\(\int (2-e x)^p (2+e x)^{p+q} \, dx\) [388]

Optimal result

Integrand size = 18, antiderivative size = 44 \[ \int (2-e x)^p (2+e x)^{p+q} \, dx=\frac {4^p (2+e x)^{1+p+q} \operatorname {Hypergeometric2F1}\left (-p,1+p+q,2+p+q,\frac {1}{4} (2+e x)\right )}{e (1+p+q)} \] Output:

4^p*(e*x+2)^(1+p+q)*hypergeom([-p, 1+p+q],[2+p+q],1/4*e*x+1/2)/e/(1+p+q)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11 \[ \int (2-e x)^p (2+e x)^{p+q} \, dx=-\frac {4^{p+q} (2-e x)^{1+p} \operatorname {Hypergeometric2F1}\left (1+p,-p-q,2+p,\frac {1}{4} (2-e x)\right )}{e (1+p)} \] Input:

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

Output:

-((4^(p + q)*(2 - e*x)^(1 + p)*Hypergeometric2F1[1 + p, -p - q, 2 + p, (2 
- e*x)/4])/(e*(1 + p)))
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.11, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {79}

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.

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

\(\Big \downarrow \) 79

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

Input:

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

Output:

-((4^(p + q)*(2 - e*x)^(1 + p)*Hypergeometric2F1[1 + p, -p - q, 2 + p, (2 
- e*x)/4])/(e*(1 + p)))
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (2-e x)^p (2+e x)^{p+q} \, dx=\frac {\left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} e q x +4 \left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} p +2 \left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} q -16 \left (\int \frac {\left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} x}{2 e^{2} p \,x^{2}+e^{2} q \,x^{2}+e^{2} x^{2}-8 p -4 q -4}d x \right ) e^{2} p^{3}-24 \left (\int \frac {\left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} x}{2 e^{2} p \,x^{2}+e^{2} q \,x^{2}+e^{2} x^{2}-8 p -4 q -4}d x \right ) e^{2} p^{2} q -8 \left (\int \frac {\left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} x}{2 e^{2} p \,x^{2}+e^{2} q \,x^{2}+e^{2} x^{2}-8 p -4 q -4}d x \right ) e^{2} p^{2}-8 \left (\int \frac {\left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} x}{2 e^{2} p \,x^{2}+e^{2} q \,x^{2}+e^{2} x^{2}-8 p -4 q -4}d x \right ) e^{2} p \,q^{2}-8 \left (\int \frac {\left (e x +2\right )^{p +q} \left (-e x +2\right )^{p} x}{2 e^{2} p \,x^{2}+e^{2} q \,x^{2}+e^{2} x^{2}-8 p -4 q -4}d x \right ) e^{2} p q}{e q \left (2 p +q +1\right )} \] Input:

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

Output:

((e*x + 2)**(p + q)*( - e*x + 2)**p*e*q*x + 4*(e*x + 2)**(p + q)*( - e*x + 
 2)**p*p + 2*(e*x + 2)**(p + q)*( - e*x + 2)**p*q - 16*int(((e*x + 2)**(p 
+ q)*( - e*x + 2)**p*x)/(2*e**2*p*x**2 + e**2*q*x**2 + e**2*x**2 - 8*p - 4 
*q - 4),x)*e**2*p**3 - 24*int(((e*x + 2)**(p + q)*( - e*x + 2)**p*x)/(2*e* 
*2*p*x**2 + e**2*q*x**2 + e**2*x**2 - 8*p - 4*q - 4),x)*e**2*p**2*q - 8*in 
t(((e*x + 2)**(p + q)*( - e*x + 2)**p*x)/(2*e**2*p*x**2 + e**2*q*x**2 + e* 
*2*x**2 - 8*p - 4*q - 4),x)*e**2*p**2 - 8*int(((e*x + 2)**(p + q)*( - e*x 
+ 2)**p*x)/(2*e**2*p*x**2 + e**2*q*x**2 + e**2*x**2 - 8*p - 4*q - 4),x)*e* 
*2*p*q**2 - 8*int(((e*x + 2)**(p + q)*( - e*x + 2)**p*x)/(2*e**2*p*x**2 + 
e**2*q*x**2 + e**2*x**2 - 8*p - 4*q - 4),x)*e**2*p*q)/(e*q*(2*p + q + 1))