Optimal. Leaf size=116 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}-\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]
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Rubi [A] time = 0.0790384, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}+\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x \left (\sqrt{a}-\sqrt{b} x^2\right )}{\sqrt [4]{a} \sqrt{a-b x^4}}\right )}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a - b*x^4]/(a*c + b*c*x^4),x]
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Rubi in Sympy [A] time = 12.6542, size = 100, normalized size = 0.86 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} x \left (\sqrt{a} + \sqrt{b} x^{2}\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} \right )}}{2 \sqrt [4]{a} \sqrt [4]{b} c} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} x \left (\sqrt{a} - \sqrt{b} x^{2}\right )}{\sqrt [4]{a} \sqrt{a - b x^{4}}} \right )}}{2 \sqrt [4]{a} \sqrt [4]{b} c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**4+a)**(1/2)/(b*c*x**4+a*c),x)
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Mathematica [C] time = 0.271918, size = 155, normalized size = 1.34 \[ \frac{5 a x \sqrt{a-b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )}{c \left (a+b x^4\right ) \left (5 a F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )-2 b x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{b x^4}{a},-\frac{b x^4}{a}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[Sqrt[a - b*x^4]/(a*c + b*c*x^4),x]
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Maple [A] time = 0.025, size = 158, normalized size = 1.4 \[ -{\frac{1}{8\,c}\ln \left ({1 \left ({\frac{-b{x}^{4}+a}{2\,{x}^{2}}}-{\frac{1}{x}\sqrt [4]{ab}\sqrt{-b{x}^{4}+a}}+\sqrt{ab} \right ) \left ({\frac{-b{x}^{4}+a}{2\,{x}^{2}}}+{\frac{1}{x}\sqrt [4]{ab}\sqrt{-b{x}^{4}+a}}+\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{ab}}}}-{\frac{1}{4\,c}\arctan \left ({\frac{1}{x}\sqrt{-b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}}+1 \right ){\frac{1}{\sqrt [4]{ab}}}}+{\frac{1}{4\,c}\arctan \left ( -{\frac{1}{x}\sqrt{-b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}}+1 \right ){\frac{1}{\sqrt [4]{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^4+a)^(1/2)/(b*c*x^4+a*c),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-b x^{4} + a}}{b c x^{4} + a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c),x, algorithm="maxima")
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Fricas [A] time = 0.586726, size = 500, normalized size = 4.31 \[ \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{2 \,{\left (2 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}}\right )}}{\sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - \sqrt{-b x^{4} + a} x^{2} +{\left (b x^{4} + a\right )} \sqrt{-\frac{1}{b}}}\right ) - \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) + \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (-\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} - \sqrt{-b x^{4} + a} a c^{2} \sqrt{-\frac{1}{a b c^{4}}} - 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (-\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \sqrt{-b x^{4} + a} x^{2}}{b x^{4} + a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{\sqrt{a - b x^{4}}}{a + b x^{4}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**4+a)**(1/2)/(b*c*x**4+a*c),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-b x^{4} + a}}{b c x^{4} + a c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(-b*x^4 + a)/(b*c*x^4 + a*c),x, algorithm="giac")
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