Optimal. Leaf size=58 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \]
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Rubi [A] time = 0.0895296, antiderivative size = 58, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{\sqrt [4]{a} b^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]/(a - b*x^2),x]
[Out]
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Rubi in Sympy [A] time = 17.6884, size = 53, normalized size = 0.91 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{\sqrt [4]{a} b^{\frac{3}{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{\sqrt [4]{a} b^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(1/2)/(-b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0323244, size = 73, normalized size = 1.26 \[ -\frac{\log \left (\sqrt [4]{a}-\sqrt [4]{b} \sqrt{x}\right )-\log \left (\sqrt [4]{a}+\sqrt [4]{b} \sqrt{x}\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]/(a - b*x^2),x]
[Out]
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Maple [A] time = 0.012, size = 66, normalized size = 1.1 \[ -{\frac{1}{b}\arctan \left ({1\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{1}{2\,b}\ln \left ({1 \left ( \sqrt{x}+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sqrt{x}-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(1/2)/(-b*x^2+a),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(-sqrt(x)/(b*x^2 - a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255073, size = 149, normalized size = 2.57 \[ 2 \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a b^{2} \left (\frac{1}{a b^{3}}\right )^{\frac{3}{4}}}{\sqrt{a b \sqrt{\frac{1}{a b^{3}}} + x} + \sqrt{x}}\right ) + \frac{1}{2} \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (a b^{2} \left (\frac{1}{a b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) - \frac{1}{2} \, \left (\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (-a b^{2} \left (\frac{1}{a b^{3}}\right )^{\frac{3}{4}} + \sqrt{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(x)/(b*x^2 - a),x, algorithm="fricas")
[Out]
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Sympy [A] time = 17.5813, size = 128, normalized size = 2.21 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{2 x^{\frac{3}{2}}}{3 a} & \text{for}\: b = 0 \\\frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\- \frac{\log{\left (- \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 \sqrt [4]{a} b^{3} \left (\frac{1}{b}\right )^{\frac{9}{4}}} + \frac{\log{\left (\sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sqrt{x} \right )}}{2 \sqrt [4]{a} b^{3} \left (\frac{1}{b}\right )^{\frac{9}{4}}} - \frac{\operatorname{atan}{\left (\frac{\sqrt{x}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{\sqrt [4]{a} b^{3} \left (\frac{1}{b}\right )^{\frac{9}{4}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(1/2)/(-b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.218923, size = 262, normalized size = 4.52 \[ \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{3}} + \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (-\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{2 \, a b^{3}} - \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{-\frac{a}{b}}\right )}{4 \, a b^{3}} + \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (-\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{-\frac{a}{b}}\right )}{4 \, a b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(x)/(b*x^2 - a),x, algorithm="giac")
[Out]