\(\int (1-e x)^m (1+e x)^m (a+c x^2)^p \, dx\) [31]

Optimal result

Integrand size = 25, antiderivative size = 54 \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right ) \] Output:

x*(c*x^2+a)^p*AppellF1(1/2,-m,-p,3/2,e^2*x^2,-c*x^2/a)/((1+c*x^2/a)^p)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(167\) vs. \(2(54)=108\).

Time = 0.39 (sec) , antiderivative size = 167, normalized size of antiderivative = 3.09 \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\frac {3 a x \left (a+c x^2\right )^p \left (1-e^2 x^2\right )^m \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right )}{3 a \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},e^2 x^2\right )+2 x^2 \left (c p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-m,\frac {5}{2},-\frac {c x^2}{a},e^2 x^2\right )-a e^2 m \operatorname {AppellF1}\left (\frac {3}{2},-p,1-m,\frac {5}{2},-\frac {c x^2}{a},e^2 x^2\right )\right )} \] Input:

Integrate[(1 - e*x)^m*(1 + e*x)^m*(a + c*x^2)^p,x]
 

Output:

(3*a*x*(a + c*x^2)^p*(1 - e^2*x^2)^m*AppellF1[1/2, -p, -m, 3/2, -((c*x^2)/ 
a), e^2*x^2])/(3*a*AppellF1[1/2, -p, -m, 3/2, -((c*x^2)/a), e^2*x^2] + 2*x 
^2*(c*p*AppellF1[3/2, 1 - p, -m, 5/2, -((c*x^2)/a), e^2*x^2] - a*e^2*m*App 
ellF1[3/2, -p, 1 - m, 5/2, -((c*x^2)/a), e^2*x^2]))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {643, 334, 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.

\(\displaystyle \int (1-e x)^m (e x+1)^m \left (a+c x^2\right )^p \, dx\)

\(\Big \downarrow \) 643

\(\displaystyle \int \left (1-e^2 x^2\right )^m \left (a+c x^2\right )^pdx\)

\(\Big \downarrow \) 334

\(\displaystyle \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \int \left (\frac {c x^2}{a}+1\right )^p \left (1-e^2 x^2\right )^mdx\)

\(\Big \downarrow \) 333

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

Input:

Int[(1 - e*x)^m*(1 + e*x)^m*(a + c*x^2)^p,x]
 

Output:

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

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 334

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

rule 643

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_.), x_Symbol] :> Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x] /; FreeQ[{a 
, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && (Integer 
Q[m] || (GtQ[c, 0] && GtQ[e, 0]))
 

Maple [F]

\[\int \left (-e x +1\right )^{m} \left (e x +1\right )^{m} \left (c \,x^{2}+a \right )^{p}d x\]

Input:

int((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x)
 

Output:

int((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x)
 

Fricas [F]

\[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + 1\right )}^{m} {\left (-e x + 1\right )}^{m} \,d x } \] Input:

integrate((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((-e*x+1)**m*(e*x+1)**m*(c*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + 1\right )}^{m} {\left (-e x + 1\right )}^{m} \,d x } \] Input:

integrate((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m, x)
 

Giac [F]

\[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\int { {\left (c x^{2} + a\right )}^{p} {\left (e x + 1\right )}^{m} {\left (-e x + 1\right )}^{m} \,d x } \] Input:

integrate((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((c*x^2 + a)^p*(e*x + 1)^m*(-e*x + 1)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx=\int {\left (c\,x^2+a\right )}^p\,{\left (1-e\,x\right )}^m\,{\left (e\,x+1\right )}^m \,d x \] Input:

int((a + c*x^2)^p*(1 - e*x)^m*(e*x + 1)^m,x)
 

Output:

int((a + c*x^2)^p*(1 - e*x)^m*(e*x + 1)^m, x)
 

Reduce [F]

\[ \int (1-e x)^m (1+e x)^m \left (a+c x^2\right )^p \, dx =\text {Too large to display} \] Input:

int((-e*x+1)^m*(e*x+1)^m*(c*x^2+a)^p,x)
 

Output:

((a + c*x**2)**p*(e*x + 1)**m*( - e*x + 1)**m*x + 4*int(((a + c*x**2)**p*( 
e*x + 1)**m*( - e*x + 1)**m*x**2)/(2*a*e**2*m*x**2 + 2*a*e**2*p*x**2 + a*e 
**2*x**2 - 2*a*m - 2*a*p - a + 2*c*e**2*m*x**4 + 2*c*e**2*p*x**4 + c*e**2* 
x**4 - 2*c*m*x**2 - 2*c*p*x**2 - c*x**2),x)*a*e**2*m*p + 4*int(((a + c*x** 
2)**p*(e*x + 1)**m*( - e*x + 1)**m*x**2)/(2*a*e**2*m*x**2 + 2*a*e**2*p*x** 
2 + a*e**2*x**2 - 2*a*m - 2*a*p - a + 2*c*e**2*m*x**4 + 2*c*e**2*p*x**4 + 
c*e**2*x**4 - 2*c*m*x**2 - 2*c*p*x**2 - c*x**2),x)*a*e**2*p**2 + 2*int(((a 
 + c*x**2)**p*(e*x + 1)**m*( - e*x + 1)**m*x**2)/(2*a*e**2*m*x**2 + 2*a*e* 
*2*p*x**2 + a*e**2*x**2 - 2*a*m - 2*a*p - a + 2*c*e**2*m*x**4 + 2*c*e**2*p 
*x**4 + c*e**2*x**4 - 2*c*m*x**2 - 2*c*p*x**2 - c*x**2),x)*a*e**2*p - 4*in 
t(((a + c*x**2)**p*(e*x + 1)**m*( - e*x + 1)**m*x**2)/(2*a*e**2*m*x**2 + 2 
*a*e**2*p*x**2 + a*e**2*x**2 - 2*a*m - 2*a*p - a + 2*c*e**2*m*x**4 + 2*c*e 
**2*p*x**4 + c*e**2*x**4 - 2*c*m*x**2 - 2*c*p*x**2 - c*x**2),x)*c*m**2 - 4 
*int(((a + c*x**2)**p*(e*x + 1)**m*( - e*x + 1)**m*x**2)/(2*a*e**2*m*x**2 
+ 2*a*e**2*p*x**2 + a*e**2*x**2 - 2*a*m - 2*a*p - a + 2*c*e**2*m*x**4 + 2* 
c*e**2*p*x**4 + c*e**2*x**4 - 2*c*m*x**2 - 2*c*p*x**2 - c*x**2),x)*c*m*p - 
 2*int(((a + c*x**2)**p*(e*x + 1)**m*( - e*x + 1)**m*x**2)/(2*a*e**2*m*x** 
2 + 2*a*e**2*p*x**2 + a*e**2*x**2 - 2*a*m - 2*a*p - a + 2*c*e**2*m*x**4 + 
2*c*e**2*p*x**4 + c*e**2*x**4 - 2*c*m*x**2 - 2*c*p*x**2 - c*x**2),x)*c*m - 
 4*int(((a + c*x**2)**p*(e*x + 1)**m*( - e*x + 1)**m)/(2*a*e**2*m*x**2 ...