3.2.78 \(\int (g x)^m (a+b \log (c x^n))^p (d+e \log (f x^r)) \, dx\) [178]

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

[Out]

-e*r*x*(g*x)^m*GAMMA(2+p,-a*(1+m)/b/n-(1+m)*ln(c*x^n)/n)*(a+b*ln(c*x^n))^p/exp(a*(1+m)/b/n)/(1+m)^2/((c*x^n)^(
(1+m)/n))/((-(1+m)*(a+b*ln(c*x^n))/b/n)^p)-e*r*x*(g*x)^m*GAMMA(1+p,-a*(1+m)/b/n-(1+m)*ln(c*x^n)/n)*(a+b*ln(c*x
^n))^(1+p)/b/exp(a*(1+m)/b/n)/(1+m)/n/((c*x^n)^((1+m)/n))/((-(1+m)*(a+b*ln(c*x^n))/b/n)^p)+(g*x)^(1+m)*GAMMA(1
+p,-(1+m)*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p*(d+e*ln(f*x^r))/exp(a*(1+m)/b/n)/g/(1+m)/((c*x^n)^((1+m)/n))/
((-(1+m)*(a+b*ln(c*x^n))/b/n)^p)

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Rubi [A]
time = 0.25, antiderivative size = 347, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2347, 2212, 2413, 12, 15, 19, 6692} \begin {gather*} \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} \end {gather*}

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 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 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 2413

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 6692

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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.67, size = 179, normalized size = 0.52 \begin {gather*} -\frac {e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} x^{-m} (g x)^m \left (a+b \log \left (c x^n\right )\right )^{-1+p} \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{1-p} \left (-b e n r \Gamma \left (2+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+(1+m) \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (b d n-a e r-b e r \log \left (c x^n\right )+b e n \log \left (f x^r\right )\right )\right )}{(1+m)^3} \end {gather*}

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 = 0.10, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{m} \left (a +b \ln \left (c \,x^{n}\right )\right )^{p} \left (d +e \ln \left (f \,x^{r}\right )\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Simplification as
suming t_nostep near 0Simplification assuming sageVARg near 0Simplification assuming t_nostep near 0Simplifica
tion assuming

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (d+e\,\ln \left (f\,x^r\right )\right )\,{\left (g\,x\right )}^m\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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