Integrand size = 31, antiderivative size = 465 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=-\frac {2 p^2 r^2}{b (a+b x)}+\frac {2 d p q r^2 \log (a+b x)}{b (b c-a d)}-\frac {d p q r^2 \log ^2(a+b x)}{b (b c-a d)}-\frac {2 d p q r^2 \log (c+d x)}{b (b c-a d)}+\frac {2 d p q r^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)}+\frac {d q^2 r^2 \log ^2(c+d x)}{b (b c-a d)}-\frac {2 d q^2 r^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)}-\frac {2 p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {2 d q r \log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac {2 d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (b c-a d)}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}-\frac {2 d q^2 r^2 \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)}+\frac {2 d p q r^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)} \]
-2*p^2*r^2/b/(b*x+a)+2*d*p*q*r^2*ln(b*x+a)/b/(-a*d+b*c)-d*p*q*r^2*ln(b*x+a )^2/b/(-a*d+b*c)-2*d*p*q*r^2*ln(d*x+c)/b/(-a*d+b*c)+2*d*p*q*r^2*ln(-d*(b*x +a)/(-a*d+b*c))*ln(d*x+c)/b/(-a*d+b*c)+d*q^2*r^2*ln(d*x+c)^2/b/(-a*d+b*c)- 2*d*q^2*r^2*ln(b*x+a)*ln(b*(d*x+c)/(-a*d+b*c))/b/(-a*d+b*c)-2*p*r*ln(e*(f* (b*x+a)^p*(d*x+c)^q)^r)/b/(b*x+a)+2*d*q*r*ln(b*x+a)*ln(e*(f*(b*x+a)^p*(d*x +c)^q)^r)/b/(-a*d+b*c)-2*d*q*r*ln(d*x+c)*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b /(-a*d+b*c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)^2/b/(b*x+a)-2*d*q^2*r^2*polylo g(2,-d*(b*x+a)/(-a*d+b*c))/b/(-a*d+b*c)+2*d*p*q*r^2*polylog(2,b*(d*x+c)/(- a*d+b*c))/b/(-a*d+b*c)
Time = 0.40 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.88 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\frac {-2 b c p^2 r^2+2 a d p^2 r^2-d p q r^2 (a+b x) \log ^2(a+b x)-2 a d p q r^2 \log (c+d x)-2 b d p q r^2 x \log (c+d x)+a d q^2 r^2 \log ^2(c+d x)+b d q^2 r^2 x \log ^2(c+d x)-2 b c p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 a d p r \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 a d q r \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-2 b d q r x \log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )-b c \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+a d \log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )+2 d q r (a+b x) \log (a+b x) \left (p r+p r \log (c+d x)-(p+q) r \log \left (\frac {b (c+d x)}{b c-a d}\right )+\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )\right )-2 d q (p+q) r^2 (a+b x) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{b (b c-a d) (a+b x)} \]
(-2*b*c*p^2*r^2 + 2*a*d*p^2*r^2 - d*p*q*r^2*(a + b*x)*Log[a + b*x]^2 - 2*a *d*p*q*r^2*Log[c + d*x] - 2*b*d*p*q*r^2*x*Log[c + d*x] + a*d*q^2*r^2*Log[c + d*x]^2 + b*d*q^2*r^2*x*Log[c + d*x]^2 - 2*b*c*p*r*Log[e*(f*(a + b*x)^p* (c + d*x)^q)^r] + 2*a*d*p*r*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 2*a*d*q *r*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - 2*b*d*q*r*x*Log[c + d*x]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - b*c*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + a*d*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2 + 2*d*q*r*(a + b*x)*Log[a + b*x]*(p*r + p*r*Log[c + d*x] - (p + q)*r*Log[(b*(c + d*x))/(b *c - a*d)] + Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) - 2*d*q*(p + q)*r^2*(a + b*x)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(b*c - a*d)*(a + b*x))
Time = 0.81 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.88, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {2984, 2981, 17, 47, 16, 2994, 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.
| \(\displaystyle \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx\) |
| \(\Big \downarrow \) 2984 |
| \(\displaystyle 2 p r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2}dx+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (c+d x)}dx}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}\) |
| \(\Big \downarrow \) 2981 |
| \(\displaystyle 2 p r \left (\frac {d q r \int \frac {1}{(a+b x) (c+d x)}dx}{b}+p r \int \frac {1}{(a+b x)^2}dx-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}\right )+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (c+d x)}dx}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle 2 p r \left (\frac {d q r \int \frac {1}{(a+b x) (c+d x)}dx}{b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}-\frac {p r}{b (a+b x)}\right )+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (c+d x)}dx}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle 2 p r \left (\frac {d q r \left (\frac {b \int \frac {1}{a+b x}dx}{b c-a d}-\frac {d \int \frac {1}{c+d x}dx}{b c-a d}\right )}{b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}-\frac {p r}{b (a+b x)}\right )+\frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (c+d x)}dx}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 d q r \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x) (c+d x)}dx}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+2 p r \left (-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {d q r \left (\frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d}\right )}{b}-\frac {p r}{b (a+b x)}\right )\) |
| \(\Big \downarrow \) 2994 |
| \(\displaystyle \frac {2 d q r \int \left (\frac {b \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d) (a+b x)}-\frac {d \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(b c-a d) (c+d x)}\right )dx}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+2 p r \left (-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {d q r \left (\frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d}\right )}{b}-\frac {p r}{b (a+b x)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 d q r \left (\frac {\log (a+b x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b c-a d}-\frac {\log (c+d x) \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b c-a d}+\frac {p r \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{b c-a d}-\frac {p r \log ^2(a+b x)}{2 (b c-a d)}+\frac {p r \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{b c-a d}-\frac {q r \operatorname {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b c-a d}+\frac {q r \log ^2(c+d x)}{2 (b c-a d)}-\frac {q r \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b c-a d}\right )}{b}-\frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+2 p r \left (-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {d q r \left (\frac {\log (a+b x)}{b c-a d}-\frac {\log (c+d x)}{b c-a d}\right )}{b}-\frac {p r}{b (a+b x)}\right )\) |
-(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^2/(b*(a + b*x))) + 2*p*r*(-((p*r)/( b*(a + b*x))) + (d*q*r*(Log[a + b*x]/(b*c - a*d) - Log[c + d*x]/(b*c - a*d )))/b - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(b*(a + b*x))) + (2*d*q*r*(-1 /2*(p*r*Log[a + b*x]^2)/(b*c - a*d) + (p*r*Log[-((d*(a + b*x))/(b*c - a*d) )]*Log[c + d*x])/(b*c - a*d) + (q*r*Log[c + d*x]^2)/(2*(b*c - a*d)) - (q*r *Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b*c - a*d) + (Log[a + b*x]* Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(b*c - a*d) - (Log[c + d*x]*Log[e*(f *(a + b*x)^p*(c + d*x)^q)^r])/(b*c - a*d) - (q*r*PolyLog[2, -((d*(a + b*x) )/(b*c - a*d))])/(b*c - a*d) + (p*r*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) /(b*c - a*d)))/b
3.1.21.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
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[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]^(s_)*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1 )*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s/(h*(m + 1))), x] + (-Simp[b*p*r*( s/(h*(m + 1))) Int[(g + h*x)^(m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r ]^(s - 1)/(a + b*x)), x], x] - Simp[d*q*r*(s/(h*(m + 1))) Int[(g + h*x)^( m + 1)*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1)/(c + d*x)), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && NeQ[m, -1]
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]^(s_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c , d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]
\[\int \frac {{\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}^{2}}{\left (b x +a \right )^{2}}d x\]
\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{2}} \,d x } \]
\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\int \frac {\log {\left (e \left (f \left (a + b x\right )^{p} \left (c + d x\right )^{q}\right )^{r} \right )}^{2}}{\left (a + b x\right )^{2}}\, dx \]
Time = 0.23 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.84 \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\frac {2 \, {\left (\frac {d f q \log \left (b x + a\right )}{b c - a d} - \frac {d f q \log \left (d x + c\right )}{b c - a d} - \frac {f p}{b x + a}\right )} r \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{b f} - \frac {{\left (\frac {2 \, d f^{2} p q \log \left (d x + c\right )}{b c - a d} + \frac {2 \, {\left (p q + q^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} d f^{2}}{b c - a d} + \frac {2 \, b c f^{2} p^{2} - 2 \, a d f^{2} p^{2} + {\left (b d f^{2} p q x + a d f^{2} p q\right )} \log \left (b x + a\right )^{2} - 2 \, {\left (b d f^{2} p q x + a d f^{2} p q\right )} \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b d f^{2} q^{2} x + a d f^{2} q^{2}\right )} \log \left (d x + c\right )^{2} - 2 \, {\left (b d f^{2} p q x + a d f^{2} p q\right )} \log \left (b x + a\right )}{a b c - a^{2} d + {\left (b^{2} c - a b d\right )} x}\right )} r^{2}}{b f^{2}} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )} b} \]
2*(d*f*q*log(b*x + a)/(b*c - a*d) - d*f*q*log(d*x + c)/(b*c - a*d) - f*p/( b*x + a))*r*log(((b*x + a)^p*(d*x + c)^q*f)^r*e)/(b*f) - (2*d*f^2*p*q*log( d*x + c)/(b*c - a*d) + 2*(p*q + q^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*d*f^2/(b*c - a*d) + (2*b* c*f^2*p^2 - 2*a*d*f^2*p^2 + (b*d*f^2*p*q*x + a*d*f^2*p*q)*log(b*x + a)^2 - 2*(b*d*f^2*p*q*x + a*d*f^2*p*q)*log(b*x + a)*log(d*x + c) - (b*d*f^2*q^2* x + a*d*f^2*q^2)*log(d*x + c)^2 - 2*(b*d*f^2*p*q*x + a*d*f^2*p*q)*log(b*x + a))/(a*b*c - a^2*d + (b^2*c - a*b*d)*x))*r^2/(b*f^2) - log(((b*x + a)^p* (d*x + c)^q*f)^r*e)^2/((b*x + a)*b)
\[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\int { \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )^{2}}{{\left (b x + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\log ^2\left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx=\int \frac {{\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )}^2}{{\left (a+b\,x\right )}^2} \,d x \]