Optimal. Leaf size=91 \[ \frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{3/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-4 a d)}{8 d x}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{4 d x} \]
[Out]
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Rubi [A] time = 0.159678, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{c (b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d}}{x \sqrt{c+\frac{d}{x^2}}}\right )}{8 d^{3/2}}+\frac{\sqrt{c+\frac{d}{x^2}} (b c-4 a d)}{8 d x}-\frac{b \left (c+\frac{d}{x^2}\right )^{3/2}}{4 d x} \]
Antiderivative was successfully verified.
[In] Int[((a + b/x^2)*Sqrt[c + d/x^2])/x^2,x]
[Out]
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Rubi in Sympy [A] time = 13.0123, size = 76, normalized size = 0.84 \[ - \frac{b \left (c + \frac{d}{x^{2}}\right )^{\frac{3}{2}}}{4 d x} - \frac{c \left (4 a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{d}}{x \sqrt{c + \frac{d}{x^{2}}}} \right )}}{8 d^{\frac{3}{2}}} - \frac{\sqrt{c + \frac{d}{x^{2}}} \left (4 a d - b c\right )}{8 d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.157091, size = 119, normalized size = 1.31 \[ -\frac{\sqrt{c+\frac{d}{x^2}} \left (c x^4 \log (x) (b c-4 a d)+\sqrt{d} \sqrt{c x^2+d} \left (4 a d x^2+b c x^2+2 b d\right )+c x^4 (4 a d-b c) \log \left (\sqrt{d} \sqrt{c x^2+d}+d\right )\right )}{8 d^{3/2} x^3 \sqrt{c x^2+d}} \]
Antiderivative was successfully verified.
[In] Integrate[((a + b/x^2)*Sqrt[c + d/x^2])/x^2,x]
[Out]
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Maple [B] time = 0.018, size = 187, normalized size = 2.1 \[ -{\frac{1}{8\,{x}^{3}}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}} \left ( -4\,ac\sqrt{c{x}^{2}+d}{x}^{4}{d}^{5/2}+b{c}^{2}\sqrt{c{x}^{2}+d}{x}^{4}{d}^{{\frac{3}{2}}}+4\,ac\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{3}+4\,a \left ( c{x}^{2}+d \right ) ^{3/2}{x}^{2}{d}^{5/2}-bc \left ( c{x}^{2}+d \right ) ^{{\frac{3}{2}}}{x}^{2}{d}^{{\frac{3}{2}}}-b{c}^{2}\ln \left ( 2\,{\frac{\sqrt{d}\sqrt{c{x}^{2}+d}+d}{x}} \right ){x}^{4}{d}^{2}+2\,b \left ( c{x}^{2}+d \right ) ^{3/2}{d}^{5/2} \right ){\frac{1}{\sqrt{c{x}^{2}+d}}}{d}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)*(c+d/x^2)^(1/2)/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)*sqrt(c + d/x^2)/x^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.238281, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c^{2} - 4 \, a c d\right )} \sqrt{d} x^{3} \log \left (\frac{2 \, d x \sqrt{\frac{c x^{2} + d}{x^{2}}} -{\left (c x^{2} + 2 \, d\right )} \sqrt{d}}{x^{2}}\right ) + 2 \,{\left (2 \, b d^{2} +{\left (b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{16 \, d^{2} x^{3}}, -\frac{{\left (b c^{2} - 4 \, a c d\right )} \sqrt{-d} x^{3} \arctan \left (\frac{\sqrt{-d}}{x \sqrt{\frac{c x^{2} + d}{x^{2}}}}\right ) +{\left (2 \, b d^{2} +{\left (b c d + 4 \, a d^{2}\right )} x^{2}\right )} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{8 \, d^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.85855, size = 144, normalized size = 1.58 \[ - \frac{a \sqrt{c} \sqrt{1 + \frac{d}{c x^{2}}}}{2 x} - \frac{a c \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{2 \sqrt{d}} - \frac{b c^{\frac{3}{2}}}{8 d x \sqrt{1 + \frac{d}{c x^{2}}}} - \frac{3 b \sqrt{c}}{8 x^{3} \sqrt{1 + \frac{d}{c x^{2}}}} + \frac{b c^{2} \operatorname{asinh}{\left (\frac{\sqrt{d}}{\sqrt{c} x} \right )}}{8 d^{\frac{3}{2}}} - \frac{b d}{4 \sqrt{c} x^{5} \sqrt{1 + \frac{d}{c x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)*(c+d/x**2)**(1/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.249531, size = 176, normalized size = 1.93 \[ -\frac{\frac{{\left (b c^{3}{\rm sign}\left (x\right ) - 4 \, a c^{2} d{\rm sign}\left (x\right )\right )} \arctan \left (\frac{\sqrt{c x^{2} + d}}{\sqrt{-d}}\right )}{\sqrt{-d} d} + \frac{{\left (c x^{2} + d\right )}^{\frac{3}{2}} b c^{3}{\rm sign}\left (x\right ) + 4 \,{\left (c x^{2} + d\right )}^{\frac{3}{2}} a c^{2} d{\rm sign}\left (x\right ) + \sqrt{c x^{2} + d} b c^{3} d{\rm sign}\left (x\right ) - 4 \, \sqrt{c x^{2} + d} a c^{2} d^{2}{\rm sign}\left (x\right )}{c^{2} d x^{4}}}{8 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^2)*sqrt(c + d/x^2)/x^2,x, algorithm="giac")
[Out]