\(\int \frac {(e x)^{-1-2 p} (a+b x^2)^p}{c+d x^2} \, dx\) [1625]

Optimal result

Integrand size = 28, antiderivative size = 63 \[ \int \frac {(e x)^{-1-2 p} \left (a+b x^2\right )^p}{c+d x^2} \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,-p,1-p,\frac {(b c-a d) x^2}{c \left (a+b x^2\right )}\right )}{2 c e p} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.09 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.44 \[ \int \frac {(e x)^{-1-2 p} \left (a+b x^2\right )^p}{c+d x^2} \, dx=-\frac {(e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (1+\frac {d x^2}{c}\right )^p \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,\frac {(-b c+a d) x^2}{a \left (c+d x^2\right )}\right )}{2 c e p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, 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.

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

\(\Big \downarrow \) 393

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

\(\Big \downarrow \) 141

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^{-1-2 p} \left (a+b x^2\right )^p}{c+d x^2} \, dx=\frac {-\left (b \,x^{2}+a \right )^{p} b +2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a c x +x^{2 p} a d \,x^{3}+x^{2 p} b c \,x^{3}+x^{2 p} b d \,x^{5}}d x \right ) a^{2} d p -2 x^{2 p} \left (\int \frac {\left (b \,x^{2}+a \right )^{p}}{x^{2 p} a c x +x^{2 p} a d \,x^{3}+x^{2 p} b c \,x^{3}+x^{2 p} b d \,x^{5}}d x \right ) a b c p}{2 x^{2 p} e^{2 p} a d e p} \] Input:

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

Output:

( - (a + b*x**2)**p*b + 2*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*c*x + x 
**(2*p)*a*d*x**3 + x**(2*p)*b*c*x**3 + x**(2*p)*b*d*x**5),x)*a**2*d*p - 2* 
x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a*c*x + x**(2*p)*a*d*x**3 + x**(2*p 
)*b*c*x**3 + x**(2*p)*b*d*x**5),x)*a*b*c*p)/(2*x**(2*p)*e**(2*p)*a*d*e*p)