Integrand size = 50, antiderivative size = 44 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=\frac {\log \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B (b c-a d) f n} \] Output:
ln(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/(-a*d+b*c)/f/n
Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=\frac {\log \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{f (b B c n-a B d n)} \] Input:
Integrate[1/((a*c*f + (b*c + a*d)*f*x + b*d*f*x^2)*(A + B*Log[(e*(a + b*x) ^n)/(c + d*x)^n])),x]
Output:
Log[A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]]/(f*(b*B*c*n - a*B*d*n))
Time = 0.53 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2973, 2974, 2961, 2739, 14}
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.
\(\displaystyle \int \frac {1}{\left (f x (a d+b c)+a c f+b d f x^2\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )} \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int \frac {1}{\left (f x (a d+b c)+a c f+b d f x^2\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}dx\) |
\(\Big \downarrow \) 2974 |
\(\displaystyle \frac {\int \frac {1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}dx}{f}\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {\int \frac {c+d x}{(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}d\frac {a+b x}{c+d x}}{f (b c-a d)}\) |
\(\Big \downarrow \) 2739 |
\(\displaystyle \frac {\int \frac {c+d x}{a+b x}d\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B f n (b c-a d)}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {\log \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{B f n (b c-a d)}\) |
Input:
Int[1/((a*c*f + (b*c + a*d)*f*x + b*d*f*x^2)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])),x]
Output:
Log[A + B*Log[e*((a + b*x)/(c + d*x))^n]]/(B*(b*c - a*d)*f*n)
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
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[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_) + (h_.)*(x_)^2)^(m_.), x_Symbol] :> Simp[h^ m/(b^m*d^m) Int[(a + b*x)^m*(c + d*x)^m*(A + B*Log[e*((a + b*x)/(c + d*x) )^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, n, p}, x] && EqQ[b* d*f - a*c*h, 0] && EqQ[b*d*g - h*(b*c + a*d), 0] && IntegerQ[m]
Time = 62.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05
method | result | size |
parallelrisch | \(-\frac {\ln \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}{B f n \left (d a -b c \right )}\) | \(46\) |
risch | \(-\frac {\ln \left (\ln \left (\left (b x +a \right )^{n}\right )+\frac {-i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}+i B \pi \,\operatorname {csgn}\left (i \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{2}-i B \pi \operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right )^{3}+i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2}-i B \pi \,\operatorname {csgn}\left (i \left (d x +c \right )^{-n} \left (b x +a \right )^{n}\right ) \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right ) \operatorname {csgn}\left (i e \right )-i B \pi \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{3}+i B \pi \operatorname {csgn}\left (i e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )^{2} \operatorname {csgn}\left (i e \right )+2 B \ln \left (e \right )-2 B \ln \left (\left (d x +c \right )^{n}\right )+2 A}{2 B}\right )}{B f n \left (d a -b c \right )}\) | \(371\) |
Input:
int(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x, method=_RETURNVERBOSE)
Output:
-ln(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/f/n/(a*d-b*c)
Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=\frac {\log \left (-B n \log \left (b x + a\right ) + B n \log \left (d x + c\right ) - B \log \left (e\right ) - A\right )}{{\left (B b c - B a d\right )} f n} \] Input:
integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^ n))),x, algorithm="fricas")
Output:
log(-B*n*log(b*x + a) + B*n*log(d*x + c) - B*log(e) - A)/((B*b*c - B*a*d)* f*n)
Timed out. \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x**2)/(A+B*ln(e*(b*x+a)**n/((d*x+c) **n))),x)
Output:
Timed out
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=\frac {\log \left (-\frac {B \log \left ({\left (b x + a\right )}^{n}\right ) - B \log \left ({\left (d x + c\right )}^{n}\right ) + B \log \left (e\right ) + A}{B}\right )}{{\left (b c f n - a d f n\right )} B} \] Input:
integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^ n))),x, algorithm="maxima")
Output:
log(-(B*log((b*x + a)^n) - B*log((d*x + c)^n) + B*log(e) + A)/B)/((b*c*f*n - a*d*f*n)*B)
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=\frac {\log \left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )}{B b c f n - B a d f n} \] Input:
integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^ n))),x, algorithm="giac")
Output:
log(B*n*log(b*x + a) - B*n*log(d*x + c) + B*log(e) + A)/(B*b*c*f*n - B*a*d *f*n)
Time = 26.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=-\frac {\ln \left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}{B\,a\,d\,f\,n-B\,b\,c\,f\,n} \] Input:
int(1/((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))*(a*c*f + f*x*(a*d + b*c) + b*d*f*x^2)),x)
Output:
-log(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))/(B*a*d*f*n - B*b*c*f*n)
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a c f+(b c+a d) f x+b d f x^2\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx=-\frac {\mathrm {log}\left (\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b +a \right )}{b f n \left (a d -b c \right )} \] Input:
int(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x )
Output:
( - log(log(((a + b*x)**n*e)/(c + d*x)**n)*b + a))/(b*f*n*(a*d - b*c))