\(\int (a-b x^n)^p (a+b x^n)^p (a^2+b^2 x^{2 n})^p \, dx\) [4]

Optimal result

Integrand size = 35, antiderivative size = 87 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4 n},-p,\frac {1}{4} \left (4+\frac {1}{n}\right ),\frac {b^4 x^{4 n}}{a^4}\right ) \] Output:

x*(a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p*hypergeom([-p, 1/4/n],[1+1/4 
/n],b^4*x^(4*n)/a^4)/((1-b^4*x^(4*n)/a^4)^p)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{4 n},-p,1+\frac {1}{4 n},\frac {b^4 x^{4 n}}{a^4}\right ) \] Input:

Integrate[(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p,x]
 

Output:

(x*(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p*Hypergeometric2F1[1/( 
4*n), -p, 1 + 1/(4*n), (b^4*x^(4*n))/a^4])/(1 - (b^4*x^(4*n))/a^4)^p
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2038, 785, 779, 778}

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.

\(\displaystyle \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx\)

\(\Big \downarrow \) 2038

\(\displaystyle \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2-b^2 x^{2 n}\right )^{-p} \int \left (a^2-b^2 x^{2 n}\right )^p \left (b^2 x^{2 n}+a^2\right )^pdx\)

\(\Big \downarrow \) 785

\(\displaystyle \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (a^4-b^4 x^{4 n}\right )^{-p} \int \left (a^4-b^4 x^{4 n}\right )^pdx\)

\(\Big \downarrow \) 779

\(\displaystyle \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^{-p} \int \left (1-\frac {b^4 x^{4 n}}{a^4}\right )^pdx\)

\(\Big \downarrow \) 778

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

Input:

Int[(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p,x]
 

Output:

(x*(a - b*x^n)^p*(a + b*x^n)^p*(a^2 + b^2*x^(2*n))^p*Hypergeometric2F1[1/( 
4*n), -p, (4 + n^(-1))/4, (b^4*x^(4*n))/a^4])/(1 - (b^4*x^(4*n))/a^4)^p
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 785

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Sy 
mbol] :> Simp[(a1 + b1*x^n)^FracPart[p]*((a2 + b2*x^n)^FracPart[p]/(a1*a2 + 
 b1*b2*x^(2*n))^FracPart[p])   Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /; Fre 
eQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]
 

rule 2038

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p 
_)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) 
^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] 
)   Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, 
 b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] &&  !(Eq 
Q[n, 2] && IGtQ[q, 0])
 

Maple [F]

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

Input:

int((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x)
 

Output:

int((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x)
 

Fricas [F]

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

integrate((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x, algorithm="fricas 
")
 

Output:

integral((b^2*x^(2*n) + a^2)^p*(b*x^n + a)^p*(-b*x^n + a)^p, x)
 

Sympy [F(-1)]

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

integrate((a-b*x**n)**p*(a+b*x**n)**p*(a**2+b**2*x**(2*n))**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x, algorithm="maxima 
")
 

Output:

integrate((b^2*x^(2*n) + a^2)^p*(b*x^n + a)^p*(-b*x^n + a)^p, x)
 

Giac [F]

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

integrate((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x, algorithm="giac")
 

Output:

integrate((b^2*x^(2*n) + a^2)^p*(b*x^n + a)^p*(-b*x^n + a)^p, x)
 

Mupad [F(-1)]

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

int((a + b*x^n)^p*(a - b*x^n)^p*(a^2 + b^2*x^(2*n))^p,x)
 

Output:

int((a + b*x^n)^p*(a - b*x^n)^p*(a^2 + b^2*x^(2*n))^p, x)
 

Reduce [F]

\[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (a^2+b^2 x^{2 n}\right )^p \, dx=\frac {\left (x^{n} b +a \right )^{p} \left (x^{2 n} b^{2}+a^{2}\right )^{p} \left (-x^{n} b +a \right )^{p} x -16 \left (\int \frac {\left (x^{n} b +a \right )^{p} \left (x^{2 n} b^{2}+a^{2}\right )^{p} \left (-x^{n} b +a \right )^{p}}{4 x^{4 n} b^{4} n p +x^{4 n} b^{4}-4 a^{4} n p -a^{4}}d x \right ) a^{4} n^{2} p^{2}-4 \left (\int \frac {\left (x^{n} b +a \right )^{p} \left (x^{2 n} b^{2}+a^{2}\right )^{p} \left (-x^{n} b +a \right )^{p}}{4 x^{4 n} b^{4} n p +x^{4 n} b^{4}-4 a^{4} n p -a^{4}}d x \right ) a^{4} n p}{4 n p +1} \] Input:

int((a-b*x^n)^p*(a+b*x^n)^p*(a^2+b^2*x^(2*n))^p,x)
 

Output:

((x**n*b + a)**p*(x**(2*n)*b**2 + a**2)**p*( - x**n*b + a)**p*x - 16*int(( 
(x**n*b + a)**p*(x**(2*n)*b**2 + a**2)**p*( - x**n*b + a)**p)/(4*x**(4*n)* 
b**4*n*p + x**(4*n)*b**4 - 4*a**4*n*p - a**4),x)*a**4*n**2*p**2 - 4*int((( 
x**n*b + a)**p*(x**(2*n)*b**2 + a**2)**p*( - x**n*b + a)**p)/(4*x**(4*n)*b 
**4*n*p + x**(4*n)*b**4 - 4*a**4*n*p - a**4),x)*a**4*n*p)/(4*n*p + 1)