Optimal. Leaf size=46 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2} \]
[Out]
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Rubi [A] time = 0.120396, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (c+\frac{d}{x^2}\right )^{3/2} (b c-a d)}{3 d^2}-\frac{b \left (c+\frac{d}{x^2}\right )^{5/2}}{5 d^2} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*Sqrt[c + d/x^2])/x^3,x]
[Out]
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Rubi in Sympy [A] time = 12.0012, size = 39, normalized size = 0.85 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5 d^{2}} - \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0597789, size = 47, normalized size = 1.02 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^2+d\right ) \left (5 a d x^2-2 b c x^2+3 b d\right )}{15 d^2 x^4} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^3,x]
[Out]
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Maple [A] time = 0.01, size = 48, normalized size = 1. \[ -{\frac{ \left ( 5\,ad{x}^{2}-2\,bc{x}^{2}+3\,bd \right ) \left ( c{x}^{2}+d \right ) }{15\,{d}^{2}{x}^{4}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*(c+d/x^2)^(1/2)/x^3,x)
[Out]
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Maxima [A] time = 1.37597, size = 66, normalized size = 1.43 \[ -\frac{1}{15} \, b{\left (\frac{3 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{5}{2}}}{d^{2}} - \frac{5 \,{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} c}{d^{2}}\right )} - \frac{a{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.228014, size = 81, normalized size = 1.76 \[ \frac{{\left ({\left (2 \, b c^{2} - 5 \, a c d\right )} x^{4} - 3 \, b d^{2} -{\left (b c d + 5 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{15 \, d^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.76533, size = 58, normalized size = 1.26 \[ - \frac{a \left (\begin{cases} \frac{\sqrt{c}}{x^{2}} & \text{for}\: d = 0 \\\frac{2 \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3 d} & \text{otherwise} \end{cases}\right )}{2} - \frac{b \left (- \frac{c \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{3} + \frac{\left (c + \frac{d}{x^{2}}\right )^{\frac{5}{2}}}{5}\right )}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.453602, size = 338, normalized size = 7.35 \[ \frac{2 \,{\left (15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{8} a c^{\frac{3}{2}}{\rm sign}\left (x\right ) + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} b c^{\frac{5}{2}}{\rm sign}\left (x\right ) - 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{6} a c^{\frac{3}{2}} d{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} b c^{\frac{5}{2}} d{\rm sign}\left (x\right ) + 20 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{4} a c^{\frac{3}{2}} d^{2}{\rm sign}\left (x\right ) + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} b c^{\frac{5}{2}} d^{2}{\rm sign}\left (x\right ) - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} a c^{\frac{3}{2}} d^{3}{\rm sign}\left (x\right ) - 2 \, b c^{\frac{5}{2}} d^{3}{\rm sign}\left (x\right ) + 5 \, a c^{\frac{3}{2}} d^{4}{\rm sign}\left (x\right )\right )}}{15 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + d}\right )}^{2} - d\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^3,x, algorithm="giac")
[Out]