\(\int (e x)^{-1-2 p} (a+b x^2)^p (c+d x^2)^2 \, dx\) [1622]

Optimal result

Integrand size = 28, antiderivative size = 176 \[ \int (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\frac {d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{1+p}}{4 b e^3}-\frac {c^2 (e x)^{-2 p} \left (a+b x^2\right )^{1+p}}{2 a e p}+\frac {\left (2 b^2 c^2+4 a b c d p-a^2 d^2 (1-p) p\right ) (e x)^{2-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{4 a b e^3 (1-p) p} \] Output:

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

Mathematica [A] (verified)

Time = 6.15 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.85 \[ \int (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (2 c d (-2+p) p x^2 \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+(-1+p) \left (d^2 p x^4 \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )+c^2 (-2+p) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )\right )\right )}{2 e (-2+p) (-1+p) p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {367, 27, 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.

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

\(\Big \downarrow \) 367

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 363

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {d (e x)^{-2 p} \left (a+b x^2\right )^{p+1} (4 b c-a d (1-p))}{2 b e}-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^2 d^2 (1-p) p+4 a b c d p+2 b^2 c^2\right ) \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^2}{a}\right )}{2 b e p}}{2 b}+\frac {d^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{4 b e^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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]
 

rule 367

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, 
x_Symbol] :> Simp[d^2*(e*x)^(m + 3)*((a + b*x^2)^(p + 1)/(b*e^3*(m + 2*p + 
5))), x] + Simp[1/(b*(m + 2*p + 5))   Int[(e*x)^m*(a + b*x^2)^p*Simp[b*c^2* 
(m + 2*p + 5) - d*(a*d*(m + 3) - 2*b*c*(m + 2*p + 5))*x^2, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + 2*p + 5, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 95.85 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.87 \[ \int (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\frac {a^{p} c^{2} e^{- 2 p - 1} x^{- 2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (1 - p\right )} + \frac {a^{p} c d e^{- 2 p - 1} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (2 - p\right )} + \frac {a^{p} d^{2} e^{- 2 p - 1} x^{4 - 2 p} \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 - p \\ 3 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (3 - p\right )} \] Input:

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

Output:

a**p*c**2*e**(-2*p - 1)*gamma(-p)*hyper((-p, -p), (1 - p,), b*x**2*exp_pol 
ar(I*pi)/a)/(2*x**(2*p)*gamma(1 - p)) + a**p*c*d*e**(-2*p - 1)*x**(2 - 2*p 
)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p,), b*x**2*exp_polar(I*pi)/a)/gamm 
a(2 - p) + a**p*d**2*e**(-2*p - 1)*x**(4 - 2*p)*gamma(2 - p)*hyper((-p, 2 
- p), (3 - p,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(3 - p))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(((a + b*x^2)^p*(c + d*x^2)^2)/(e*x)^(2*p + 1),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x^2)^2)/(e*x)^(2*p + 1), x)
 

Reduce [F]

\[ \int (e x)^{-1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\frac {\left (b \,x^{2}+a \right )^{p} a \,d^{2} p^{2} x^{2}-2 \left (b \,x^{2}+a \right )^{p} b \,c^{2}+4 \left (b \,x^{2}+a \right )^{p} b c d p \,x^{2}+\left (b \,x^{2}+a \right )^{p} b \,d^{2} p \,x^{4}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{2} p^{3}-2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} d^{2} p^{2}+8 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a b c d \,p^{2}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) b^{2} c^{2} p}{4 x^{2 p} e^{2 p} b e p} \] Input:

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

Output:

((a + b*x**2)**p*a*d**2*p**2*x**2 - 2*(a + b*x**2)**p*b*c**2 + 4*(a + b*x* 
*2)**p*b*c*d*p*x**2 + (a + b*x**2)**p*b*d**2*p*x**4 + 2*x**(2*p)*int(((a + 
 b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*d**2*p**3 - 2*x**(2* 
p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*d**2*p** 
2 + 8*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a 
*b*c*d*p**2 + 4*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b* 
x**2),x)*b**2*c**2*p)/(4*x**(2*p)*e**(2*p)*b*e*p)