\(\int \frac {(e x)^{1-2 p} (a+b x^2)^p}{(c+d x^2)^2} \, dx\) [1612]

Optimal result

Integrand size = 28, antiderivative size = 74 \[ \int \frac {(e x)^{1-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\frac {a (e x)^{2-2 p} \left (a+b x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (2,1-p,2-p,\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{2 c^2 e (1-p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.10 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.42 \[ \int \frac {(e x)^{1-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=-\frac {e x^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (1+\frac {d x^2}{c}\right )^p \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )}{2 c (-1+p) \left (c+d x^2\right )} \] 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*(1 + (d*x^2)/c)^p*Hypergeometric2F1[1 - p, -p, 2 
 - p, ((-(b*c) + a*d)*x^2)/(a*(c + d*x^2))])/(c*(-1 + p)*(e*x)^(2*p)*(1 + 
(b*x^2)/a)^p*(c + d*x^2))
 

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {393, 141}

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.

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

\(\Big \downarrow \) 393

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 141

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^( 
n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2, (-(d*e - c*f 
))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, 
p}, x] && EqQ[m + n + p + 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !Su 
mSimplerQ[p, 1]) &&  !ILtQ[m, 0]
 

rule 393

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

Maple [F]

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

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]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \frac {(e x)^{1-2 p} \left (a+b x^2\right )^p}{\left (c+d x^2\right )^2} \, dx=\text {too large to display} \] 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*b*x**2 + 2*x**(2*p)*int(((a + b*x**2)**p*x**3)/(x** 
(2*p)*a**2*c**2*d*p + x**(2*p)*a**2*c**2*d + 2*x**(2*p)*a**2*c*d**2*p*x**2 
 + 2*x**(2*p)*a**2*c*d**2*x**2 + x**(2*p)*a**2*d**3*p*x**4 + x**(2*p)*a**2 
*d**3*x**4 - 2*x**(2*p)*a*b*c**3 + x**(2*p)*a*b*c**2*d*p*x**2 - 3*x**(2*p) 
*a*b*c**2*d*x**2 + 2*x**(2*p)*a*b*c*d**2*p*x**4 + x**(2*p)*a*b*d**3*p*x**6 
 + x**(2*p)*a*b*d**3*x**6 - 2*x**(2*p)*b**2*c**3*x**2 - 4*x**(2*p)*b**2*c* 
*2*d*x**4 - 2*x**(2*p)*b**2*c*d**2*x**6),x)*a**2*b*c*d**2*p + 2*x**(2*p)*i 
nt(((a + b*x**2)**p*x**3)/(x**(2*p)*a**2*c**2*d*p + x**(2*p)*a**2*c**2*d + 
 2*x**(2*p)*a**2*c*d**2*p*x**2 + 2*x**(2*p)*a**2*c*d**2*x**2 + x**(2*p)*a* 
*2*d**3*p*x**4 + x**(2*p)*a**2*d**3*x**4 - 2*x**(2*p)*a*b*c**3 + x**(2*p)* 
a*b*c**2*d*p*x**2 - 3*x**(2*p)*a*b*c**2*d*x**2 + 2*x**(2*p)*a*b*c*d**2*p*x 
**4 + x**(2*p)*a*b*d**3*p*x**6 + x**(2*p)*a*b*d**3*x**6 - 2*x**(2*p)*b**2* 
c**3*x**2 - 4*x**(2*p)*b**2*c**2*d*x**4 - 2*x**(2*p)*b**2*c*d**2*x**6),x)* 
a**2*b*c*d**2 + 2*x**(2*p)*int(((a + b*x**2)**p*x**3)/(x**(2*p)*a**2*c**2* 
d*p + x**(2*p)*a**2*c**2*d + 2*x**(2*p)*a**2*c*d**2*p*x**2 + 2*x**(2*p)*a* 
*2*c*d**2*x**2 + x**(2*p)*a**2*d**3*p*x**4 + x**(2*p)*a**2*d**3*x**4 - 2*x 
**(2*p)*a*b*c**3 + x**(2*p)*a*b*c**2*d*p*x**2 - 3*x**(2*p)*a*b*c**2*d*x**2 
 + 2*x**(2*p)*a*b*c*d**2*p*x**4 + x**(2*p)*a*b*d**3*p*x**6 + x**(2*p)*a*b* 
d**3*x**6 - 2*x**(2*p)*b**2*c**3*x**2 - 4*x**(2*p)*b**2*c**2*d*x**4 - 2*x* 
*(2*p)*b**2*c*d**2*x**6),x)*a**2*b*d**3*p*x**2 + 2*x**(2*p)*int(((a + b...