\(\int (a g+b g x)^m (c i+d i x)^{-2-m} (A+B \log (e (a+b x)^n (c+d x)^{-n}))^p \, dx\) [226]

Optimal result

Integrand size = 50, antiderivative size = 193 \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\frac {e^{-\frac {A (1+m)}{B n}} (a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \left (-\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right )^{-p}}{(b c-a d) i^2 (1+m) (c+d x)} \]

[Out]

(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)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2573, 2563, 2347, 2212} \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\frac {(a+b x) e^{-\frac {A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac {m+1}{n}} \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p \left (-\frac {(m+1) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n}\right )^{-p} \Gamma \left (p+1,-\frac {(m+1) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right )}{i^2 (m+1) (c+d x) (b c-a d)} \]

[In]

Int[(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]

[Out]

((a + b*x)*(g*(a + b*x))^m*Gamma[1 + p, -(((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n))]*(A + B*Lo
g[(e*(a + b*x)^n)/(c + d*x)^n])^p)/((b*c - a*d)*E^((A*(1 + m))/(B*n))*i^2*(1 + m)*(c + d*x)*(i*(c + d*x))^m*((
e*(a + b*x)^n)/(c + d*x)^n)^((1 + m)/n)*(-(((1 + m)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n]))/(B*n)))^p)

Rule 2212

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]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(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]

Rule 2563

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[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]

Rule 2573

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)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^p \, dx,e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = \text {Subst}\left (\frac {\left ((g (a+b x))^m \left (\frac {a+b x}{c+d x}\right )^{-m} (i (c+d x))^{-m}\right ) \text {Subst}\left (\int x^m \left (A+B \log \left (e x^n\right )\right )^p \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) i^2},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = \text {Subst}\left (\frac {\left ((a+b x) (g (a+b x))^m \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )^{-\frac {1+m}{n}} (i (c+d x))^{-m}\right ) \text {Subst}\left (\int e^{\frac {(1+m) x}{n}} (A+B x)^p \, dx,x,\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) i^2 n (c+d x)},e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right ) \\ & = \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)} \\ \end{align*}

Mathematica [F]

\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx \]

[In]

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]

[Out]

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]

Maple [F]

\[\int \left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m} {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}^{p}d x\]

[In]

int((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p,x)

[Out]

int((b*g*x+a*g)^m*(d*i*x+c*i)^(-2-m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p,x)

Fricas [F]

\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int { {\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p} \,d x } \]

[In]

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")

[Out]

integral((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)

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\text {Timed out} \]

[In]

integrate((b*g*x+a*g)**m*(d*i*x+c*i)**(-2-m)*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**p,x)

[Out]

Timed out

Maxima [F]

\[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int { {\left (b g x + a g\right )}^{m} {\left (d i x + c i\right )}^{-m - 2} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p} \,d x } \]

[In]

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")

[Out]

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)

Giac [F(-2)]

Exception generated. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \]

[In]

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")

[Out]

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,

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^m\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{{\left (c\,i+d\,i\,x\right )}^{m+2}} \,d x \]

[In]

int(((a*g + b*g*x)^m*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p)/(c*i + d*i*x)^(m + 2),x)

[Out]

int(((a*g + b*g*x)^m*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p)/(c*i + d*i*x)^(m + 2), x)