3.178 \(\int (g x)^m (a+b \log (c x^n))^p (d+e \log (f x^r)) \, dx\)

Optimal. Leaf size=347 \[ \frac{(g x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{g (m+1)}-\frac{e r x (g x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+2,-\frac{a (m+1)}{b n}-\frac{(m+1) \log \left (c x^n\right )}{n}\right )}{(m+1)^2}-\frac{e r x (g x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a (m+1)}{b n}-\frac{(m+1) \log \left (c x^n\right )}{n}\right )}{b (m+1) n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.358171, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2310, 2181, 2366, 12, 15, 19, 6557} \[ \frac{(g x)^{m+1} e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (d+e \log \left (f x^r\right )\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{g (m+1)}-\frac{e r x (g x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+2,-\frac{a (m+1)}{b n}-\frac{(m+1) \log \left (c x^n\right )}{n}\right )}{(m+1)^2}-\frac{e r x (g x)^m e^{-\frac{a (m+1)}{b n}} \left (c x^n\right )^{-\frac{m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^{p+1} \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text{Gamma}\left (p+1,-\frac{a (m+1)}{b n}-\frac{(m+1) \log \left (c x^n\right )}{n}\right )}{b (m+1) n} \]

Antiderivative was successfully verified.

[In]

Int[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]),x]

[Out]

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

Rule 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 2366

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 19

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[(a^(m + n)*(b*v)^n)/(a*v)^n, Int[u*v^(m + n),
x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] && IntegerQ[m + n]

Rule 6557

Int[Gamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((a + b*x)*Gamma[n, a + b*x])/b, x] - Simp[Gamma[n + 1, a
+ b*x]/b, x] /; FreeQ[{a, b, n}, x]

Rubi steps

\begin{align*} \int (g x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (d+e \log \left (f x^r\right )\right ) \, dx &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}-(e r) \int \frac{e^{-\frac{a (1+m)}{b n}} (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m} \, dx\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}-\frac{\left (e e^{-\frac{a (1+m)}{b n}} r\right ) \int (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \, dx}{1+m}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}-\frac{\left (e e^{-\frac{a (1+m)}{b n}} r x^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \int x^{-1-m} (g x)^m \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \, dx}{1+m}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}-\frac{\left (e e^{-\frac{a (1+m)}{b n}} r x (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}}\right ) \int \frac{\Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{x} \, dx}{1+m}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}-\frac{\left (e e^{-\frac{a (1+m)}{b n}} r x (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}\right ) \int \frac{\Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{x} \, dx}{1+m}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}-\frac{\left (e e^{-\frac{a (1+m)}{b n}} r x (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}\right ) \operatorname{Subst}\left (\int \Gamma \left (1+p,-\frac{(1+m) (a+b x)}{b n}\right ) \, dx,x,\log \left (c x^n\right )\right )}{(1+m) n}\\ &=\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}+\frac{\left (e e^{-\frac{a (1+m)}{b n}} r x (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}\right ) \operatorname{Subst}\left (\int \Gamma (1+p,x) \, dx,x,-\frac{a (1+m)}{b n}-\frac{(1+m) \log \left (c x^n\right )}{n}\right )}{(1+m)^2}\\ &=-\frac{e e^{-\frac{a (1+m)}{b n}} r x (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (2+p,-\frac{a (1+m)}{b n}-\frac{(1+m) \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{(1+m)^2}-\frac{e e^{-\frac{a (1+m)}{b n}} r x (g x)^m \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{a (1+m)}{b n}-\frac{(1+m) \log \left (c x^n\right )}{n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (\frac{a}{b n}+\frac{\log \left (c x^n\right )}{n}\right )}{1+m}+\frac{e^{-\frac{a (1+m)}{b n}} (g x)^{1+m} \left (c x^n\right )^{-\frac{1+m}{n}} \Gamma \left (1+p,-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac{(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \left (d+e \log \left (f x^r\right )\right )}{g (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.610759, size = 179, normalized size = 0.52 \[ -\frac{x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right )^{p-1} \exp \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{1-p} \left ((m+1) \text{Gamma}\left (p+1,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-a e r-b e r \log \left (c x^n\right )+b d n+b e n \log \left (f x^r\right )\right )-b e n r \text{Gamma}\left (p+2,-\frac{(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )}{(m+1)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(g*x)^m*(a + b*Log[c*x^n])^p*(d + e*Log[f*x^r]),x]

[Out]

-(((g*x)^m*(a + b*Log[c*x^n])^(-1 + p)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^(1 - p)*(-(b*e*n*r*Gamma[2 + p,
-(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]) + (1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(b*d*n
- a*e*r - b*e*r*Log[c*x^n] + b*e*n*Log[f*x^r])))/(E^(((1 + m)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m)^
3*x^m))

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Maple [F]  time = 0.827, size = 0, normalized size = 0. \begin{align*} \int \left ( gx \right ) ^{m} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{p} \left ( d+e\ln \left ( f{x}^{r} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x)^m*(a+b*ln(c*x^n))^p*(d+e*ln(f*x^r)),x)

[Out]

int((g*x)^m*(a+b*ln(c*x^n))^p*(d+e*ln(f*x^r)),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(a+b*log(c*x^n))^p*(d+e*log(f*x^r)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (\left (g x\right )^{m} e \log \left (f x^{r}\right ) + \left (g x\right )^{m} d\right )}{\left (b \log \left (c x^{n}\right ) + a\right )}^{p}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(a+b*log(c*x^n))^p*(d+e*log(f*x^r)),x, algorithm="fricas")

[Out]

integral(((g*x)^m*e*log(f*x^r) + (g*x)^m*d)*(b*log(c*x^n) + a)^p, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)**m*(a+b*ln(c*x**n))**p*(d+e*ln(f*x**r)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x)^m*(a+b*log(c*x^n))^p*(d+e*log(f*x^r)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError