Integrand size = 24, antiderivative size = 105 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\frac {\left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {d}{c x^2}\right )^{-q} (e x)^{1+m} \operatorname {AppellF1}\left (\frac {1}{2} (-1-m),-p,-q,\frac {1-m}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{e (1+m)} \]
(a+b/x^2)^p*(c+d/x^2)^q*(e*x)^(1+m)*AppellF1(-1/2-1/2*m,-p,-q,-1/2*m+1/2,- b/a/x^2,-d/c/x^2)/e/(1+m)/((1+b/a/x^2)^p)/((1+d/c/x^2)^q)
Time = 0.34 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.10 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\frac {\left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q x (e x)^m \left (1+\frac {a x^2}{b}\right )^{-p} \left (1+\frac {c x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2} (1+m-2 p-2 q),-p,-q,\frac {1}{2} (3+m-2 p-2 q),-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{1+m-2 p-2 q} \]
((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)
Time = 0.27 (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 |
\(\displaystyle \frac {x (e x)^m \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2} (-m-1),-p,-q,\frac {1-m}{2},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{m+1}\) |
((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)
3.10.86.3.1 Defintions of rubi rules used
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])
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])
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]
\[\int \left (a +\frac {b}{x^{2}}\right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q} \left (e x \right )^{m}d x\]
\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\int { \left (e x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\text {Timed out} \]
\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\int { \left (e x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\int { \left (e x\right )^{m} {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q (e x)^m \, dx=\int {\left (e\,x\right )}^m\,{\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q \,d x \]