Optimal. Leaf size=71 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{16 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2} \]
[Out]
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Rubi [A] time = 0.131929, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{3 a^2 \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right )}{16 \sqrt{b}}-\frac{3 a \sqrt{a+\frac{b}{x^4}}}{16 x^2}-\frac{\left (a+\frac{b}{x^4}\right )^{3/2}}{8 x^2} \]
Antiderivative was successfully verified.
[In] Int[(a + b/x^4)^(3/2)/x^3,x]
[Out]
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Rubi in Sympy [A] time = 8.35633, size = 66, normalized size = 0.93 \[ - \frac{3 a^{2} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{16 \sqrt{b}} - \frac{3 a \sqrt{a + \frac{b}{x^{4}}}}{16 x^{2}} - \frac{\left (a + \frac{b}{x^{4}}\right )^{\frac{3}{2}}}{8 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**4)**(3/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.141109, size = 70, normalized size = 0.99 \[ \frac{\sqrt{a+\frac{b}{x^4}} \left (-\frac{3 a^2 x^8 \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )}{\sqrt{b} \sqrt{a x^4+b}}-5 a x^4-2 b\right )}{16 x^6} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b/x^4)^(3/2)/x^3,x]
[Out]
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Maple [A] time = 0.028, size = 93, normalized size = 1.3 \[ -{\frac{1}{16\,{x}^{2}} \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{{\frac{3}{2}}} \left ( 3\,{a}^{2}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ){x}^{8}+5\,a\sqrt{a{x}^{4}+b}{x}^{4}\sqrt{b}+2\,{b}^{3/2}\sqrt{a{x}^{4}+b} \right ) \left ( a{x}^{4}+b \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^4)^(3/2)/x^3,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^4)^(3/2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.25221, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, a^{2} \sqrt{b} x^{6} \log \left (-\frac{2 \, b x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + 2 \, b\right )} \sqrt{b}}{x^{4}}\right ) - 2 \,{\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{32 \, b x^{6}}, -\frac{3 \, a^{2} \sqrt{-b} x^{6} \arctan \left (\frac{b}{\sqrt{-b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}\right ) +{\left (5 \, a b x^{4} + 2 \, b^{2}\right )} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{16 \, b x^{6}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^4)^(3/2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 13.076, size = 75, normalized size = 1.06 \[ - \frac{5 a^{\frac{3}{2}} \sqrt{1 + \frac{b}{a x^{4}}}}{16 x^{2}} - \frac{\sqrt{a} b \sqrt{1 + \frac{b}{a x^{4}}}}{8 x^{6}} - \frac{3 a^{2} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{16 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**4)**(3/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.23397, size = 82, normalized size = 1.15 \[ \frac{1}{16} \, a^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (a x^{4} + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{a x^{4} + b} b}{a^{2} x^{8}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a + b/x^4)^(3/2)/x^3,x, algorithm="giac")
[Out]