Integrand size = 18, antiderivative size = 117 \[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=2^{-1-p} e^{-\frac {2 a}{b m n}} x^2 \left (c \left (d x^m\right )^n\right )^{-\frac {2}{m n}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac {a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \] Output:
2^(-1-p)*x^2*GAMMA(p+1,(-2*a-2*b*ln(c*(d*x^m)^n))/b/m/n)*(a+b*ln(c*(d*x^m) ^n))^p/exp(2*a/b/m/n)/((c*(d*x^m)^n)^(2/m/n))/((-(a+b*ln(c*(d*x^m)^n))/b/m /n)^p)
Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=2^{-1-p} e^{-\frac {2 a}{b m n}} x^2 \left (c \left (d x^m\right )^n\right )^{-\frac {2}{m n}} \Gamma \left (1+p,-\frac {2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac {a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \] Input:
Integrate[x*(a + b*Log[c*(d*x^m)^n])^p,x]
Output:
(2^(-1 - p)*x^2*Gamma[1 + p, (-2*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)]*(a + b *Log[c*(d*x^m)^n])^p)/(E^((2*a)/(b*m*n))*(c*(d*x^m)^n)^(2/(m*n))*(-((a + b *Log[c*(d*x^m)^n])/(b*m*n)))^p)
Time = 0.44 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2895, 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.
\(\displaystyle \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^pdx\) |
\(\Big \downarrow \) 2747 |
\(\displaystyle \frac {x^2 \left (c \left (d x^m\right )^n\right )^{-\frac {2}{m n}} \int \left (c \left (d x^m\right )^n\right )^{\frac {2}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^pd\log \left (c \left (d x^m\right )^n\right )}{m n}\) |
\(\Big \downarrow \) 2612 |
\(\displaystyle 2^{-p-1} x^2 e^{-\frac {2 a}{b m n}} \left (c \left (d x^m\right )^n\right )^{-\frac {2}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (-\frac {a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \Gamma \left (p+1,-\frac {2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )\) |
Input:
Int[x*(a + b*Log[c*(d*x^m)^n])^p,x]
Output:
(2^(-1 - p)*x^2*Gamma[1 + p, (-2*(a + b*Log[c*(d*x^m)^n]))/(b*m*n)]*(a + b *Log[c*(d*x^m)^n])^p)/(E^((2*a)/(b*m*n))*(c*(d*x^m)^n)^(2/(m*n))*(-((a + b *Log[c*(d*x^m)^n])/(b*m*n)))^p)
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[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(u_.), x_Symbol] :> Subst[Int[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && !IntegerQ[n] && !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[ IntHide[u*(a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x]]
\[\int x {\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )}^{p}d x\]
Input:
int(x*(a+b*ln(c*(d*x^m)^n))^p,x)
Output:
int(x*(a+b*ln(c*(d*x^m)^n))^p,x)
\[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=\int { {\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d*x^m)^n))^p,x, algorithm="fricas")
Output:
integral((b*log((d*x^m)^n*c) + a)^p*x, x)
\[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=\int x \left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{p}\, dx \] Input:
integrate(x*(a+b*ln(c*(d*x**m)**n))**p,x)
Output:
Integral(x*(a + b*log(c*(d*x**m)**n))**p, x)
Exception generated. \[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*(a+b*log(c*(d*x^m)^n))^p,x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
\[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=\int { {\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d*x^m)^n))^p,x, algorithm="giac")
Output:
integrate((b*log((d*x^m)^n*c) + a)^p*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^p \,d x \] Input:
int(x*(a + b*log(c*(d*x^m)^n))^p,x)
Output:
int(x*(a + b*log(c*(d*x^m)^n))^p, x)
\[ \int x \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \, dx=\frac {{\left (\mathrm {log}\left (x^{m n} d^{n} c \right ) b +a \right )}^{p} a \,x^{2}+2 \left (\int \frac {{\left (\mathrm {log}\left (x^{m n} d^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{m n} d^{n} c \right ) x}{2 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a b +\mathrm {log}\left (x^{m n} d^{n} c \right ) b^{2} m n p +2 a^{2}+a b m n p}d x \right ) a \,b^{2} m n p +\left (\int \frac {{\left (\mathrm {log}\left (x^{m n} d^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{m n} d^{n} c \right ) x}{2 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a b +\mathrm {log}\left (x^{m n} d^{n} c \right ) b^{2} m n p +2 a^{2}+a b m n p}d x \right ) b^{3} m^{2} n^{2} p^{2}}{b m n p +2 a} \] Input:
int(x*(a+b*log(c*(d*x^m)^n))^p,x)
Output:
((log(x**(m*n)*d**n*c)*b + a)**p*a*x**2 + 2*int(((log(x**(m*n)*d**n*c)*b + a)**p*log(x**(m*n)*d**n*c)*x)/(2*log(x**(m*n)*d**n*c)*a*b + log(x**(m*n)* d**n*c)*b**2*m*n*p + 2*a**2 + a*b*m*n*p),x)*a*b**2*m*n*p + int(((log(x**(m *n)*d**n*c)*b + a)**p*log(x**(m*n)*d**n*c)*x)/(2*log(x**(m*n)*d**n*c)*a*b + log(x**(m*n)*d**n*c)*b**2*m*n*p + 2*a**2 + a*b*m*n*p),x)*b**3*m**2*n**2* p**2)/(2*a + b*m*n*p)