\(\int (e x)^{-5-4 p} (a+b x^2)^p (c+d x^2)^p \, dx\) [1655]

Optimal result

Integrand size = 28, antiderivative size = 167 \[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=-\frac {(e x)^{-4 (1+p)} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^{1+p}}{4 a c e (1+p)}+\frac {(b c+a d) (e x)^{-2-4 p} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^p \left (\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{4 a^2 c e^3 (1+2 p)} \] Output:

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

Mathematica [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx \] Input:

Integrate[(e*x)^(-5 - 4*p)*(a + b*x^2)^p*(c + d*x^2)^p,x]
 

Output:

Integrate[(e*x)^(-5 - 4*p)*(a + b*x^2)^p*(c + d*x^2)^p, x]
 

Rubi [A] (warning: unable to verify)

Time = 19.50 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.30, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {395, 395, 1012}

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.

\(\displaystyle \int (e x)^{-4 p-5} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx\)

\(\Big \downarrow \) 395

\(\displaystyle \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \int (e x)^{-4 p-5} \left (\frac {b x^2}{a}+1\right )^p \left (d x^2+c\right )^pdx\)

\(\Big \downarrow \) 395

\(\displaystyle \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^p \left (\frac {d x^2}{c}+1\right )^{-p} \int (e x)^{-4 p-5} \left (\frac {b x^2}{a}+1\right )^p \left (\frac {d x^2}{c}+1\right )^pdx\)

\(\Big \downarrow \) 1012

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

Input:

Int[(e*x)^(-5 - 4*p)*(a + b*x^2)^p*(c + d*x^2)^p,x]
 

Output:

(4^(-2 - p)*(a + b*x^2)^(-1 + p)*(c + d*x^2)^p*(1 + (d*x^2)/c)*Gamma[-1/2 
- p]*(c*(a + b*x^2)*(c*(1 + 2*p) - d*x^2)*Gamma[1 - 2*p]*Gamma[-p]*Hyperge 
ometric2F1[1, -p, -2*p, ((b*c - a*d)*x^2)/(c*(a + b*x^2))] - (b*c - a*d)*x 
^2*(c + d*x^2)*Gamma[1 - p]*Gamma[-2*p]*Hypergeometric2F1[2, 1 - p, 1 - 2* 
p, ((b*c - a*d)*x^2)/(c*(a + b*x^2))]))/(c^2*e*(1 + p)*Sqrt[Pi]*(e*x)^(4*( 
1 + p))*Gamma[1 - 2*p]*Gamma[-2*p])
 

Defintions of rubi rules used

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])
 

rule 1012

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

Maple [F]

\[\int \left (e x \right )^{-5-4 p} \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )^{p}d x\]

Input:

int((e*x)^(-5-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Output:

int((e*x)^(-5-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Fricas [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{p} \left (e x\right )^{-4 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-4*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 - 5), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-5-4*p)*(b*x**2+a)**p*(d*x**2+c)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{p} \left (e x\right )^{-4 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-4*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 - 5), x)
 

Giac [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x^{2} + c\right )}^{p} \left (e x\right )^{-4 \, p - 5} \,d x } \] Input:

integrate((e*x)^(-5-4*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 - 5), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^p}{{\left (e\,x\right )}^{4\,p+5}} \,d x \] Input:

int(((a + b*x^2)^p*(c + d*x^2)^p)/(e*x)^(4*p + 5),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x^2)^p)/(e*x)^(4*p + 5), x)
 

Reduce [F]

\[ \int (e x)^{-5-4 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\text {too large to display} \] Input:

int((e*x)^(-5-4*p)*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Output:

( - 2*(c + d*x**2)**p*(a + b*x**2)**p*a*c*p - (c + d*x**2)**p*(a + b*x**2) 
**p*a*c - (c + d*x**2)**p*(a + b*x**2)**p*a*d*p*x**2 - (c + d*x**2)**p*(a 
+ b*x**2)**p*b*c*p*x**2 + (c + d*x**2)**p*(a + b*x**2)**p*b*d*x**4 - 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**3*x**4 - 6*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**4 - 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**(...