Integrand size = 25, antiderivative size = 294 \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\frac {c^2 x^{-3 n} (e x)^{3 n} \left (c+d x^n\right ) \left (a+b \left (c+d x^n\right )^q\right )^p \left (1+\frac {b \left (c+d x^n\right )^q}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1}{q},1+\frac {1}{q},-\frac {b \left (c+d x^n\right )^q}{a}\right )}{d^3 e n}-\frac {c x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^2 \left (a+b \left (c+d x^n\right )^q\right )^p \left (1+\frac {b \left (c+d x^n\right )^q}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {2}{q},\frac {2+q}{q},-\frac {b \left (c+d x^n\right )^q}{a}\right )}{d^3 e n}+\frac {x^{-3 n} (e x)^{3 n} \left (c+d x^n\right )^3 \left (a+b \left (c+d x^n\right )^q\right )^p \left (1+\frac {b \left (c+d x^n\right )^q}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {3}{q},\frac {3+q}{q},-\frac {b \left (c+d x^n\right )^q}{a}\right )}{3 d^3 e n} \] Output:
c^2*(e*x)^(3*n)*(c+d*x^n)*(a+b*(c+d*x^n)^q)^p*hypergeom([-p, 1/q],[1+1/q], -b*(c+d*x^n)^q/a)/d^3/e/n/(x^(3*n))/((1+b*(c+d*x^n)^q/a)^p)-c*(e*x)^(3*n)* (c+d*x^n)^2*(a+b*(c+d*x^n)^q)^p*hypergeom([-p, 2/q],[(2+q)/q],-b*(c+d*x^n) ^q/a)/d^3/e/n/(x^(3*n))/((1+b*(c+d*x^n)^q/a)^p)+1/3*(e*x)^(3*n)*(c+d*x^n)^ 3*(a+b*(c+d*x^n)^q)^p*hypergeom([-p, 3/q],[(3+q)/q],-b*(c+d*x^n)^q/a)/d^3/ e/n/(x^(3*n))/((1+b*(c+d*x^n)^q/a)^p)
\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx \] Input:
Integrate[(e*x)^(-1 + 3*n)*(a + b*(c + d*x^n)^q)^p,x]
Output:
Integrate[(e*x)^(-1 + 3*n)*(a + b*(c + d*x^n)^q)^p, x]
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.
\(\displaystyle \int (e x)^{3 n-1} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int (e x)^{3 n-1} \left (a+b \left (c+d x^n\right )^q\right )^pdx\) |
Input:
Int[(e*x)^(-1 + 3*n)*(a + b*(c + d*x^n)^q)^p,x]
Output:
$Aborted
\[\int \left (e x \right )^{-1+3 n} {\left (a +b \left (c +d \,x^{n}\right )^{q}\right )}^{p}d x\]
Input:
int((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^q)^p,x)
Output:
int((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^q)^p,x)
\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{q} b + a\right )}^{p} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:
integrate((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^q)^p,x, algorithm="fricas")
Output:
integral(((d*x^n + c)^q*b + a)^p*(e*x)^(3*n - 1), x)
Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\text {Timed out} \] Input:
integrate((e*x)**(-1+3*n)*(a+b*(c+d*x**n)**q)**p,x)
Output:
Timed out
\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{q} b + a\right )}^{p} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:
integrate((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^q)^p,x, algorithm="maxima")
Output:
integrate(((d*x^n + c)^q*b + a)^p*(e*x)^(3*n - 1), x)
\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{q} b + a\right )}^{p} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:
integrate((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^q)^p,x, algorithm="giac")
Output:
integrate(((d*x^n + c)^q*b + a)^p*(e*x)^(3*n - 1), x)
Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\int {\left (a+b\,{\left (c+d\,x^n\right )}^q\right )}^p\,{\left (e\,x\right )}^{3\,n-1} \,d x \] Input:
int((a + b*(c + d*x^n)^q)^p*(e*x)^(3*n - 1),x)
Output:
int((a + b*(c + d*x^n)^q)^p*(e*x)^(3*n - 1), x)
\[ \int (e x)^{-1+3 n} \left (a+b \left (c+d x^n\right )^q\right )^p \, dx=\text {too large to display} \] Input:
int((e*x)^(-1+3*n)*(a+b*(c+d*x^n)^q)^p,x)
Output:
(e**(3*n)*(x**(3*n)*((x**n*d + c)**q*b + a)**p*d**3*p**2*q**2 + 3*x**(3*n) *((x**n*d + c)**q*b + a)**p*d**3*p*q + 2*x**(3*n)*((x**n*d + c)**q*b + a)* *p*d**3 + x**(2*n)*((x**n*d + c)**q*b + a)**p*c*d**2*p**2*q**2 + x**(2*n)* ((x**n*d + c)**q*b + a)**p*c*d**2*p*q - 2*x**n*((x**n*d + c)**q*b + a)**p* c**2*d*p*q + 2*((x**n*d + c)**q*b + a)**p*c**3 + int((x**(3*n)*((x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q*b*p**3*q**3*x + 6*(x**n*d + c)**q*b*p**2 *q**2*x + 11*(x**n*d + c)**q*b*p*q*x + 6*(x**n*d + c)**q*b*x + a*p**3*q**3 *x + 6*a*p**2*q**2*x + 11*a*p*q*x + 6*a*x),x)*a*d**3*n*p**6*q**6 + 9*int(( x**(3*n)*((x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q*b*p**3*q**3*x + 6*(x **n*d + c)**q*b*p**2*q**2*x + 11*(x**n*d + c)**q*b*p*q*x + 6*(x**n*d + c)* *q*b*x + a*p**3*q**3*x + 6*a*p**2*q**2*x + 11*a*p*q*x + 6*a*x),x)*a*d**3*n *p**5*q**5 + 31*int((x**(3*n)*((x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q *b*p**3*q**3*x + 6*(x**n*d + c)**q*b*p**2*q**2*x + 11*(x**n*d + c)**q*b*p* q*x + 6*(x**n*d + c)**q*b*x + a*p**3*q**3*x + 6*a*p**2*q**2*x + 11*a*p*q*x + 6*a*x),x)*a*d**3*n*p**4*q**4 + 51*int((x**(3*n)*((x**n*d + c)**q*b + a) **p)/((x**n*d + c)**q*b*p**3*q**3*x + 6*(x**n*d + c)**q*b*p**2*q**2*x + 11 *(x**n*d + c)**q*b*p*q*x + 6*(x**n*d + c)**q*b*x + a*p**3*q**3*x + 6*a*p** 2*q**2*x + 11*a*p*q*x + 6*a*x),x)*a*d**3*n*p**3*q**3 + 40*int((x**(3*n)*(( x**n*d + c)**q*b + a)**p)/((x**n*d + c)**q*b*p**3*q**3*x + 6*(x**n*d + c)* *q*b*p**2*q**2*x + 11*(x**n*d + c)**q*b*p*q*x + 6*(x**n*d + c)**q*b*x +...