Optimal. Leaf size=52 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a-b x^4}}{4 a x^4} \]
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Rubi [A] time = 0.0787222, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a-b x^4}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Int[1/(x^5*Sqrt[a - b*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 7.78037, size = 42, normalized size = 0.81 \[ - \frac{\sqrt{a - b x^{4}}}{4 a x^{4}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a - b x^{4}}}{\sqrt{a}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**5/(-b*x**4+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0984556, size = 52, normalized size = 1. \[ -\frac{b \tanh ^{-1}\left (\frac{\sqrt{a-b x^4}}{\sqrt{a}}\right )}{4 a^{3/2}}-\frac{\sqrt{a-b x^4}}{4 a x^4} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^5*Sqrt[a - b*x^4]),x]
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Maple [A] time = 0.018, size = 50, normalized size = 1. \[ -{\frac{1}{4\,a{x}^{4}}\sqrt{-b{x}^{4}+a}}-{\frac{b}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{-b{x}^{4}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^5/(-b*x^4+a)^(1/2),x)
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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(1/(sqrt(-b*x^4 + a)*x^5),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287458, size = 1, normalized size = 0.02 \[ \left [\frac{b x^{4} \log \left (\frac{{\left (b x^{4} - 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{-b x^{4} + a} a}{x^{4}}\right ) - 2 \, \sqrt{-b x^{4} + a} \sqrt{a}}{8 \, a^{\frac{3}{2}} x^{4}}, \frac{b x^{4} \arctan \left (\frac{a}{\sqrt{-b x^{4} + a} \sqrt{-a}}\right ) - \sqrt{-b x^{4} + a} \sqrt{-a}}{4 \, \sqrt{-a} a x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x^4 + a)*x^5),x, algorithm="fricas")
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Sympy [A] time = 8.47815, size = 129, normalized size = 2.48 \[ \begin{cases} - \frac{\sqrt{b} \sqrt{\frac{a}{b x^{4}} - 1}}{4 a x^{2}} - \frac{b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \left |{\frac{a}{b x^{4}}}\right | > 1 \\\frac{i}{4 \sqrt{b} x^{6} \sqrt{- \frac{a}{b x^{4}} + 1}} - \frac{i \sqrt{b}}{4 a x^{2} \sqrt{- \frac{a}{b x^{4}} + 1}} + \frac{i b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**5/(-b*x**4+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.214263, size = 69, normalized size = 1.33 \[ \frac{1}{4} \, b{\left (\frac{\arctan \left (\frac{\sqrt{-b x^{4} + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} - \frac{\sqrt{-b x^{4} + a}}{a b x^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(-b*x^4 + a)*x^5),x, algorithm="giac")
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