^{2} b^{2} c d^{3} - 13 \, a^{3} b d^{4}\right )} q r x + 12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} r \log \left (f\right ) + {\left (3 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} p + 2 \, {\left (2 \, a b^{3} c^{3} d - 9 \, a^{2} b^{2} c^{2} d^{2} + 18 \, a^{3} b c d^{3} - 11 \, a^{4} d^{4}\right )} q\right )} r + 12 \, {\left (b^{4} d^{4} q r x^{4} + 4 \, a b^{3} d^{4} q r x^{3} + 6 \, a^{2} b^{2} d^{4} q r x^{2} + 4 \, a^{3} b d^{4} q r x + {\left (a^{4} d^{4} q + {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} p\right )} r\right )} \log \left (b x + a\right ) - 12 \, {\left (b^{4} d^{4} q r x^{4} + 4 \, a b^{3} d^{4} q r x^{3} + 6 \, a^{2} b^{2} d^{4} q r x^{2} + 4 \, a^{3} b d^{4} q r x - {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3}\right )} q r\right )} \log \left (d x + c\right ) + 12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \log \left (e\right )}{48 \, {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4} + {\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} x\right )}} \]
-1/48*(12*(b^4*c*d^3 - a*b^3*d^4)*q*r*x^3 - 6*(b^4*c^2*d^2 - 8*a*b^3*c*d^3 + 7*a^2*b^2*d^4)*q*r*x^2 + 4*(b^4*c^3*d - 6*a*b^3*c^2*d^2 + 18*a^2*b^2*c* d^3 - 13*a^3*b*d^4)*q*r*x + 12*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^ 2 - 4*a^3*b*c*d^3 + a^4*d^4)*r*log(f) + (3*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^ 2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*p + 2*(2*a*b^3*c^3*d - 9*a^2*b^2* c^2*d^2 + 18*a^3*b*c*d^3 - 11*a^4*d^4)*q)*r + 12*(b^4*d^4*q*r*x^4 + 4*a*b^ 3*d^4*q*r*x^3 + 6*a^2*b^2*d^4*q*r*x^2 + 4*a^3*b*d^4*q*r*x + (a^4*d^4*q + ( b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*p)* r)*log(b*x + a) - 12*(b^4*d^4*q*r*x^4 + 4*a*b^3*d^4*q*r*x^3 + 6*a^2*b^2*d^ 4*q*r*x^2 + 4*a^3*b*d^4*q*r*x - (b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d ^2 - 4*a^3*b*c*d^3)*q*r)*log(d*x + c) + 12*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^ 2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*log(e))/(a^4*b^5*c^4 - 4*a^5*b^4* c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4 + (b^9*c^4 - 4*a*b ^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*x^4 + 4*(a*b ^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d ^4)*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4 *c*d^3 + a^6*b^3*d^4)*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c ^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*x)
Timed out. \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 459 vs. \(2 (179) = 358\).
Time = 0.20 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.38 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=-\frac {{\left (2 \, {\left (\frac {6 \, d^{3} \log \left (b x + a\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}} + \frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3} + {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c^{3} - 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{3} b^{3} c d^{2} - a^{4} b^{2} d^{3}\right )} x^{2} + 3 \, {\left (a^{2} b^{4} c^{3} - 3 \, a^{3} b^{3} c^{2} d + 3 \, a^{4} b^{2} c d^{2} - a^{5} b d^{3}\right )} x}\right )} d f q + \frac {3 \, b f p}{b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b}\right )} r}{48 \, b f} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{4 \, {\left (b x + a\right )}^{4} b} \]
-1/48*(2*(6*d^3*log(b*x + a)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) - 6*d^3*log(d*x + c)/(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4) + (6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/(a^3*b^3*c^3 - 3*a^ 4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3 + (b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b ^4*c*d^2 - a^3*b^3*d^3)*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c *d^2 - a^4*b^2*d^3)*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d ^2 - a^5*b*d^3)*x))*d*f*q + 3*b*f*p/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b))*r/(b*f) - 1/4*log(((b*x + a)^p*(d*x + c)^q*f)^r*e )/((b*x + a)^4*b)
Leaf count of result is larger than twice the leaf count of optimal. 756 vs. \(2 (179) = 358\).
Time = 0.29 (sec) , antiderivative size = 756, normalized size of antiderivative = 3.92 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=-\frac {d^{4} q r \log \left (b x + a\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} + \frac {d^{4} q r \log \left (d x + c\right )}{4 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )}} - \frac {p r \log \left (b x + a\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {q r \log \left (d x + c\right )}{4 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {12 \, b^{3} d^{3} q r x^{3} - 6 \, b^{3} c d^{2} q r x^{2} + 42 \, a b^{2} d^{3} q r x^{2} + 4 \, b^{3} c^{2} d q r x - 20 \, a b^{2} c d^{2} q r x + 52 \, a^{2} b d^{3} q r x + 3 \, b^{3} c^{3} p r - 9 \, a b^{2} c^{2} d p r + 9 \, a^{2} b c d^{2} p r - 3 \, a^{3} d^{3} p r + 4 \, a b^{2} c^{2} d q r - 14 \, a^{2} b c d^{2} q r + 22 \, a^{3} d^{3} q r + 12 \, b^{3} c^{3} r \log \left (f\right ) - 36 \, a b^{2} c^{2} d r \log \left (f\right ) + 36 \, a^{2} b c d^{2} r \log \left (f\right ) - 12 \, a^{3} d^{3} r \log \left (f\right ) + 12 \, b^{3} c^{3} \log \left (e\right ) - 36 \, a b^{2} c^{2} d \log \left (e\right ) + 36 \, a^{2} b c d^{2} \log \left (e\right ) - 12 \, a^{3} d^{3} \log \left (e\right )}{48 \, {\left (b^{8} c^{3} x^{4} - 3 \, a b^{7} c^{2} d x^{4} + 3 \, a^{2} b^{6} c d^{2} x^{4} - a^{3} b^{5} d^{3} x^{4} + 4 \, a b^{7} c^{3} x^{3} - 12 \, a^{2} b^{6} c^{2} d x^{3} + 12 \, a^{3} b^{5} c d^{2} x^{3} - 4 \, a^{4} b^{4} d^{3} x^{3} + 6 \, a^{2} b^{6} c^{3} x^{2} - 18 \, a^{3} b^{5} c^{2} d x^{2} + 18 \, a^{4} b^{4} c d^{2} x^{2} - 6 \, a^{5} b^{3} d^{3} x^{2} + 4 \, a^{3} b^{5} c^{3} x - 12 \, a^{4} b^{4} c^{2} d x + 12 \, a^{5} b^{3} c d^{2} x - 4 \, a^{6} b^{2} d^{3} x + a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )}} \]
-1/4*d^4*q*r*log(b*x + a)/(b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4 *a^3*b^2*c*d^3 + a^4*b*d^4) + 1/4*d^4*q*r*log(d*x + c)/(b^5*c^4 - 4*a*b^4* c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4) - 1/4*p*r*log(b*x + a)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^4*b) - 1/4* q*r*log(d*x + c)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3*x^2 + 4*a^3*b^2*x + a^ 4*b) - 1/48*(12*b^3*d^3*q*r*x^3 - 6*b^3*c*d^2*q*r*x^2 + 42*a*b^2*d^3*q*r*x ^2 + 4*b^3*c^2*d*q*r*x - 20*a*b^2*c*d^2*q*r*x + 52*a^2*b*d^3*q*r*x + 3*b^3 *c^3*p*r - 9*a*b^2*c^2*d*p*r + 9*a^2*b*c*d^2*p*r - 3*a^3*d^3*p*r + 4*a*b^2 *c^2*d*q*r - 14*a^2*b*c*d^2*q*r + 22*a^3*d^3*q*r + 12*b^3*c^3*r*log(f) - 3 6*a*b^2*c^2*d*r*log(f) + 36*a^2*b*c*d^2*r*log(f) - 12*a^3*d^3*r*log(f) + 1 2*b^3*c^3*log(e) - 36*a*b^2*c^2*d*log(e) + 36*a^2*b*c*d^2*log(e) - 12*a^3* d^3*log(e))/(b^8*c^3*x^4 - 3*a*b^7*c^2*d*x^4 + 3*a^2*b^6*c*d^2*x^4 - a^3*b ^5*d^3*x^4 + 4*a*b^7*c^3*x^3 - 12*a^2*b^6*c^2*d*x^3 + 12*a^3*b^5*c*d^2*x^3 - 4*a^4*b^4*d^3*x^3 + 6*a^2*b^6*c^3*x^2 - 18*a^3*b^5*c^2*d*x^2 + 18*a^4*b ^4*c*d^2*x^2 - 6*a^5*b^3*d^3*x^2 + 4*a^3*b^5*c^3*x - 12*a^4*b^4*c^2*d*x + 12*a^5*b^3*c*d^2*x - 4*a^6*b^2*d^3*x + a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a ^6*b^2*c*d^2 - a^7*b*d^3)
Time = 4.79 (sec) , antiderivative size = 526, normalized size of antiderivative = 2.73 \[ \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^5} \, dx=\frac {\frac {3\,b^3\,c^3\,p\,r-3\,a^3\,d^3\,p\,r+22\,a^3\,d^3\,q\,r-9\,a\,b^2\,c^2\,d\,p\,r+9\,a^2\,b\,c\,d^2\,p\,r+4\,a\,b^2\,c^2\,d\,q\,r-14\,a^2\,b\,c\,d^2\,q\,r}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {x\,\left (13\,q\,r\,a^2\,b\,d^3-5\,q\,r\,a\,b^2\,c\,d^2+q\,r\,b^3\,c^2\,d\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x^2\,\left (7\,a\,b^2\,d^2\,q\,r-b^3\,c\,d\,q\,r\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {b^3\,d^3\,q\,r\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b+16\,a^3\,b^2\,x+24\,a^2\,b^3\,x^2+16\,a\,b^4\,x^3+4\,b^5\,x^4}-\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (\frac {x}{4}+\frac {a}{4\,b}\right )}{{\left (a+b\,x\right )}^5}+\frac {d^4\,q\,r\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4+8\,a^3\,b^2\,c\,d^3-8\,a\,b^4\,c^3\,d+4\,b^5\,c^4}{4\,b\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,{\left (a\,d-b\,c\right )}^4} \]
((3*b^3*c^3*p*r - 3*a^3*d^3*p*r + 22*a^3*d^3*q*r - 9*a*b^2*c^2*d*p*r + 9*a ^2*b*c*d^2*p*r + 4*a*b^2*c^2*d*q*r - 14*a^2*b*c*d^2*q*r)/(12*(a^3*d^3 - b^ 3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (x*(13*a^2*b*d^3*q*r + b^3*c^2*d *q*r - 5*a*b^2*c*d^2*q*r))/(3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b *c*d^2)) + (d*x^2*(7*a*b^2*d^2*q*r - b^3*c*d*q*r))/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (b^3*d^3*q*r*x^3)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(4*a^4*b + 4*b^5*x^4 + 16*a^3*b^2*x + 16*a *b^4*x^3 + 24*a^2*b^3*x^2) - (log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(x/4 + a/(4*b)))/(a + b*x)^5 + (d^4*q*r*atanh((4*b^5*c^4 - 4*a^4*b*d^4 + 8*a^3*b^ 2*c*d^3 - 8*a*b^4*c^3*d)/(4*b*(a*d - b*c)^4) - (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c)^4))/(2*b*(a*d - b*c)^4)