\(\int (a+\frac {b}{x^2})^p (c+\frac {d}{x^2})^q \sqrt {e x} \, dx\) [206]

Optimal result

Integrand size = 26, antiderivative size = 91 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\frac {2 \left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {d}{c x^2}\right )^{-q} (e x)^{3/2} \operatorname {AppellF1}\left (-\frac {3}{4},-p,-q,\frac {1}{4},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{3 e} \] Output:

2/3*(a+b/x^2)^p*(c+d/x^2)^q*(e*x)^(3/2)*AppellF1(-3/4,-p,-q,1/4,-b/a/x^2,- 
d/c/x^2)/e/((1+b/a/x^2)^p)/((1+d/c/x^2)^q)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.22 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=-\frac {2 \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q x \sqrt {e x} \left (1+\frac {a x^2}{b}\right )^{-p} \left (1+\frac {c x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{4}-p-q,-p,-q,\frac {7}{4}-p-q,-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{-3+4 p+4 q} \] Input:

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

Output:

(-2*(a + b/x^2)^p*(c + d/x^2)^q*x*Sqrt[e*x]*AppellF1[3/4 - p - q, -p, -q, 
7/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-3 + 4*p + 4*q)*(1 + (a*x^2)/b 
)^p*(1 + (c*x^2)/d)^q)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {998, 1013, 1013, 1012}

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.

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

\(\Big \downarrow \) 998

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle \frac {2 (e x)^{3/2} \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} \operatorname {AppellF1}\left (-\frac {3}{4},-p,-q,\frac {1}{4},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{3 e}\)

Input:

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

Output:

(2*(a + b/x^2)^p*(c + d/x^2)^q*(e*x)^(3/2)*AppellF1[-3/4, -p, -q, 1/4, -(b 
/(a*x^2)), -(d/(c*x^2))])/(3*e*(1 + b/(a*x^2))^p*(1 + d/(c*x^2))^q)
 

Defintions of rubi rules used

rule 998

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> With[{g = Denominator[m]}, Simp[-g/e   Subst[Int[(a + 
b/(e^n*x^(g*n)))^p*((c + d/(e^n*x^(g*n)))^q/x^(g*(m + 1) + 1)), x], x, 1/(e 
*x)^(1/g)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && ILtQ[n, 0] && Fractio 
nQ[m]
 

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1013

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^ 
n/a))^FracPart[p])   Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(e)*(sqrt(x)*(c*x**2 + d)**q*(a*x**2 + b)**p*x + 2*x**(2*p + 2*q)*i 
nt((sqrt(x)*(c*x**2 + d)**q*(a*x**2 + b)**p*x**2)/(x**(2*p + 2*q)*a*c*x**4 
 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d) 
,x)*a*d*q + 2*x**(2*p + 2*q)*int((sqrt(x)*(c*x**2 + d)**q*(a*x**2 + b)**p* 
x**2)/(x**(2*p + 2*q)*a*c*x**4 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)* 
b*c*x**2 + x**(2*p + 2*q)*b*d),x)*b*c*p + 2*x**(2*p + 2*q)*int((sqrt(x)*(c 
*x**2 + d)**q*(a*x**2 + b)**p)/(x**(2*p + 2*q)*a*c*x**4 + x**(2*p + 2*q)*a 
*d*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d),x)*b*d*p + 2*x**(2 
*p + 2*q)*int((sqrt(x)*(c*x**2 + d)**q*(a*x**2 + b)**p)/(x**(2*p + 2*q)*a* 
c*x**4 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q 
)*b*d),x)*b*d*q))/(3*x**(2*p + 2*q))