\(\int (2+e x^2)^q (4-e^2 x^4)^p \, dx\) [185]

Optimal result

Integrand size = 22, antiderivative size = 42 \[ \int \left (2+e x^2\right )^q \left (4-e^2 x^4\right )^p \, dx=2^{2 p+q} x \operatorname {AppellF1}\left (\frac {1}{2},-p,-p-q,\frac {3}{2},\frac {e x^2}{2},-\frac {e x^2}{2}\right ) \] 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 [F]

\[ \int \left (2+e x^2\right )^q \left (4-e^2 x^4\right )^p \, dx=\int \left (2+e x^2\right )^q \left (4-e^2 x^4\right )^p \, dx \] Input:

Integrate[(2 + e*x^2)^q*(4 - e^2*x^4)^p,x]
 

Output:

Integrate[(2 + e*x^2)^q*(4 - e^2*x^4)^p, x]
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1388, 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.

\(\displaystyle \int \left (4-e^2 x^4\right )^p \left (e x^2+2\right )^q \, dx\)

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 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)^q*(4 - e^2*x^4)^p,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])
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

Maple [F]

\[\int \left (e \,x^{2}+2\right )^{q} \left (-e^{2} x^{4}+4\right )^{p}d x\]

Input:

int((e*x^2+2)^q*(-e^2*x^4+4)^p,x)
 

Output:

int((e*x^2+2)^q*(-e^2*x^4+4)^p,x)
 

Fricas [F]

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

integrate((e*x^2+2)^q*(-e^2*x^4+4)^p,x, algorithm="fricas")
 

Output:

integral((-e^2*x^4 + 4)^p*(e*x^2 + 2)^q, x)
 

Sympy [F]

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

integrate((e*x**2+2)**q*(-e**2*x**4+4)**p,x)
 

Output:

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

Maxima [F]

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

integrate((e*x^2+2)^q*(-e^2*x^4+4)^p,x, algorithm="maxima")
 

Output:

integrate((-e^2*x^4 + 4)^p*(e*x^2 + 2)^q, x)
 

Giac [F]

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

integrate((e*x^2+2)^q*(-e^2*x^4+4)^p,x, algorithm="giac")
 

Output:

integrate((-e^2*x^4 + 4)^p*(e*x^2 + 2)^q, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (2+e x^2\right )^q \left (4-e^2 x^4\right )^p \, dx=\int {\left (4-e^2\,x^4\right )}^p\,{\left (e\,x^2+2\right )}^q \,d x \] Input:

int((4 - e^2*x^4)^p*(e*x^2 + 2)^q,x)
 

Output:

int((4 - e^2*x^4)^p*(e*x^2 + 2)^q, x)
 

Reduce [F]

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

int((e*x^2+2)^q*(-e^2*x^4+4)^p,x)
 

Output:

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