Optimal. Leaf size=52 \[ \frac{b c-a d}{c d \sqrt{c+\frac{d}{x^2}}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{c^{3/2}} \]
[Out]
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Rubi [A] time = 0.148333, antiderivative size = 52, 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 \[ \frac{b c-a d}{c d \sqrt{c+\frac{d}{x^2}}}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c+\frac{d}{x^2}}}{\sqrt{c}}\right )}{c^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^2)/((c + d/x^2)^(3/2)*x),x]
[Out]
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Rubi in Sympy [A] time = 12.1826, size = 42, normalized size = 0.81 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c + \frac{d}{x^{2}}}}{\sqrt{c}} \right )}}{c^{\frac{3}{2}}} - \frac{a d - b c}{c d \sqrt{c + \frac{d}{x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x,x)
[Out]
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Mathematica [A] time = 0.0669372, size = 75, normalized size = 1.44 \[ \frac{\sqrt{c} x (b c-a d)+a d \sqrt{c x^2+d} \log \left (\sqrt{c} \sqrt{c x^2+d}+c x\right )}{c^{3/2} d x \sqrt{c+\frac{d}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^2)/((c + d/x^2)^(3/2)*x),x]
[Out]
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Maple [A] time = 0.013, size = 75, normalized size = 1.4 \[{\frac{c{x}^{2}+d}{d{x}^{3}} \left ( xb{c}^{{\frac{5}{2}}}-axd{c}^{{\frac{3}{2}}}+a\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) d\sqrt{c{x}^{2}+d}c \right ) \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}{c}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)/(c+d/x^2)^(3/2)/x,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),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.229632, size = 1, normalized size = 0.02 \[ \left [\frac{2 \,{\left (b c^{2} - a c d\right )} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} +{\left (a c d x^{2} + a d^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (2 \, c x^{2} + d\right )} \sqrt{c}\right )}{2 \,{\left (c^{3} d x^{2} + c^{2} d^{2}\right )}}, \frac{{\left (b c^{2} - a c d\right )} x^{2} \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (a c d x^{2} + a d^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c}}{\sqrt{\frac{c x^{2} + d}{x^{2}}}}\right )}{c^{3} d x^{2} + c^{2} d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 15.6087, size = 218, normalized size = 4.19 \[ a \left (- \frac{2 c^{3} x^{2} \sqrt{1 + \frac{d}{c x^{2}}}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{3} x^{2} \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{3} x^{2} \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} - \frac{c^{2} d \log{\left (\frac{d}{c x^{2}} \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d} + \frac{2 c^{2} d \log{\left (\sqrt{1 + \frac{d}{c x^{2}}} + 1 \right )}}{2 c^{\frac{9}{2}} x^{2} + 2 c^{\frac{7}{2}} d}\right ) + b \left (\begin{cases} \frac{1}{d \sqrt{c + \frac{d}{x^{2}}}} & \text{for}\: d \neq 0 \\- \frac{1}{2 c^{\frac{3}{2}} x^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)/(c+d/x**2)**(3/2)/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + \frac{b}{x^{2}}}{{\left (c + \frac{d}{x^{2}}\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)/((c + d/x^2)^(3/2)*x),x, algorithm="giac")
[Out]