\(\int (e x)^{-3-4 p} (a+b x^2)^p (c+d x^2)^p \, dx\) [1656]

Optimal result

Integrand size = 28, antiderivative size = 107 \[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=-\frac {(e x)^{-2-4 p} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^p \left (\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{2 a e (1+2 p)} \] Output:

-1/2*(e*x)^(-2-4*p)*(b*x^2+a)^(p+1)*(d*x^2+c)^p*hypergeom([-p, -1-2*p],[-2 
*p],(-a*d+b*c)*x^2/c/(b*x^2+a))/a/e/(1+2*p)/((a*(d*x^2+c)/c/(b*x^2+a))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=-\frac {(e x)^{-4 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^{1+p} \left (1+\frac {d x^2}{c}\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )}{2 e^3 (c+2 c p) x^2} \] Input:

Integrate[(e*x)^(-3 - 4*p)*(a + b*x^2)^p*(c + d*x^2)^p,x]
 

Output:

-1/2*((a + b*x^2)^p*(c + d*x^2)^(1 + p)*(1 + (d*x^2)/c)^p*Hypergeometric2F 
1[-1 - 2*p, -p, -2*p, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(e^3*(c + 2*c 
*p)*x^2*(e*x)^(4*p)*(1 + (b*x^2)/a)^p)
 

Rubi [A] (warning: unable to verify)

Time = 0.24 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {395, 395, 394}

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.

\(\displaystyle \int (e x)^{-4 p-3} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx\)

\(\Big \downarrow \) 395

\(\displaystyle \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-4 p-3} \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^pdx\)

\(\Big \downarrow \) 395

\(\displaystyle \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^p \left (\frac {d x^2}{c}+1\right )^{-p} \int (e x)^{-4 p-3} \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^pdx\)

\(\Big \downarrow \) 394

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

Input:

Int[(e*x)^(-3 - 4*p)*(a + b*x^2)^p*(c + d*x^2)^p,x]
 

Output:

-1/2*((e*x)^(-2 - 4*p)*(a + b*x^2)^p*(c + d*x^2)^p*(1 + (d*x^2)/c)^(1 + p) 
*Hypergeometric2F1[-1 - 2*p, -p, -2*p, -((c*((b*x^2)/a - (d*x^2)/c))/(c + 
d*x^2))])/(e*(1 + 2*p)*(1 + (b*x^2)/a)^p)
 

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

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

Maple [F]

\[\int \left (e x \right )^{-3-4 p} \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )^{p}d x\]

Input:

int((e*x)^(-3-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Output:

int((e*x)^(-3-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Fricas [F]

\[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{p} \left (e x\right )^{-4 \, p - 3} \,d x } \] Input:

integrate((e*x)^(-3-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(d*x^2 + c)^p*(e*x)^(-4*p - 3), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-3-4*p)*(b*x**2+a)**p*(d*x**2+c)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{p} \left (e x\right )^{-4 \, p - 3} \,d x } \] Input:

integrate((e*x)^(-3-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*(d*x^2 + c)^p*(e*x)^(-4*p - 3), x)
 

Giac [F]

\[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{p} \left (e x\right )^{-4 \, p - 3} \,d x } \] Input:

integrate((e*x)^(-3-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*(d*x^2 + c)^p*(e*x)^(-4*p - 3), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^p}{{\left (e\,x\right )}^{4\,p+3}} \,d x \] Input:

int(((a + b*x^2)^p*(c + d*x^2)^p)/(e*x)^(4*p + 3),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x^2)^p)/(e*x)^(4*p + 3), x)
 

Reduce [F]

\[ \int (e x)^{-3-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\text {too large to display} \] Input:

int((e*x)^(-3-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Output:

( - (c + d*x**2)**p*(a + b*x**2)**p*a*d - (c + d*x**2)**p*(a + b*x**2)**p* 
b*c - 2*(c + d*x**2)**p*(a + b*x**2)**p*b*d*x**2 + 4*x**(4*p)*int(((c + d* 
x**2)**p*(a + b*x**2)**p)/(2*x**(4*p)*a**2*c*d*p*x + x**(4*p)*a**2*c*d*x + 
 2*x**(4*p)*a**2*d**2*p*x**3 + x**(4*p)*a**2*d**2*x**3 + 2*x**(4*p)*a*b*c* 
*2*p*x + x**(4*p)*a*b*c**2*x + 4*x**(4*p)*a*b*c*d*p*x**3 + 2*x**(4*p)*a*b* 
c*d*x**3 + 2*x**(4*p)*a*b*d**2*p*x**5 + x**(4*p)*a*b*d**2*x**5 + 2*x**(4*p 
)*b**2*c**2*p*x**3 + x**(4*p)*b**2*c**2*x**3 + 2*x**(4*p)*b**2*c*d*p*x**5 
+ x**(4*p)*b**2*c*d*x**5),x)*a**3*d**3*p**2*x**2 + 2*x**(4*p)*int(((c + d* 
x**2)**p*(a + b*x**2)**p)/(2*x**(4*p)*a**2*c*d*p*x + x**(4*p)*a**2*c*d*x + 
 2*x**(4*p)*a**2*d**2*p*x**3 + x**(4*p)*a**2*d**2*x**3 + 2*x**(4*p)*a*b*c* 
*2*p*x + x**(4*p)*a*b*c**2*x + 4*x**(4*p)*a*b*c*d*p*x**3 + 2*x**(4*p)*a*b* 
c*d*x**3 + 2*x**(4*p)*a*b*d**2*p*x**5 + x**(4*p)*a*b*d**2*x**5 + 2*x**(4*p 
)*b**2*c**2*p*x**3 + x**(4*p)*b**2*c**2*x**3 + 2*x**(4*p)*b**2*c*d*p*x**5 
+ x**(4*p)*b**2*c*d*x**5),x)*a**3*d**3*p*x**2 - 4*x**(4*p)*int(((c + d*x** 
2)**p*(a + b*x**2)**p)/(2*x**(4*p)*a**2*c*d*p*x + x**(4*p)*a**2*c*d*x + 2* 
x**(4*p)*a**2*d**2*p*x**3 + x**(4*p)*a**2*d**2*x**3 + 2*x**(4*p)*a*b*c**2* 
p*x + x**(4*p)*a*b*c**2*x + 4*x**(4*p)*a*b*c*d*p*x**3 + 2*x**(4*p)*a*b*c*d 
*x**3 + 2*x**(4*p)*a*b*d**2*p*x**5 + x**(4*p)*a*b*d**2*x**5 + 2*x**(4*p)*b 
**2*c**2*p*x**3 + x**(4*p)*b**2*c**2*x**3 + 2*x**(4*p)*b**2*c*d*p*x**5 + x 
**(4*p)*b**2*c*d*x**5),x)*a**2*b*c*d**2*p**2*x**2 - 2*x**(4*p)*int(((c ...