Integrand size = 50, antiderivative size = 193 \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\frac {e^{-\frac {A (1+m)}{B n}} (a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \left (-\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right )^{-p}}{(b c-a d) i^2 (1+m) (c+d x)} \]
(b*x+a)*(g*(b*x+a))^m*GAMMA(p+1,-(1+m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B /n)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p/(-a*d+b*c)/exp(A*(1+m)/B/n)/i^2/(1 +m)/(d*x+c)/((i*(d*x+c))^m)/((e*(b*x+a)^n/((d*x+c)^n))^((1+m)/n))/((-(1+m) *(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/n)^p)
\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx \]
Integrate[(a*g + b*g*x)^m*(c*i + d*i*x)^(-2 - m)*(A + B*Log[(e*(a + b*x)^n )/(c + d*x)^n])^p,x]
Integrate[(a*g + b*g*x)^m*(c*i + d*i*x)^(-2 - m)*(A + B*Log[(e*(a + b*x)^n )/(c + d*x)^n])^p, x]
Time = 0.74 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2973, 2963, 2747, 2612}
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.
\(\displaystyle \int (a g+b g x)^m (c i+d i x)^{-m-2} \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int (a g+b g x)^m (c i+d i x)^{-m-2} \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^pdx\) |
\(\Big \downarrow \) 2963 |
\(\displaystyle \frac {(g (a+b x))^m (i (c+d x))^{-m} \left (\frac {a+b x}{c+d x}\right )^{-m} \int \left (\frac {a+b x}{c+d x}\right )^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^pd\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {(a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \int \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{\frac {m+1}{n}} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^pd\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{i^2 n (c+d x) (b c-a d)}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle \frac {(a+b x) e^{-\frac {A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {m+1}{n}} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^p \left (-\frac {(m+1) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{B n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{i^2 (m+1) (c+d x) (b c-a d)}\) |
((a + b*x)*(g*(a + b*x))^m*Gamma[1 + p, -(((1 + m)*(A + B*Log[e*((a + b*x) /(c + d*x))^n]))/(B*n))]*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^p)/((b*c - a*d)*E^((A*(1 + m))/(B*n))*i^2*(1 + m)*(e*((a + b*x)/(c + d*x))^n)^((1 + m)/n)*(c + d*x)*(i*(c + d*x))^m*(-(((1 + m)*(A + B*Log[e*((a + b*x)/(c + d *x))^n]))/(B*n)))^p)
3.3.26.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) )^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol ] :> Simp[(d*x)^(m + 1)/(d*n*(c*x^n)^((m + 1)/n)) Subst[Int[E^(((m + 1)/n )*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, 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[d^2*((g*((a + b*x)/b))^m/(i^2*(b*c - a*d)*(i*((c + d*x)/d))^m*((a + b*x)/(c + d*x))^m)) Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b* x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x ] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && EqQ[m + q + 2, 0]
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]
\[\int \left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m} {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{p}d x\]
\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int { {\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p} \,d x } \]
integrate((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)*(A+B*log(e*(b*x+a)^n/((d*x+c)^n )))^p,x, algorithm="fricas")
Timed out. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\text {Timed out} \]
\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int { {\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p} \,d x } \]
integrate((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)*(A+B*log(e*(b*x+a)^n/((d*x+c)^n )))^p,x, algorithm="maxima")
integrate((b*g*x + a*g)^m*(d*i*x + c*i)^(-m - 2)*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p, x)
Exception generated. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \]
integrate((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)*(A+B*log(e*(b*x+a)^n/((d*x+c)^n )))^p,x, algorithm="giac")
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 ,0,5,5,0,2,2,3,3,0,0,0,2]%%%}+%%%{-2,[0,0,5,4,1,3,1,3,3,0,0,0,2]%%%}+%%%{1 ,[0,0,5,3,2,
Timed out. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^m\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{{\left (c\,i+d\,i\,x\right )}^{m+2}} \,d x \]