Integrand size = 45, antiderivative size = 620 \[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{4 (b c-a d) n}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{4 (b c-a d) n}-\frac {m \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{4 (b c-a d) n}+\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {m \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {3 m n \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}+\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}-\frac {6 m n^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (4,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d}-\frac {6 m n^3 \operatorname {PolyLog}\left (5,\frac {d (a+b x)}{b (c+d x)}\right )}{b c-a d}+\frac {6 m n^3 \operatorname {PolyLog}\left (5,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{b c-a d} \]
1/4*m*ln(e*((b*x+a)/(d*x+c))^n)^4*ln((-a*d+b*c)/b/(d*x+c))/(-a*d+b*c)/n+1/ 4*ln(e*((b*x+a)/(d*x+c))^n)^4*ln(h*(g*x+f)^m)/(-a*d+b*c)/n-1/4*m*ln(e*((b* x+a)/(d*x+c))^n)^4*ln(1-(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)/ n+m*ln(e*((b*x+a)/(d*x+c))^n)^3*polylog(2,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)- m*ln(e*((b*x+a)/(d*x+c))^n)^3*polylog(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x +c))/(-a*d+b*c)-3*m*n*ln(e*((b*x+a)/(d*x+c))^n)^2*polylog(3,d*(b*x+a)/b/(d *x+c))/(-a*d+b*c)+3*m*n*ln(e*((b*x+a)/(d*x+c))^n)^2*polylog(3,(-c*g+d*f)*( b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)+6*m*n^2*ln(e*((b*x+a)/(d*x+c))^n)*po lylog(4,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)-6*m*n^2*ln(e*((b*x+a)/(d*x+c))^n)* polylog(4,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*d+b*c)-6*m*n^3*polylo g(5,d*(b*x+a)/b/(d*x+c))/(-a*d+b*c)+6*m*n^3*polylog(5,(-c*g+d*f)*(b*x+a)/( -a*g+b*f)/(d*x+c))/(-a*d+b*c)
\[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx \]
Time = 1.03 (sec) , antiderivative size = 647, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {2989, 2953, 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.
| \(\displaystyle \int \frac {\log \left (h (f+g x)^m\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 2989 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac {g m \int \frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x}dx}{4 n (b c-a d)}\) |
| \(\Big \downarrow \) 2953 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac {g m \int \frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right ) \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{4 n}\) |
| \(\Big \downarrow \) 2804 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac {g m \int \left (\frac {d \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) g \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(c g-d f) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d) g \left (b f-a g-\frac {(d f-c g) (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{4 n}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log \left (h (f+g x)^m\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{4 n (b c-a d)}-\frac {g m \left (\frac {24 n^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (4,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}-\frac {12 n^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (3,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}+\frac {4 n \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}+\frac {\log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{g (b c-a d)}-\frac {24 n^3 \operatorname {PolyLog}\left (4,\frac {d (a+b x)}{b (c+d x)}\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}+\frac {12 n^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right ) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {4 n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {\log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \log ^4\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g (b c-a d)}-\frac {24 n^4 \operatorname {PolyLog}\left (5,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{g (b c-a d)}+\frac {24 n^4 \operatorname {PolyLog}\left (5,\frac {d (a+b x)}{b (c+d x)}\right )}{g (b c-a d)}\right )}{4 n}\) |
(Log[e*((a + b*x)/(c + d*x))^n]^4*Log[h*(f + g*x)^m])/(4*(b*c - a*d)*n) - (g*m*(-((Log[e*((a + b*x)/(c + d*x))^n]^4*Log[1 - (d*(a + b*x))/(b*(c + d* x))])/((b*c - a*d)*g)) + (Log[e*((a + b*x)/(c + d*x))^n]^4*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g) - (4*n*Log[e*(( a + b*x)/(c + d*x))^n]^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)*g) + (4*n*Log[e*((a + b*x)/(c + d*x))^n]^3*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g) + (12*n^2*Log[e*((a + b *x)/(c + d*x))^n]^2*PolyLog[3, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)* g) - (12*n^2*Log[e*((a + b*x)/(c + d*x))^n]^2*PolyLog[3, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g) - (24*n^3*Log[e*((a + b*x )/(c + d*x))^n]*PolyLog[4, (d*(a + b*x))/(b*(c + d*x))])/((b*c - a*d)*g) + (24*n^3*Log[e*((a + b*x)/(c + d*x))^n]*PolyLog[4, ((d*f - c*g)*(a + b*x)) /((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g) + (24*n^4*PolyLog[5, (d*(a + b* x))/(b*(c + d*x))])/((b*c - a*d)*g) - (24*n^4*PolyLog[5, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/((b*c - a*d)*g)))/(4*n)
3.1.67.3.1 Defintions of rubi rules used
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[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_S ymbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[k*Log[i*(j*(g + h*x)^t)^u]*(Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(p*r*(s + 1)*(b*c - a*d))), x] - Simp[k*h*t*(u/(p*r*(s + 1)*(b*c - a*d))) Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]] /; FreeQ[ {a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] & & EqQ[p + q, 0] && NeQ[s, -1]
\[\int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{3} \ln \left (h \left (g x +f \right )^{m}\right )}{\left (b x +a \right ) \left (d x +c \right )}d x\]
\[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
integrate(log(e*((b*x+a)/(d*x+c))^n)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="fricas")
integral(log((g*x + f)^m*h)*log(e*((b*x + a)/(d*x + c))^n)^3/(b*d*x^2 + a* c + (b*c + a*d)*x), x)
Timed out. \[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\text {Timed out} \]
\[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
integrate(log(e*((b*x+a)/(d*x+c))^n)^3*log(h*(g*x+f)^m)/(b*x+a)/(d*x+c),x, algorithm="maxima")
-1/4*(n^3*log(b*x + a)^4 + n^3*log(d*x + c)^4 - 4*n^2*log(b*x + a)^3*log(e ) + 6*n*log(b*x + a)^2*log(e)^2 - 4*(n^3*log(b*x + a) - n^2*log(e))*log(d* x + c)^3 - 4*(log(b*x + a) - log(d*x + c))*log((b*x + a)^n)^3 + 4*(log(b*x + a) - log(d*x + c))*log((d*x + c)^n)^3 - 4*log(b*x + a)*log(e)^3 + 6*(n^ 3*log(b*x + a)^2 - 2*n^2*log(b*x + a)*log(e) + n*log(e)^2)*log(d*x + c)^2 + 6*(n*log(b*x + a)^2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))*log (d*x + c) - 2*log(b*x + a)*log(e))*log((b*x + a)^n)^2 + 6*(n*log(b*x + a)^ 2 + n*log(d*x + c)^2 - 2*(n*log(b*x + a) - log(e))*log(d*x + c) - 2*(log(b *x + a) - log(d*x + c))*log((b*x + a)^n) - 2*log(b*x + a)*log(e))*log((d*x + c)^n)^2 - 4*(n^3*log(b*x + a)^3 - 3*n^2*log(b*x + a)^2*log(e) + 3*n*log (b*x + a)*log(e)^2 - log(e)^3)*log(d*x + c) - 4*(n^2*log(b*x + a)^3 - n^2* log(d*x + c)^3 - 3*n*log(b*x + a)^2*log(e) + 3*(n^2*log(b*x + a) - n*log(e ))*log(d*x + c)^2 + 3*log(b*x + a)*log(e)^2 - 3*(n^2*log(b*x + a)^2 - 2*n* log(b*x + a)*log(e) + log(e)^2)*log(d*x + c))*log((b*x + a)^n) + 4*(n^2*lo g(b*x + a)^3 - n^2*log(d*x + c)^3 - 3*n*log(b*x + a)^2*log(e) + 3*(n^2*log (b*x + a) - n*log(e))*log(d*x + c)^2 + 3*(log(b*x + a) - log(d*x + c))*log ((b*x + a)^n)^2 + 3*log(b*x + a)*log(e)^2 - 3*(n^2*log(b*x + a)^2 - 2*n*lo g(b*x + a)*log(e) + log(e)^2)*log(d*x + c) - 3*(n*log(b*x + a)^2 + n*log(d *x + c)^2 - 2*(n*log(b*x + a) - log(e))*log(d*x + c) - 2*log(b*x + a)*log( e))*log((b*x + a)^n))*log((d*x + c)^n))*log((g*x + f)^m)/(b*c - a*d) + ...
\[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int { \frac {\log \left ({\left (g x + f\right )}^{m} h\right ) \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{3}}{{\left (b x + a\right )} {\left (d x + c\right )}} \,d x } \]
Timed out. \[ \int \frac {\log ^3\left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (h (f+g x)^m\right )}{(a+b x) (c+d x)} \, dx=\int \frac {\ln \left (h\,{\left (f+g\,x\right )}^m\right )\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^3}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]