Integrand size = 46, antiderivative size = 55 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^{1+p}}{B (b c-a d) f g n (1+p)} \] Output:
(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^(p+1)/B/(-a*d+b*c)/f/g/n/(p+1)
Time = 0.06 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.93 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^{1+p}}{(b B c f g n-a B d f g n) (1+p)} \] Input:
Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p/((a*f + b*f*x)*(c*g + d*g*x)),x]
Output:
(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^(1 + p)/((b*B*c*f*g*n - a*B*d*f*g *n)*(1 + p))
Time = 0.62 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2973, 2961, 2739, 15}
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.
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p}{(a f+b f x) (c g+d g x)} \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int \frac {\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p}{(a f+b f x) (c g+d g x)}dx\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^p}{a+b x}d\frac {a+b x}{c+d x}}{f g (b c-a d)}\) |
\(\Big \downarrow \) 2739 |
\(\displaystyle \frac {\int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^pd\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B f g n (b c-a d)}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^{p+1}}{B f g n (p+1) (b c-a d)}\) |
Input:
Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p/((a*f + b*f*x)*(c*g + d*g*x )),x]
Output:
(A + B*Log[e*((a + b*x)/(c + d*x))^n])^(1 + p)/(B*(b*c - a*d)*f*g*n*(1 + p ))
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04
\[-\frac {{\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{p +1}}{f g n \left (d a -b c \right ) B \left (p +1\right )}\]
Input:
int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p/(b*f*x+a*f)/(d*g*x+c*g),x)
Output:
-1/f/g/n/(a*d-b*c)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^(p+1)/B/(p+1)
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.55 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\frac {{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )} {\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )}^{p}}{{\left (B b c - B a d\right )} f g n p + {\left (B b c - B a d\right )} f g n} \] Input:
integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p/(b*f*x+a*f)/(d*g*x+c*g),x, algorithm="fricas")
Output:
(B*n*log(b*x + a) - B*n*log(d*x + c) + B*log(e) + A)*(B*n*log(b*x + a) - B *n*log(d*x + c) + B*log(e) + A)^p/((B*b*c - B*a*d)*f*g*n*p + (B*b*c - B*a* d)*f*g*n)
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\text {Timed out} \] Input:
integrate((A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**p/(b*f*x+a*f)/(d*g*x+c*g),x )
Output:
Timed out
\[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{{\left (b f x + a f\right )} {\left (d g x + c g\right )}} \,d x } \] Input:
integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p/(b*f*x+a*f)/(d*g*x+c*g),x, algorithm="maxima")
Output:
integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p/((b*f*x + a*f)*(d*g*x + c*g)), x)
Time = 0.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\frac {{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )}^{p + 1}}{{\left (B b c f g n - B a d f g n\right )} {\left (p + 1\right )}} \] Input:
integrate((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p/(b*f*x+a*f)/(d*g*x+c*g),x, algorithm="giac")
Output:
(B*n*log(b*x + a) - B*n*log(d*x + c) + B*log(e) + A)^(p + 1)/((B*b*c*f*g*n - B*a*d*f*g*n)*(p + 1))
Timed out. \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{\left (a\,f+b\,f\,x\right )\,\left (c\,g+d\,g\,x\right )} \,d x \] Input:
int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p/((a*f + b*f*x)*(c*g + d*g*x )),x)
Output:
int((A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p/((a*f + b*f*x)*(c*g + d*g*x )), x)
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.47 \[ \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p}{(a f+b f x) (c g+d g x)} \, dx=-\frac {{\left (\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b +a \right )}^{p} \left (\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) b +a \right )}{b f g n \left (a d p -b c p +a d -b c \right )} \] Input:
int((A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p/(b*f*x+a*f)/(d*g*x+c*g),x)
Output:
( - (log(((a + b*x)**n*e)/(c + d*x)**n)*b + a)**p*(log(((a + b*x)**n*e)/(c + d*x)**n)*b + a))/(b*f*g*n*(a*d*p + a*d - b*c*p - b*c))