\(\int (e x)^{1-2 p} (a+b x^2)^p (c+d x^2)^2 \, dx\) [1608]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\frac {c^2 (e x)^{2-2 p} \left (a+b x^2\right )^{1+p}}{2 a e (1-p)}+\frac {d^2 (e x)^{4-2 p} \left (a+b x^2\right )^{1+p}}{6 b e^3}-\frac {\left (6 b^2 c^2-6 a b c d (1-p)+a^2 d^2 \left (2-3 p+p^2\right )\right ) (e x)^{4-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )}{6 a b e^3 (1-p) (2-p)} \] Output:

1/2*c^2*(e*x)^(2-2*p)*(b*x^2+a)^(p+1)/a/e/(-p+1)+1/6*d^2*(e*x)^(4-2*p)*(b* 
x^2+a)^(p+1)/b/e^3-1/6*(6*b^2*c^2-6*a*b*c*d*(-p+1)+a^2*d^2*(p^2-3*p+2))*(e 
*x)^(4-2*p)*(b*x^2+a)^p*hypergeom([-p, 2-p],[3-p],-b*x^2/a)/a/b/e^3/(-p+1) 
/(2-p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 5.31 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.83 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=-\frac {e x^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c^2 \left (6-5 p+p^2\right ) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )+d (-1+p) x^2 \left (2 c (-3+p) \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )+d (-2+p) x^2 \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^2}{a}\right )\right )\right )}{2 (-3+p) (-2+p) (-1+p)} \] Input:

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

Output:

-1/2*(e*x^2*(a + b*x^2)^p*(c^2*(6 - 5*p + p^2)*Hypergeometric2F1[1 - p, -p 
, 2 - p, -((b*x^2)/a)] + d*(-1 + p)*x^2*(2*c*(-3 + p)*Hypergeometric2F1[2 
- p, -p, 3 - p, -((b*x^2)/a)] + d*(-2 + p)*x^2*Hypergeometric2F1[3 - p, -p 
, 4 - p, -((b*x^2)/a)])))/((-3 + p)*(-2 + p)*(-1 + p)*(e*x)^(2*p)*(1 + (b* 
x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 367

\(\displaystyle \frac {\int 2 (e x)^{1-2 p} \left (b x^2+a\right )^p \left (3 b c^2+d (6 b c-a d (2-p)) x^2\right )dx}{6 b}+\frac {d^2 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{1-2 p} \left (b x^2+a\right )^p \left (3 b c^2+d (6 b c-a d (2-p)) x^2\right )dx}{3 b}+\frac {d^2 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {\frac {\left (6 b^2 c^2-a d (1-p) (6 b c-a d (2-p))\right ) \int (e x)^{1-2 p} \left (b x^2+a\right )^pdx}{2 b}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (2-p))}{4 b e}}{3 b}+\frac {d^2 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (6 b^2 c^2-a d (1-p) (6 b c-a d (2-p))\right ) \int (e x)^{1-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{2 b}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (2-p))}{4 b e}}{3 b}+\frac {d^2 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {(e x)^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (6 b^2 c^2-a d (1-p) (6 b c-a d (2-p))\right ) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{4 b e (1-p)}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (2-p))}{4 b e}}{3 b}+\frac {d^2 (e x)^{4-2 p} \left (a+b x^2\right )^{p+1}}{6 b e^3}\)

Input:

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

Output:

(d^2*(e*x)^(4 - 2*p)*(a + b*x^2)^(1 + p))/(6*b*e^3) + ((d*(6*b*c - a*d*(2 
- p))*(e*x)^(2 - 2*p)*(a + b*x^2)^(1 + p))/(4*b*e) + ((6*b^2*c^2 - a*d*(6* 
b*c - a*d*(2 - p))*(1 - p))*(e*x)^(2 - 2*p)*(a + b*x^2)^p*Hypergeometric2F 
1[1 - p, -p, 2 - p, -((b*x^2)/a)])/(4*b*e*(1 - p)*(1 + (b*x^2)/a)^p))/(3*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 = 110.62 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.79 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\frac {a^{p} c^{2} e^{1 - 2 p} x^{2 - 2 p} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (2 - p\right )} + \frac {a^{p} c d e^{1 - 2 p} x^{4 - 2 p} \Gamma \left (2 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 2 - p \\ 3 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (3 - p\right )} + \frac {a^{p} d^{2} e^{1 - 2 p} x^{6 - 2 p} \Gamma \left (3 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 3 - p \\ 4 - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (4 - p\right )} \] Input:

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

Output:

a**p*c**2*e**(1 - 2*p)*x**(2 - 2*p)*gamma(1 - p)*hyper((-p, 1 - p), (2 - p 
,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(2 - p)) + a**p*c*d*e**(1 - 2*p)*x** 
(4 - 2*p)*gamma(2 - p)*hyper((-p, 2 - p), (3 - p,), b*x**2*exp_polar(I*pi) 
/a)/gamma(3 - p) + a**p*d**2*e**(1 - 2*p)*x**(6 - 2*p)*gamma(3 - p)*hyper( 
(-p, 3 - p), (4 - p,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(4 - 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. \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\int {\left (e\,x\right )}^{1-2\,p}\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \, dx=\frac {e \left (\left (b \,x^{2}+a \right )^{p} a^{2} d^{2} p^{2} x^{2}-2 \left (b \,x^{2}+a \right )^{p} a^{2} d^{2} p \,x^{2}+6 \left (b \,x^{2}+a \right )^{p} a b c d p \,x^{2}+\left (b \,x^{2}+a \right )^{p} a b \,d^{2} p \,x^{4}+6 \left (b \,x^{2}+a \right )^{p} b^{2} c^{2} x^{2}+6 \left (b \,x^{2}+a \right )^{p} b^{2} c d \,x^{4}+2 \left (b \,x^{2}+a \right )^{p} b^{2} d^{2} x^{6}+2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{3} d^{2} p^{3}-6 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{3} d^{2} p^{2}+4 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{3} d^{2} p +12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} b c d \,p^{2}-12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a^{2} b c d p +12 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p} x}{x^{2 p} a +x^{2 p} b \,x^{2}}d x \right ) a \,b^{2} c^{2} p \right )}{12 x^{2 p} e^{2 p} b^{2}} \] Input:

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

Output:

(e*((a + b*x**2)**p*a**2*d**2*p**2*x**2 - 2*(a + b*x**2)**p*a**2*d**2*p*x* 
*2 + 6*(a + b*x**2)**p*a*b*c*d*p*x**2 + (a + b*x**2)**p*a*b*d**2*p*x**4 + 
6*(a + b*x**2)**p*b**2*c**2*x**2 + 6*(a + b*x**2)**p*b**2*c*d*x**4 + 2*(a 
+ b*x**2)**p*b**2*d**2*x**6 + 2*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p) 
*a + x**(2*p)*b*x**2),x)*a**3*d**2*p**3 - 6*x**(2*p)*int(((a + b*x**2)**p* 
x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*d**2*p**2 + 4*x**(2*p)*int(((a + 
 b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*d**2*p + 12*x**(2*p) 
*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*b*c*d*p**2 
 - 12*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a 
**2*b*c*d*p + 12*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b 
*x**2),x)*a*b**2*c**2*p))/(12*x**(2*p)*e**(2*p)*b**2)