Integrand size = 28, antiderivative size = 28 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\text {Int}\left ((g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n,x\right ) \] Output:
Defer(Int)((h*x+g)^m*(a+b*ln(c*(d*(f*x+e)^p)^q))^n,x)
Not integrable
Time = 0.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx \] Input:
Integrate[(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^n,x]
Output:
Integrate[(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^n, x]
Not integrable
Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2896}
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.
| \(\displaystyle \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx\) |
| \(\Big \downarrow \) 2896 |
| \(\displaystyle \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^ndx\) |
Input:
Int[(g + h*x)^m*(a + b*Log[c*(d*(e + f*x)^p)^q])^n,x]
Output:
$Aborted
Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_. )*(AFx_), x_Symbol] :> Unintegrable[AFx*(a + b*Log[c*(d*(e + f*x)^m)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && AlgebraicFunctionQ[AFx, x, True]
Not integrable
Time = 0.66 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00
\[\int \left (h x +g \right )^{m} {\left (a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )\right )}^{n}d x\]
Input:
int((h*x+g)^m*(a+b*ln(c*(d*(f*x+e)^p)^q))^n,x)
Output:
int((h*x+g)^m*(a+b*ln(c*(d*(f*x+e)^p)^q))^n,x)
Not integrable
Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int { {\left (h x + g\right )}^{m} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{n} \,d x } \] Input:
integrate((h*x+g)^m*(a+b*log(c*(d*(f*x+e)^p)^q))^n,x, algorithm="fricas")
Output:
integral((h*x + g)^m*(b*log(((f*x + e)^p*d)^q*c) + a)^n, x)
Exception generated. \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((h*x+g)**m*(a+b*ln(c*(d*(f*x+e)**p)**q))**n,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
Exception generated. \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((h*x+g)^m*(a+b*log(c*(d*(f*x+e)^p)^q))^n,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
Exception generated. \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((h*x+g)^m*(a+b*log(c*(d*(f*x+e)^p)^q))^n,x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Unable to divide, perhaps due to rounding error%%%{1,[0 ,0,7,4,0,5,0,3,3,3,0,2,0,0,0]%%%}+%%%{5,[0,0,6,4,0,4,1,3,3,3,0,2,0,0,0]%%% }+%%%{2,[0,0
Not integrable
Time = 26.78 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx=\int {\left (g+h\,x\right )}^m\,{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^n \,d x \] Input:
int((g + h*x)^m*(a + b*log(c*(d*(e + f*x)^p)^q))^n,x)
Output:
int((g + h*x)^m*(a + b*log(c*(d*(e + f*x)^p)^q))^n, x)
Not integrable
Time = 1.43 (sec) , antiderivative size = 40428, normalized size of antiderivative = 1443.86 \[ \int (g+h x)^m \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^n \, dx =\text {Too large to display} \] Input:
int((h*x+g)^m*(a+b*log(c*(d*(f*x+e)^p)^q))^n,x)
Output:
((g + h*x)**m*(log(d**q*(e + f*x)**(p*q)*c)*b + a)**n*a*f*g*m**2 + (g + h* x)**m*(log(d**q*(e + f*x)**(p*q)*c)*b + a)**n*a*f*g*m + (g + h*x)**m*(log( d**q*(e + f*x)**(p*q)*c)*b + a)**n*a*f*h*m**2*x + (g + h*x)**m*(log(d**q*( e + f*x)**(p*q)*c)*b + a)**n*a*f*h*m*x + (g + h*x)**m*(log(d**q*(e + f*x)* *(p*q)*c)*b + a)**n*b*e*h*m*n*p*q + (g + h*x)**m*(log(d**q*(e + f*x)**(p*q )*c)*b + a)**n*b*e*h*n*p*q + (g + h*x)**m*(log(d**q*(e + f*x)**(p*q)*c)*b + a)**n*b*f*g*m*n*p*q + (g + h*x)**m*(log(d**q*(e + f*x)**(p*q)*c)*b + a)* *n*b*f*h*m*n*p*q*x - int(((g + h*x)**m*(log(d**q*(e + f*x)**(p*q)*c)*b + a )**n*x**2)/(log(d**q*(e + f*x)**(p*q)*c)*a*b*e*g*m**2 + 2*log(d**q*(e + f* x)**(p*q)*c)*a*b*e*g*m + log(d**q*(e + f*x)**(p*q)*c)*a*b*e*g + log(d**q*( e + f*x)**(p*q)*c)*a*b*e*h*m**2*x + 2*log(d**q*(e + f*x)**(p*q)*c)*a*b*e*h *m*x + log(d**q*(e + f*x)**(p*q)*c)*a*b*e*h*x + log(d**q*(e + f*x)**(p*q)* c)*a*b*f*g*m**2*x + 2*log(d**q*(e + f*x)**(p*q)*c)*a*b*f*g*m*x + log(d**q* (e + f*x)**(p*q)*c)*a*b*f*g*x + log(d**q*(e + f*x)**(p*q)*c)*a*b*f*h*m**2* x**2 + 2*log(d**q*(e + f*x)**(p*q)*c)*a*b*f*h*m*x**2 + log(d**q*(e + f*x)* *(p*q)*c)*a*b*f*h*x**2 + log(d**q*(e + f*x)**(p*q)*c)*b**2*e*g*m*n*p*q + l og(d**q*(e + f*x)**(p*q)*c)*b**2*e*g*n*p*q + log(d**q*(e + f*x)**(p*q)*c)* b**2*e*h*m*n*p*q*x + log(d**q*(e + f*x)**(p*q)*c)*b**2*e*h*n*p*q*x + log(d **q*(e + f*x)**(p*q)*c)*b**2*f*g*m*n*p*q*x + log(d**q*(e + f*x)**(p*q)*c)* b**2*f*g*n*p*q*x + log(d**q*(e + f*x)**(p*q)*c)*b**2*f*h*m*n*p*q*x**2 +...