Integrand size = 26, antiderivative size = 91 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\frac {2 \left (a+\frac {b}{x^2}\right )^p \left (1+\frac {b}{a x^2}\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (1+\frac {d}{c x^2}\right )^{-q} (e x)^{3/2} \operatorname {AppellF1}\left (-\frac {3}{4},-p,-q,\frac {1}{4},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{3 e} \]
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)
Time = 0.43 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.22 \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=-\frac {2 \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q x \sqrt {e x} \left (1+\frac {a x^2}{b}\right )^{-p} \left (1+\frac {c x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{4}-p-q,-p,-q,\frac {7}{4}-p-q,-\frac {a x^2}{b},-\frac {c x^2}{d}\right )}{-3+4 p+4 q} \]
(-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)
Time = 0.28 (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 |
\(\displaystyle \frac {2 (e x)^{3/2} \left (a+\frac {b}{x^2}\right )^p \left (\frac {b}{a x^2}+1\right )^{-p} \left (c+\frac {d}{x^2}\right )^q \left (\frac {d}{c x^2}+1\right )^{-q} \operatorname {AppellF1}\left (-\frac {3}{4},-p,-q,\frac {1}{4},-\frac {b}{a x^2},-\frac {d}{c x^2}\right )}{3 e}\) |
(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)
3.10.98.3.1 Defintions of rubi rules used
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]
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])
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])
\[\int \left (a +\frac {b}{x^{2}}\right )^{p} \left (c +\frac {d}{x^{2}}\right )^{q} \sqrt {e x}d x\]
\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\int { \sqrt {e x} {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\text {Timed out} \]
\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\int { \sqrt {e x} {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
\[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\int { \sqrt {e x} {\left (a + \frac {b}{x^{2}}\right )}^{p} {\left (c + \frac {d}{x^{2}}\right )}^{q} \,d x } \]
Timed out. \[ \int \left (a+\frac {b}{x^2}\right )^p \left (c+\frac {d}{x^2}\right )^q \sqrt {e x} \, dx=\int \sqrt {e\,x}\,{\left (a+\frac {b}{x^2}\right )}^p\,{\left (c+\frac {d}{x^2}\right )}^q \,d x \]