Optimal. Leaf size=120 \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]
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Rubi [A] time = 0.0723392, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 52.7652, size = 116, normalized size = 0.97 \[ \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b x \sqrt{- a}} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b x \sqrt{- a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(1/4)/(b*x**2+2*a),x)
[Out]
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Mathematica [C] time = 0.236927, size = 165, normalized size = 1.38 \[ \frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right ) \left (6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]
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Maple [F] time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{2}+2\,a}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(b*x^2+a)^(1/4)/(b*x^2+2*a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 2 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + 2*a)*(b*x^2 + a)^(1/4)),x, algorithm="maxima")
[Out]
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + 2*a)*(b*x^2 + a)^(1/4)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt [4]{a + b x^{2}} \left (2 a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(1/4)/(b*x**2+2*a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 2 \, a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + 2*a)*(b*x^2 + a)^(1/4)),x, algorithm="giac")
[Out]