\(\int (a-b x^n)^p (a+b x^n)^p (c+d x^{2 n})^q \, dx\) [3]

Optimal result

Integrand size = 31, antiderivative size = 113 \[ \int \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (c+d x^{2 n}\right )^q \, dx=x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (1+\frac {d x^{2 n}}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,-q,\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {b^2 x^{2 n}}{a^2},-\frac {d x^{2 n}}{c}\right ) \] Output:

x*(a-b*x^n)^p*(a+b*x^n)^p*(c+d*x^(2*n))^q*AppellF1(1/2/n,-p,-q,1+1/2/n,b^2 
*x^(2*n)/a^2,-d*x^(2*n)/c)/((1-b^2*x^(2*n)/a^2)^p)/((1+d*x^(2*n)/c)^q)
 

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2038, 937, 937, 936}

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.

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

\(\Big \downarrow \) 2038

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

\(\displaystyle \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac {d x^{2 n}}{c}+1\right )^{-q} \int \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^p \left (\frac {d x^{2 n}}{c}+1\right )^qdx\)

\(\Big \downarrow \) 936

\(\displaystyle x \left (a-b x^n\right )^p \left (a+b x^n\right )^p \left (1-\frac {b^2 x^{2 n}}{a^2}\right )^{-p} \left (c+d x^{2 n}\right )^q \left (\frac {d x^{2 n}}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2 n},-p,-q,\frac {1}{2} \left (2+\frac {1}{n}\right ),\frac {b^2 x^{2 n}}{a^2},-\frac {d x^{2 n}}{c}\right )\)

Input:

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

Output:

(x*(a - b*x^n)^p*(a + b*x^n)^p*(c + d*x^(2*n))^q*AppellF1[1/(2*n), -p, -q, 
 (2 + n^(-1))/2, (b^2*x^(2*n))/a^2, -((d*x^(2*n))/c)])/((1 - (b^2*x^(2*n)) 
/a^2)^p*(1 + (d*x^(2*n))/c)^q)
 

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 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*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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