\(\int \frac {(a+b \log (c (d x^m)^n))^p}{x^3} \, dx\) [254]

Optimal result

Integrand size = 20, antiderivative size = 117 \[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=-\frac {2^{-1-p} e^{\frac {2 a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac {2}{m n}} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac {a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p}}{x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=-\frac {2^{-1-p} e^{\frac {2 a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac {2}{m n}} \Gamma \left (1+p,\frac {2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right ) \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac {a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p}}{x^2} \] Input:

Integrate[(a + b*Log[c*(d*x^m)^n])^p/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx\)

\(\Big \downarrow \) 2895

\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3}dx\)

\(\Big \downarrow \) 2747

\(\displaystyle \frac {\left (c \left (d x^m\right )^n\right )^{\frac {2}{m n}} \int \left (c \left (d x^m\right )^n\right )^{-\frac {2}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^pd\log \left (c \left (d x^m\right )^n\right )}{m n x^2}\)

\(\Big \downarrow \) 2612

\(\displaystyle -\frac {2^{-p-1} e^{\frac {2 a}{b m n}} \left (c \left (d x^m\right )^n\right )^{\frac {2}{m n}} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p \left (\frac {a+b \log \left (c \left (d x^m\right )^n\right )}{b m n}\right )^{-p} \Gamma \left (p+1,\frac {2 \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{b m n}\right )}{x^2}\)

Input:

Int[(a + b*Log[c*(d*x^m)^n])^p/x^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2612

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 2747

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]
 

rule 2895

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]]
 

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )}^{p}}{x^{3}}d x\]

Input:

int((a+b*ln(c*(d*x^m)^n))^p/x^3,x)
 

Output:

int((a+b*ln(c*(d*x^m)^n))^p/x^3,x)
 

Fricas [F]

\[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*x^m)^n))^p/x^3,x, algorithm="fricas")
 

Output:

integral((b*log((d*x^m)^n*c) + a)^p/x^3, x)
 

Sympy [F]

\[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d x^{m}\right )^{n} \right )}\right )^{p}}{x^{3}}\, dx \] Input:

integrate((a+b*ln(c*(d*x**m)**n))**p/x**3,x)
 

Output:

Integral((a + b*log(c*(d*x**m)**n))**p/x**3, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((a+b*log(c*(d*x^m)^n))^p/x^3,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
 

Giac [F]

\[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=\int { \frac {{\left (b \log \left (\left (d x^{m}\right )^{n} c\right ) + a\right )}^{p}}{x^{3}} \,d x } \] Input:

integrate((a+b*log(c*(d*x^m)^n))^p/x^3,x, algorithm="giac")
 

Output:

integrate((b*log((d*x^m)^n*c) + a)^p/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right )}^p}{x^3} \,d x \] Input:

int((a + b*log(c*(d*x^m)^n))^p/x^3,x)
 

Output:

int((a + b*log(c*(d*x^m)^n))^p/x^3, x)
 

Reduce [F]

\[ \int \frac {\left (a+b \log \left (c \left (d x^m\right )^n\right )\right )^p}{x^3} \, dx=\frac {-{\left (\mathrm {log}\left (x^{m n} d^{n} c \right ) b +a \right )}^{p} a -2 \left (\int \frac {{\left (\mathrm {log}\left (x^{m n} d^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{m n} d^{n} c \right )}{2 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a b \,x^{3}-\mathrm {log}\left (x^{m n} d^{n} c \right ) b^{2} m n p \,x^{3}+2 a^{2} x^{3}-a b m n p \,x^{3}}d x \right ) a \,b^{2} m n p \,x^{2}+\left (\int \frac {{\left (\mathrm {log}\left (x^{m n} d^{n} c \right ) b +a \right )}^{p} \mathrm {log}\left (x^{m n} d^{n} c \right )}{2 \,\mathrm {log}\left (x^{m n} d^{n} c \right ) a b \,x^{3}-\mathrm {log}\left (x^{m n} d^{n} c \right ) b^{2} m n p \,x^{3}+2 a^{2} x^{3}-a b m n p \,x^{3}}d x \right ) b^{3} m^{2} n^{2} p^{2} x^{2}}{x^{2} \left (-b m n p +2 a \right )} \] Input:

int((a+b*log(c*(d*x^m)^n))^p/x^3,x)
 

Output:

( - (log(x**(m*n)*d**n*c)*b + a)**p*a - 2*int(((log(x**(m*n)*d**n*c)*b + a 
)**p*log(x**(m*n)*d**n*c))/(2*log(x**(m*n)*d**n*c)*a*b*x**3 - log(x**(m*n) 
*d**n*c)*b**2*m*n*p*x**3 + 2*a**2*x**3 - a*b*m*n*p*x**3),x)*a*b**2*m*n*p*x 
**2 + int(((log(x**(m*n)*d**n*c)*b + a)**p*log(x**(m*n)*d**n*c))/(2*log(x* 
*(m*n)*d**n*c)*a*b*x**3 - log(x**(m*n)*d**n*c)*b**2*m*n*p*x**3 + 2*a**2*x* 
*3 - a*b*m*n*p*x**3),x)*b**3*m**2*n**2*p**2*x**2)/(x**2*(2*a - b*m*n*p))