\(\int (2-e x^2)^p (2+e x^2)^{p+q} \, dx\) [184]

Optimal result

Integrand size = 22, antiderivative size = 42 \[ \int \left (2-e x^2\right )^p \left (2+e x^2\right )^{p+q} \, dx=2^{2 p+q} x \operatorname {AppellF1}\left (\frac {1}{2},-p,-p-q,\frac {3}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right ) \] Output:

2^(2*p+q)*x*AppellF1(1/2,-p-q,-p,3/2,-1/2*e*x^2,1/2*e*x^2)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(177\) vs. \(2(42)=84\).

Time = 0.24 (sec) , antiderivative size = 177, normalized size of antiderivative = 4.21 \[ \int \left (2-e x^2\right )^p \left (2+e x^2\right )^{p+q} \, dx=\frac {3 x \left (2-e x^2\right )^p \left (2+e x^2\right )^{p+q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p-q,\frac {3}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right )}{3 \operatorname {AppellF1}\left (\frac {1}{2},-p,-p-q,\frac {3}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right )+e x^2 \left (-p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-p-q,\frac {5}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right )+(p+q) \operatorname {AppellF1}\left (\frac {3}{2},-p,1-p-q,\frac {5}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right )\right )} \] Input:

Integrate[(2 - e*x^2)^p*(2 + e*x^2)^(p + q),x]
 

Output:

(3*x*(2 - e*x^2)^p*(2 + e*x^2)^(p + q)*AppellF1[1/2, -p, -p - q, 3/2, (e*x 
^2)/2, -1/2*(e*x^2)])/(3*AppellF1[1/2, -p, -p - q, 3/2, (e*x^2)/2, -1/2*(e 
*x^2)] + e*x^2*(-(p*AppellF1[3/2, 1 - p, -p - q, 5/2, (e*x^2)/2, -1/2*(e*x 
^2)]) + (p + q)*AppellF1[3/2, -p, 1 - p - q, 5/2, (e*x^2)/2, -1/2*(e*x^2)] 
))
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {333}

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.

\(\displaystyle \int \left (2-e x^2\right )^p \left (e x^2+2\right )^{p+q} \, dx\)

\(\Big \downarrow \) 333

\(\displaystyle x 2^{2 p+q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-p-q,\frac {3}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right )\)

Input:

Int[(2 - e*x^2)^p*(2 + e*x^2)^(p + q),x]
 

Output:

2^(2*p + q)*x*AppellF1[1/2, -p, -p - q, 3/2, (e*x^2)/2, -1/2*(e*x^2)]
 

Defintions of rubi rules used

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

Maple [F]

\[\int \left (-e \,x^{2}+2\right )^{p} \left (e \,x^{2}+2\right )^{p +q}d x\]

Input:

int((-e*x^2+2)^p*(e*x^2+2)^(p+q),x)
 

Output:

int((-e*x^2+2)^p*(e*x^2+2)^(p+q),x)
 

Fricas [F]

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

integrate((-e*x^2+2)^p*(e*x^2+2)^(p+q),x, algorithm="fricas")
 

Output:

integral((e*x^2 + 2)^(p + q)*(-e*x^2 + 2)^p, x)
 

Sympy [F]

\[ \int \left (2-e x^2\right )^p \left (2+e x^2\right )^{p+q} \, dx=\int \left (- e x^{2} + 2\right )^{p} \left (e x^{2} + 2\right )^{p + q}\, dx \] Input:

integrate((-e*x**2+2)**p*(e*x**2+2)**(p+q),x)
 

Output:

Integral((-e*x**2 + 2)**p*(e*x**2 + 2)**(p + q), x)
 

Maxima [F]

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

integrate((-e*x^2+2)^p*(e*x^2+2)^(p+q),x, algorithm="maxima")
 

Output:

integrate((e*x^2 + 2)^(p + q)*(-e*x^2 + 2)^p, x)
 

Giac [F]

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

integrate((-e*x^2+2)^p*(e*x^2+2)^(p+q),x, algorithm="giac")
 

Output:

integrate((e*x^2 + 2)^(p + q)*(-e*x^2 + 2)^p, x)
 

Mupad [F(-1)]

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

int((2 - e*x^2)^p*(e*x^2 + 2)^(p + q),x)
 

Output:

int((2 - e*x^2)^p*(e*x^2 + 2)^(p + q), x)
 

Reduce [F]

\[ \int \left (2-e x^2\right )^p \left (2+e x^2\right )^{p+q} \, dx=\frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p} x +16 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p} x^{2}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) e p q +8 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p} x^{2}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) e \,q^{2}+4 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p} x^{2}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) e q -64 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) p^{2}-64 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) p q -16 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) p -16 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) q^{2}-8 \left (\int \frac {\left (e \,x^{2}+2\right )^{p +q} \left (-e \,x^{2}+2\right )^{p}}{4 e^{2} p \,x^{4}+2 e^{2} q \,x^{4}+e^{2} x^{4}-16 p -8 q -4}d x \right ) q}{4 p +2 q +1} \] Input:

int((-e*x^2+2)^p*(e*x^2+2)^(p+q),x)
 

Output:

((e*x**2 + 2)**(p + q)*( - e*x**2 + 2)**p*x + 16*int(((e*x**2 + 2)**(p + q 
)*( - e*x**2 + 2)**p*x**2)/(4*e**2*p*x**4 + 2*e**2*q*x**4 + e**2*x**4 - 16 
*p - 8*q - 4),x)*e*p*q + 8*int(((e*x**2 + 2)**(p + q)*( - e*x**2 + 2)**p*x 
**2)/(4*e**2*p*x**4 + 2*e**2*q*x**4 + e**2*x**4 - 16*p - 8*q - 4),x)*e*q** 
2 + 4*int(((e*x**2 + 2)**(p + q)*( - e*x**2 + 2)**p*x**2)/(4*e**2*p*x**4 + 
 2*e**2*q*x**4 + e**2*x**4 - 16*p - 8*q - 4),x)*e*q - 64*int(((e*x**2 + 2) 
**(p + q)*( - e*x**2 + 2)**p)/(4*e**2*p*x**4 + 2*e**2*q*x**4 + e**2*x**4 - 
 16*p - 8*q - 4),x)*p**2 - 64*int(((e*x**2 + 2)**(p + q)*( - e*x**2 + 2)** 
p)/(4*e**2*p*x**4 + 2*e**2*q*x**4 + e**2*x**4 - 16*p - 8*q - 4),x)*p*q - 1 
6*int(((e*x**2 + 2)**(p + q)*( - e*x**2 + 2)**p)/(4*e**2*p*x**4 + 2*e**2*q 
*x**4 + e**2*x**4 - 16*p - 8*q - 4),x)*p - 16*int(((e*x**2 + 2)**(p + q)*( 
 - e*x**2 + 2)**p)/(4*e**2*p*x**4 + 2*e**2*q*x**4 + e**2*x**4 - 16*p - 8*q 
 - 4),x)*q**2 - 8*int(((e*x**2 + 2)**(p + q)*( - e*x**2 + 2)**p)/(4*e**2*p 
*x**4 + 2*e**2*q*x**4 + e**2*x**4 - 16*p - 8*q - 4),x)*q)/(4*p + 2*q + 1)