\(\int (d-e x)^m (d+e x)^m (a+c x^2)^p \, dx\) [33]

Optimal result

Integrand size = 25, antiderivative size = 89 \[ \int (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \, dx=x (d-e x)^m (d+e x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \left (1-\frac {e^2 x^2}{d^2}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-p,-m,\frac {3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right ) \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {648, 334, 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.

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 334

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

\(\Big \downarrow \) 334

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

\(\Big \downarrow \) 333

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

Input:

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

Output:

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

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 648

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c 
*e + d*f*x^2)^FracPart[m])   Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && 
  !(EqQ[p, 2] && LtQ[m, -1])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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