Optimal. Leaf size=59 \[ \frac{b c-a d}{c d x \sqrt{c+\frac{d}{x^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{d^{3/2}} \]
[Out]
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Rubi [A] time = 0.122652, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{b c-a d}{c d x \sqrt{c+\frac{d}{x^2}}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{d^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x^2),x]
[Out]
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Rubi in Sympy [A] time = 12.0733, size = 48, normalized size = 0.81 \[ - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{d^{\frac{3}{2}}} - \frac{a d - b c}{c d x \sqrt{c + \frac{d}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.107482, size = 89, normalized size = 1.51 \[ \frac{\sqrt{d} (b c-a d)+b c \log (x) \sqrt{c x^2+d}-b c \sqrt{c x^2+d} \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )}{c d^{3/2} x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x^2),x]
[Out]
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Maple [A] time = 0.016, size = 79, normalized size = 1.3 \[ -{\frac{c{x}^{2}+d}{c{x}^{3}} \left ( a{d}^{{\frac{5}{2}}}-bc{d}^{{\frac{3}{2}}}+b\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ) c\sqrt{c{x}^{2}+d}d \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{d}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/(c+d/x^2)^(3/2)/x^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255286, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (b c d - a d^{2}\right )} x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (b c^{2} x^{2} + b c d\right )} \sqrt{d} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right )}{2 \,{\left (c^{2} d^{2} x^{2} + c d^{3}\right )}}, \frac{{\left (b c d - a d^{2}\right )} x \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (b c^{2} x^{2} + b c d\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{c^{2} d^{2} x^{2} + c d^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 18.2554, size = 206, normalized size = 3.49 \[ - \frac{a}{c \sqrt{d} \sqrt{\frac{c x^{2}}{d} + 1}} + b \left (\frac{c d^{2} x^{2} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 c d^{2} x^{2} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{2 d^{3} \sqrt{\frac{c x^{2}}{d} + 1}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} + \frac{d^{3} \log{\left (\frac{c x^{2}}{d} \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}} - \frac{2 d^{3} \log{\left (\sqrt{\frac{c x^{2}}{d} + 1} + 1 \right )}}{2 c d^{\frac{7}{2}} x^{2} + 2 d^{\frac{9}{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x**2,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x^2),x, algorithm="giac")
[Out]