Integrand size = 26, antiderivative size = 268 \[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\frac {d^2 (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2 n},-p,1+\frac {1+m}{2 n},-\frac {c x^{2 n}}{a}\right )}{f (1+m)}+\frac {2 d e x^n (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{2 n},-p,\frac {1+m+3 n}{2 n},-\frac {c x^{2 n}}{a}\right )}{f (1+m+n)}+\frac {e^2 x^{2 n} (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{2 n},-p,\frac {1+m+4 n}{2 n},-\frac {c x^{2 n}}{a}\right )}{f (1+m+2 n)} \] Output:
d^2*(f*x)^(1+m)*(a+c*x^(2*n))^p*hypergeom([-p, 1/2*(1+m)/n],[1+1/2*(1+m)/n ],-c*x^(2*n)/a)/f/(1+m)/((1+c*x^(2*n)/a)^p)+2*d*e*x^n*(f*x)^(1+m)*(a+c*x^( 2*n))^p*hypergeom([-p, 1/2*(1+m+n)/n],[1/2*(1+m+3*n)/n],-c*x^(2*n)/a)/f/(1 +m+n)/((1+c*x^(2*n)/a)^p)+e^2*x^(2*n)*(f*x)^(1+m)*(a+c*x^(2*n))^p*hypergeo m([-p, 1/2*(1+m+2*n)/n],[1/2*(1+m+4*n)/n],-c*x^(2*n)/a)/f/(1+m+2*n)/((1+c* x^(2*n)/a)^p)
Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.71 \[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (\frac {d^2 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2 n},-p,1+\frac {1+m}{2 n},-\frac {c x^{2 n}}{a}\right )}{1+m}+e x^n \left (\frac {2 d \operatorname {Hypergeometric2F1}\left (\frac {1+m+n}{2 n},-p,\frac {1+m+3 n}{2 n},-\frac {c x^{2 n}}{a}\right )}{1+m+n}+\frac {e x^n \operatorname {Hypergeometric2F1}\left (\frac {1+m+2 n}{2 n},-p,\frac {1+m+4 n}{2 n},-\frac {c x^{2 n}}{a}\right )}{1+m+2 n}\right )\right ) \] Input:
Integrate[(f*x)^m*(d + e*x^n)^2*(a + c*x^(2*n))^p,x]
Output:
(x*(f*x)^m*(a + c*x^(2*n))^p*((d^2*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)])/(1 + m) + e*x^n*((2*d*Hypergeometric2F 1[(1 + m + n)/(2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/(1 + m + n) + (e*x^n*Hypergeometric2F1[(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n) , -((c*x^(2*n))/a)])/(1 + m + 2*n))))/(1 + (c*x^(2*n))/a)^p
Time = 0.40 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1885, 2009}
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.
\(\displaystyle \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx\) |
\(\Big \downarrow \) 1885 |
\(\displaystyle \int \left (d^2 (f x)^m \left (a+c x^{2 n}\right )^p+2 d e x^n (f x)^m \left (a+c x^{2 n}\right )^p+e^2 x^{2 n} (f x)^m \left (a+c x^{2 n}\right )^p\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d^2 (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2 n},-p,\frac {m+1}{2 n}+1,-\frac {c x^{2 n}}{a}\right )}{f (m+1)}+\frac {2 d e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+n+1}{2 n},-p,\frac {m+3 n+1}{2 n},-\frac {c x^{2 n}}{a}\right )}{m+n+1}+\frac {e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2 n+1}{2 n},-p,\frac {m+4 n+1}{2 n},-\frac {c x^{2 n}}{a}\right )}{m+2 n+1}\) |
Input:
Int[(f*x)^m*(d + e*x^n)^2*(a + c*x^(2*n))^p,x]
Output:
(d^2*(f*x)^(1 + m)*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m)/(2*n), -p, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a)])/(f*(1 + m)*(1 + (c*x^(2*n))/a)^p) + (2*d*e*x^(1 + n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1[(1 + m + n)/( 2*n), -p, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + n)*(1 + (c*x^( 2*n))/a)^p) + (e^2*x^(1 + 2*n)*(f*x)^m*(a + c*x^(2*n))^p*Hypergeometric2F1 [(1 + m + 2*n)/(2*n), -p, (1 + m + 4*n)/(2*n), -((c*x^(2*n))/a)])/((1 + m + 2*n)*(1 + (c*x^(2*n))/a)^p)
Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ (n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^n)^q*(a + c* x^(2*n))^p, x], x] /; FreeQ[{a, c, d, e, f, m, n, p, q}, x] && EqQ[n2, 2*n] && !RationalQ[n] && (IGtQ[p, 0] || IGtQ[q, 0])
\[\int \left (f x \right )^{m} \left (d +e \,x^{n}\right )^{2} \left (a +c \,x^{2 n}\right )^{p}d x\]
Input:
int((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x)
Output:
int((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x)
\[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (e x^{n} + d\right )}^{2} {\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m} \,d x } \] Input:
integrate((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x, algorithm="fricas")
Output:
integral((e^2*x^(2*n) + 2*d*e*x^n + d^2)*(c*x^(2*n) + a)^p*(f*x)^m, x)
Timed out. \[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\text {Timed out} \] Input:
integrate((f*x)**m*(d+e*x**n)**2*(a+c*x**(2*n))**p,x)
Output:
Timed out
\[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (e x^{n} + d\right )}^{2} {\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m} \,d x } \] Input:
integrate((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x, algorithm="maxima")
Output:
integrate((e*x^n + d)^2*(c*x^(2*n) + a)^p*(f*x)^m, x)
Exception generated. \[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:
integrate((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{64,[1,0,4,3,0,1,4,3,1,1]%%%}+%%%{256,[1,0,4,3,0,1,3,3,1,1] %%%}+%%%{
Timed out. \[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\int {\left (a+c\,x^{2\,n}\right )}^p\,{\left (f\,x\right )}^m\,{\left (d+e\,x^n\right )}^2 \,d x \] Input:
int((a + c*x^(2*n))^p*(f*x)^m*(d + e*x^n)^2,x)
Output:
int((a + c*x^(2*n))^p*(f*x)^m*(d + e*x^n)^2, x)
\[ \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\text {too large to display} \] Input:
int((f*x)^m*(d+e*x^n)^2*(a+c*x^(2*n))^p,x)
Output:
(f**m*(x**(m + 2*n)*(x**(2*n)*c + a)**p*c*e**2*m**2*x + 4*x**(m + 2*n)*(x* *(2*n)*c + a)**p*c*e**2*m*n*p*x + x**(m + 2*n)*(x**(2*n)*c + a)**p*c*e**2* m*n*x + 2*x**(m + 2*n)*(x**(2*n)*c + a)**p*c*e**2*m*x + 4*x**(m + 2*n)*(x* *(2*n)*c + a)**p*c*e**2*n**2*p**2*x + 2*x**(m + 2*n)*(x**(2*n)*c + a)**p*c *e**2*n**2*p*x + 4*x**(m + 2*n)*(x**(2*n)*c + a)**p*c*e**2*n*p*x + x**(m + 2*n)*(x**(2*n)*c + a)**p*c*e**2*n*x + x**(m + 2*n)*(x**(2*n)*c + a)**p*c* e**2*x + 2*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*m**2*x + 8*x**(m + n)*(x** (2*n)*c + a)**p*c*d*e*m*n*p*x + 4*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*m*n *x + 4*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*m*x + 8*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*n**2*p**2*x + 8*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*n**2*p *x + 8*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*n*p*x + 4*x**(m + n)*(x**(2*n) *c + a)**p*c*d*e*n*x + 2*x**(m + n)*(x**(2*n)*c + a)**p*c*d*e*x + 2*x**m*( x**(2*n)*c + a)**p*a*e**2*m*n*p*x + 4*x**m*(x**(2*n)*c + a)**p*a*e**2*n**2 *p**2*x + 2*x**m*(x**(2*n)*c + a)**p*a*e**2*n**2*p*x + 2*x**m*(x**(2*n)*c + a)**p*a*e**2*n*p*x + x**m*(x**(2*n)*c + a)**p*c*d**2*m**2*x + 4*x**m*(x* *(2*n)*c + a)**p*c*d**2*m*n*p*x + 3*x**m*(x**(2*n)*c + a)**p*c*d**2*m*n*x + 2*x**m*(x**(2*n)*c + a)**p*c*d**2*m*x + 4*x**m*(x**(2*n)*c + a)**p*c*d** 2*n**2*p**2*x + 6*x**m*(x**(2*n)*c + a)**p*c*d**2*n**2*p*x + 2*x**m*(x**(2 *n)*c + a)**p*c*d**2*n**2*x + 4*x**m*(x**(2*n)*c + a)**p*c*d**2*n*p*x + 3* x**m*(x**(2*n)*c + a)**p*c*d**2*n*x + x**m*(x**(2*n)*c + a)**p*c*d**2*x...