\(\int (a+\frac {b}{x^2})^p (c+\frac {d}{x^2})^q (e x)^m \, dx\) [210]

Optimal result

Integrand size = 24, antiderivative size = 105 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\frac {\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {d}{c x^2}\right )^{-q} (e x)^{1+m} \operatorname {AppellF1}\left (\frac {1}{2} (-1-m),-p,-q,\frac {1-m}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q x (e x)^m \left (1+\frac {a x^2}{b}\right )^{-p} \left (1+\frac {c x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2} (1+m-2 p-2 q),-p,-q,\frac {1}{2} (3+m-2 p-2 q),-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{1+m-2 p-2 q} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {999, 395, 395, 394}

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.

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

\(\Big \downarrow \) 999

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

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

rule 999

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) 
)^(q_), x_Symbol] :> Simp[(-(e*x)^m)*(x^(-1))^m   Subst[Int[(a + b/x^n)^p*( 
(c + d/x^n)^q/x^(m + 2)), x], x, 1/x], x] /; FreeQ[{a, b, c, d, e, m, p, q} 
, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0] &&  !RationalQ[m]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx =\text {Too large to display} \] Input:

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

Output:

(e**m*(x**m*(c*x**2 + d)**q*(a*x**2 + b)**p*x + 2*x**(2*p + 2*q)*int((x**m 
*(c*x**2 + d)**q*(a*x**2 + b)**p*x**2)/(x**(2*p + 2*q)*a*c*m*x**4 + x**(2* 
p + 2*q)*a*c*x**4 + x**(2*p + 2*q)*a*d*m*x**2 + x**(2*p + 2*q)*a*d*x**2 + 
x**(2*p + 2*q)*b*c*m*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d*m 
 + x**(2*p + 2*q)*b*d),x)*a*d*m*q + 2*x**(2*p + 2*q)*int((x**m*(c*x**2 + d 
)**q*(a*x**2 + b)**p*x**2)/(x**(2*p + 2*q)*a*c*m*x**4 + x**(2*p + 2*q)*a*c 
*x**4 + x**(2*p + 2*q)*a*d*m*x**2 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2* 
q)*b*c*m*x**2 + x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d*m + x**(2*p + 
 2*q)*b*d),x)*a*d*q + 2*x**(2*p + 2*q)*int((x**m*(c*x**2 + d)**q*(a*x**2 + 
 b)**p*x**2)/(x**(2*p + 2*q)*a*c*m*x**4 + x**(2*p + 2*q)*a*c*x**4 + x**(2* 
p + 2*q)*a*d*m*x**2 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*m*x**2 
+ x**(2*p + 2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d*m + x**(2*p + 2*q)*b*d),x)* 
b*c*m*p + 2*x**(2*p + 2*q)*int((x**m*(c*x**2 + d)**q*(a*x**2 + b)**p*x**2) 
/(x**(2*p + 2*q)*a*c*m*x**4 + x**(2*p + 2*q)*a*c*x**4 + x**(2*p + 2*q)*a*d 
*m*x**2 + x**(2*p + 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*m*x**2 + x**(2*p + 
2*q)*b*c*x**2 + x**(2*p + 2*q)*b*d*m + x**(2*p + 2*q)*b*d),x)*b*c*p + 2*x* 
*(2*p + 2*q)*int((x**m*(c*x**2 + d)**q*(a*x**2 + b)**p)/(x**(2*p + 2*q)*a* 
c*m*x**4 + x**(2*p + 2*q)*a*c*x**4 + x**(2*p + 2*q)*a*d*m*x**2 + x**(2*p + 
 2*q)*a*d*x**2 + x**(2*p + 2*q)*b*c*m*x**2 + x**(2*p + 2*q)*b*c*x**2 + x** 
(2*p + 2*q)*b*d*m + x**(2*p + 2*q)*b*d),x)*b*d*m*p + 2*x**(2*p + 2*q)*i...