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

Optimal. Leaf size=82 \[ -\frac{\left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{x} \]

[Out]

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

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Rubi [A]  time = 0.181357, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{\left (a+\frac{b}{x^2}\right )^p \left (\frac{b}{a x^2}+1\right )^{-p} \left (c+\frac{d}{x^2}\right )^q \left (\frac{d}{c x^2}+1\right )^{-q} F_1\left (\frac{1}{2};-p,-q;\frac{3}{2};-\frac{b}{a x^2},-\frac{d}{c x^2}\right )}{x} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 32.9574, size = 63, normalized size = 0.77 \[ - \frac{\left (1 + \frac{b}{a x^{2}}\right )^{- p} \left (1 + \frac{d}{c x^{2}}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (\frac{1}{2},- p,- q,\frac{3}{2},- \frac{b}{a x^{2}},- \frac{d}{c x^{2}} \right )}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(1 + b/(a*x**2))**(-p)*(1 + d/(c*x**2))**(-q)*(a + b/x**2)**p*(c + d/x**2)**q*a
ppellf1(1/2, -p, -q, 3/2, -b/(a*x**2), -d/(c*x**2))/x

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Mathematica [B]  time = 0.704765, size = 254, normalized size = 3.1 \[ \frac{b d (2 p+2 q-1) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q-\frac{1}{2};-p,-q;-p-q+\frac{1}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{x (2 p+2 q+1) \left (2 x^2 \left (a d p F_1\left (-p-q+\frac{1}{2};1-p,-q;-p-q+\frac{3}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+\frac{1}{2};-p,1-q;-p-q+\frac{3}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )+b d (-2 p-2 q+1) F_1\left (-p-q-\frac{1}{2};-p,-q;-p-q+\frac{1}{2};-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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