\(\int (e x)^{-2-2 p} (a+b x^2)^p (c+d x^2) \, dx\) [1630]

Optimal result

Integrand size = 26, antiderivative size = 128 \[ \int (e x)^{-2-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=-\frac {c (e x)^{-1-2 p} \left (a+b x^2\right )^{1+p}}{a e (1+2 p)}+\frac {(b c+a d (1+2 p)) (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{a e^3 \left (1-4 p^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.91 \[ \int (e x)^{-2-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=-\frac {x (e x)^{-2 (1+p)} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c (-1+2 p) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,-p,\frac {1}{2}-p,-\frac {b x^2}{a}\right )+d (1+2 p) x^2 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )\right )}{-1+4 p^2} \] Input:

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

Output:

-((x*(a + b*x^2)^p*(c*(-1 + 2*p)*Hypergeometric2F1[-1/2 - p, -p, 1/2 - p, 
-((b*x^2)/a)] + d*(1 + 2*p)*x^2*Hypergeometric2F1[1/2 - p, -p, 3/2 - p, -( 
(b*x^2)/a)]))/((-1 + 4*p^2)*(e*x)^(2*(1 + p))*(1 + (b*x^2)/a)^p))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {363, 279, 278}

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.

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 46.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.93 \[ \int (e x)^{-2-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {a^{p} c e^{- 2 p - 2} x^{- 2 p - 1} \Gamma \left (- p - \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p - \frac {1}{2} \\ \frac {1}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {1}{2} - p\right )} + \frac {a^{p} d e^{- 2 p - 2} x^{1 - 2 p} \Gamma \left (\frac {1}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {1}{2} - p \\ \frac {3}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {3}{2} - p\right )} \] Input:

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

Output:

a**p*c*e**(-2*p - 2)*x**(-2*p - 1)*gamma(-p - 1/2)*hyper((-p, -p - 1/2), ( 
1/2 - p,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(1/2 - p)) + a**p*d*e**(-2*p 
- 2)*x**(1 - 2*p)*gamma(1/2 - p)*hyper((-p, 1/2 - p), (3/2 - p,), b*x**2*e 
xp_polar(I*pi)/a)/(2*gamma(3/2 - p))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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