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

Optimal. Leaf size=89 \[ \frac{2 \sqrt{e x} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.324556, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{2 \sqrt{e x} \left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (-\frac{1}{4};-p,-q;\frac{3}{4};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 49.9559, size = 71, normalized size = 0.8 \[ \frac{2 \sqrt{e x} \left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (1 + \frac{d}{c x^{2}}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (- \frac{1}{4},- p,- q,\frac{3}{4},- \frac{b}{a x^{2}},- \frac{d}{c x^{2}} \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b/x**2)**p*(c+d/x**2)**q/(e*x)**(1/2),x)

[Out]

2*sqrt(e*x)*(1 + b/(a*x**2))**(-p)*(1 + d/(c*x**2))**(-q)*(a + b/x**2)**p*(c + d
/x**2)**q*appellf1(-1/4, -p, -q, 3/4, -b/(a*x**2), -d/(c*x**2))/e

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Mathematica [B]  time = 0.810978, size = 260, normalized size = 2.92 \[ \frac{2 b d x (4 p+4 q-5) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q+\frac{1}{4};-p,-q;-p-q+\frac{5}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{\sqrt{e x} (4 p+4 q-1) \left (4 x^2 \left (a d p F_1\left (-p-q+\frac{5}{4};1-p,-q;-p-q+\frac{9}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+\frac{5}{4};-p,1-q;-p-q+\frac{9}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )+b d (-4 p-4 q+5) F_1\left (-p-q+\frac{1}{4};-p,-q;-p-q+\frac{5}{4};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*b*d*(-5 + 4*p + 4*q)*(a + b/x^2)^p*(c + d/x^2)^q*x*AppellF1[1/4 - p - q, -p,
-q, 5/4 - p - q, -((a*x^2)/b), -((c*x^2)/d)])/((-1 + 4*p + 4*q)*Sqrt[e*x]*(b*d*(
5 - 4*p - 4*q)*AppellF1[1/4 - p - q, -p, -q, 5/4 - p - q, -((a*x^2)/b), -((c*x^2
)/d)] + 4*x^2*(a*d*p*AppellF1[5/4 - p - q, 1 - p, -q, 9/4 - p - q, -((a*x^2)/b),
-((c*x^2)/d)] + b*c*q*AppellF1[5/4 - p - q, -p, 1 - q, 9/4 - p - q, -((a*x^2)/b
), -((c*x^2)/d)])))

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{1 \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q}{\frac{1}{\sqrt{ex}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{\sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (\frac{a x^{2} + b}{x^{2}}\right )^{p} \left (\frac{c x^{2} + d}{x^{2}}\right )^{q}}{\sqrt{e x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{\sqrt{e x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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