Optimal. Leaf size=97 \[ \frac{\left (a+\frac{b}{x^2}\right )^{p+1} \left (c+\frac{d}{x^2}\right )^q \left (\frac{b \left (c+\frac{d}{x^2}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,1;p+2;-\frac{d \left (a+\frac{b}{x^2}\right )}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right )}{2 a (p+1)} \]
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Rubi [A] time = 0.21558, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\left (a+\frac{b}{x^2}\right )^{p+1} \left (c+\frac{d}{x^2}\right )^q \left (\frac{b \left (c+\frac{d}{x^2}\right )}{b c-a d}\right )^{-q} F_1\left (p+1;-q,1;p+2;-\frac{d \left (a+\frac{b}{x^2}\right )}{b c-a d},\frac{a+\frac{b}{x^2}}{a}\right )}{2 a (p+1)} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)^p*(c + d/x^2)^q)/x,x]
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Rubi in Sympy [A] time = 28.1379, size = 71, normalized size = 0.73 \[ \frac{\left (\frac{b \left (- c - \frac{d}{x^{2}}\right )}{a d - b c}\right )^{- q} \left (a + \frac{b}{x^{2}}\right )^{p + 1} \left (c + \frac{d}{x^{2}}\right )^{q} \operatorname{appellf_{1}}{\left (p + 1,1,- q,p + 2,\frac{a + \frac{b}{x^{2}}}{a},\frac{d \left (a + \frac{b}{x^{2}}\right )}{a d - b c} \right )}}{2 a \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**p*(c+d/x**2)**q/x,x)
[Out]
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Mathematica [B] time = 0.592541, size = 223, normalized size = 2.3 \[ -\frac{b d (p+q-1) \left (a+\frac{b}{x^2}\right )^p \left (c+\frac{d}{x^2}\right )^q F_1\left (-p-q;-p,-q;-p-q+1;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )}{2 (p+q) \left (b d (p+q-1) F_1\left (-p-q;-p,-q;-p-q+1;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )-x^2 \left (a d p F_1\left (-p-q+1;1-p,-q;-p-q+2;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )+b c q F_1\left (-p-q+1;-p,1-q;-p-q+2;-\frac{a x^2}{b},-\frac{c x^2}{d}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((a + b/x^2)^p*(c + d/x^2)^q)/x,x]
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Maple [F] time = 0.077, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \left ( c+{\frac{d}{{x}^{2}}} \right ) ^{q}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^p*(c+d/x^2)^q/x,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^p*(c + d/x^2)^q/x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\left (\frac{a x^{2} + b}{x^{2}}\right )^{p} \left (\frac{c x^{2} + d}{x^{2}}\right )^{q}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^p*(c + d/x^2)^q/x,x, algorithm="fricas")
[Out]
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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,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (a + \frac{b}{x^{2}}\right )}^{p}{\left (c + \frac{d}{x^{2}}\right )}^{q}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)^p*(c + d/x^2)^q/x,x, algorithm="giac")
[Out]