\(\int (d+e x^n)^2 (a+c x^{2 n})^p \, dx\) [48]

Optimal result

Integrand size = 21, antiderivative size = 201 \[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\frac {e^2 x \left (a+c x^{2 n}\right )^{1+p}}{c (1+2 n (1+p))}-\frac {\left (a e^2-c d^2 (1+2 n (1+p))\right ) x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{c (1+2 n (1+p))}+\frac {2 d e x^{1+n} \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{1+n} \] Output:

e^2*x*(a+c*x^(2*n))^(p+1)/c/(1+2*n*(p+1))-(a*e^2-c*d^2*(1+2*n*(p+1)))*x*(a 
+c*x^(2*n))^p*hypergeom([-p, 1/2/n],[1+1/2/n],-c*x^(2*n)/a)/c/(1+2*n*(p+1) 
)/((1+c*x^(2*n)/a)^p)+2*d*e*x^(1+n)*(a+c*x^(2*n))^p*hypergeom([-p, 1/2*(1+ 
n)/n],[3/2+1/2/n],-c*x^(2*n)/a)/(1+n)/((1+c*x^(2*n)/a)^p)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.85 \[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\frac {x \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \left (e^2 (1+n) x^{2 n} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+d (1+2 n) \left (d (1+n) \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+2 e x^n \operatorname {Hypergeometric2F1}\left (\frac {1+n}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )\right )\right )}{(1+n) (1+2 n)} \] Input:

Integrate[(d + e*x^n)^2*(a + c*x^(2*n))^p,x]
 

Output:

(x*(a + c*x^(2*n))^p*(e^2*(1 + n)*x^(2*n)*Hypergeometric2F1[(2 + n^(-1))/2 
, -p, (4 + n^(-1))/2, -((c*x^(2*n))/a)] + d*(1 + 2*n)*(d*(1 + n)*Hypergeom 
etric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x^(2*n))/a)] + 2*e*x^n*Hypergeo 
metric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])))/((1 + n) 
*(1 + 2*n)*(1 + (c*x^(2*n))/a)^p)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1767, 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.

\(\displaystyle \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx\)

\(\Big \downarrow \) 1767

\(\displaystyle \int \left (d^2 \left (a+c x^{2 n}\right )^p+2 d e x^n \left (a+c x^{2 n}\right )^p+e^2 x^{2 n} \left (a+c x^{2 n}\right )^p\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle d^2 x \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2 n},-p,\frac {1}{2} \left (2+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )+\frac {2 d e x^{n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {n+1}{2 n},-p,\frac {1}{2} \left (3+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{n+1}+\frac {e^2 x^{2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} \left (2+\frac {1}{n}\right ),-p,\frac {1}{2} \left (4+\frac {1}{n}\right ),-\frac {c x^{2 n}}{a}\right )}{2 n+1}\)

Input:

Int[(d + e*x^n)^2*(a + c*x^(2*n))^p,x]
 

Output:

(e^2*x^(1 + 2*n)*(a + c*x^(2*n))^p*Hypergeometric2F1[(2 + n^(-1))/2, -p, ( 
4 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 + 2*n)*(1 + (c*x^(2*n))/a)^p) + (d^2 
*x*(a + c*x^(2*n))^p*Hypergeometric2F1[1/(2*n), -p, (2 + n^(-1))/2, -((c*x 
^(2*n))/a)])/(1 + (c*x^(2*n))/a)^p + (2*d*e*x^(1 + n)*(a + c*x^(2*n))^p*Hy 
pergeometric2F1[(1 + n)/(2*n), -p, (3 + n^(-1))/2, -((c*x^(2*n))/a)])/((1 
+ n)*(1 + (c*x^(2*n))/a)^p)
 

Defintions of rubi rules used

rule 1767

Int[((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] 
 :> Int[ExpandIntegrand[(d + e*x^n)^q*(a + c*x^(2*n))^p, x], x] /; FreeQ[{a 
, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && ((Integ 
ersQ[p, q] &&  !IntegerQ[n]) || IGtQ[p, 0] || (IGtQ[q, 0] &&  !IntegerQ[n]) 
)
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

\[\int \left (d +e \,x^{n}\right )^{2} \left (a +c \,x^{2 n}\right )^{p}d x\]

Input:

int((d+e*x^n)^2*(a+c*x^(2*n))^p,x)
 

Output:

int((d+e*x^n)^2*(a+c*x^(2*n))^p,x)
 

Fricas [F]

\[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (e x^{n} + d\right )}^{2} {\left (c x^{2 \, n} + a\right )}^{p} \,d x } \] Input:

integrate((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, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\text {Timed out} \] Input:

integrate((d+e*x**n)**2*(a+c*x**(2*n))**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\int { {\left (e x^{n} + d\right )}^{2} {\left (c x^{2 \, n} + a\right )}^{p} \,d x } \] Input:

integrate((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, x)
 

Giac [F(-2)]

Exception generated. \[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:

integrate((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,[1,0,4,3,1,4,3,1,1]%%%}+%%%{256,[1,0,4,3,1,3,3,1,1]%%%} 
+%%%{384,
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\int {\left (a+c\,x^{2\,n}\right )}^p\,{\left (d+e\,x^n\right )}^2 \,d x \] Input:

int((a + c*x^(2*n))^p*(d + e*x^n)^2,x)
 

Output:

int((a + c*x^(2*n))^p*(d + e*x^n)^2, x)
 

Reduce [F]

\[ \int \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx=\text {too large to display} \] Input:

int((d+e*x^n)^2*(a+c*x^(2*n))^p,x)
 

Output:

(4*x**(2*n)*(x**(2*n)*c + a)**p*c*e**2*n**2*p**2*x + 2*x**(2*n)*(x**(2*n)* 
c + a)**p*c*e**2*n**2*p*x + 4*x**(2*n)*(x**(2*n)*c + a)**p*c*e**2*n*p*x + 
x**(2*n)*(x**(2*n)*c + a)**p*c*e**2*n*x + x**(2*n)*(x**(2*n)*c + a)**p*c*e 
**2*x + 8*x**n*(x**(2*n)*c + a)**p*c*d*e*n**2*p**2*x + 8*x**n*(x**(2*n)*c 
+ a)**p*c*d*e*n**2*p*x + 8*x**n*(x**(2*n)*c + a)**p*c*d*e*n*p*x + 4*x**n*( 
x**(2*n)*c + a)**p*c*d*e*n*x + 2*x**n*(x**(2*n)*c + a)**p*c*d*e*x + 4*(x** 
(2*n)*c + a)**p*a*e**2*n**2*p**2*x + 2*(x**(2*n)*c + a)**p*a*e**2*n**2*p*x 
 + 2*(x**(2*n)*c + a)**p*a*e**2*n*p*x + 4*(x**(2*n)*c + a)**p*c*d**2*n**2* 
p**2*x + 6*(x**(2*n)*c + a)**p*c*d**2*n**2*p*x + 2*(x**(2*n)*c + a)**p*c*d 
**2*n**2*x + 4*(x**(2*n)*c + a)**p*c*d**2*n*p*x + 3*(x**(2*n)*c + a)**p*c* 
d**2*n*x + (x**(2*n)*c + a)**p*c*d**2*x - 32*int((x**(2*n)*c + a)**p/(8*x* 
*(2*n)*c*n**3*p**3 + 12*x**(2*n)*c*n**3*p**2 + 4*x**(2*n)*c*n**3*p + 12*x* 
*(2*n)*c*n**2*p**2 + 12*x**(2*n)*c*n**2*p + 2*x**(2*n)*c*n**2 + 6*x**(2*n) 
*c*n*p + 3*x**(2*n)*c*n + x**(2*n)*c + 8*a*n**3*p**3 + 12*a*n**3*p**2 + 4* 
a*n**3*p + 12*a*n**2*p**2 + 12*a*n**2*p + 2*a*n**2 + 6*a*n*p + 3*a*n + a), 
x)*a**2*e**2*n**5*p**5 - 64*int((x**(2*n)*c + a)**p/(8*x**(2*n)*c*n**3*p** 
3 + 12*x**(2*n)*c*n**3*p**2 + 4*x**(2*n)*c*n**3*p + 12*x**(2*n)*c*n**2*p** 
2 + 12*x**(2*n)*c*n**2*p + 2*x**(2*n)*c*n**2 + 6*x**(2*n)*c*n*p + 3*x**(2* 
n)*c*n + x**(2*n)*c + 8*a*n**3*p**3 + 12*a*n**3*p**2 + 4*a*n**3*p + 12*a*n 
**2*p**2 + 12*a*n**2*p + 2*a*n**2 + 6*a*n*p + 3*a*n + a),x)*a**2*e**2*n...