Optimal. Leaf size=88 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+\frac{a \tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+b \tan ^{-1}\left (\sqrt [4]{-x^2-1}\right )-b \tanh ^{-1}\left (\sqrt [4]{-x^2-1}\right ) \]
[Out]
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Rubi [A] time = 0.130639, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+\frac{a \tanh ^{-1}\left (\frac{x}{\sqrt{2} \sqrt [4]{-x^2-1}}\right )}{2 \sqrt{2}}+b \tan ^{-1}\left (\sqrt [4]{-x^2-1}\right )-b \tanh ^{-1}\left (\sqrt [4]{-x^2-1}\right ) \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/((-1 - x^2)^(1/4)*(2 + x^2)),x]
[Out]
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Rubi in Sympy [A] time = 29.508, size = 199, normalized size = 2.26 \[ - \frac{\sqrt{2} a x \left (1 - i\right ) \Pi \left (i; \operatorname{asin}{\left (\frac{\sqrt{2} \left (1 + i\right ) \sqrt [4]{- x^{2} - 1}}{2} \right )}\middle | -1\right )}{2 \sqrt{- i \sqrt{- x^{2} - 1} + 1} \sqrt{i \sqrt{- x^{2} - 1} + 1}} - \frac{\sqrt{2} a \sqrt{- x^{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt [4]{- x^{2} - 1}}{\sqrt{- x^{2}}} \right )}}{4 x} - \frac{a \sqrt{- \frac{x^{2}}{\left (\sqrt{- x^{2} - 1} + 1\right )^{2}}} \left (\sqrt{- x^{2} - 1} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{- x^{2} - 1} \right )}\middle | \frac{1}{2}\right )}{4 x} + b \operatorname{atan}{\left (\sqrt [4]{- x^{2} - 1} \right )} - b \operatorname{atanh}{\left (\sqrt [4]{- x^{2} - 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(-x**2-1)**(1/4)/(x**2+2),x)
[Out]
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Mathematica [C] time = 0.554591, size = 221, normalized size = 2.51 \[ \frac{2 x \left (-\frac{3 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-x^2,-\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-x^2,-\frac{x^2}{2}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-x^2,-\frac{x^2}{2}\right )\right )-6 F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-x^2,-\frac{x^2}{2}\right )}-\frac{2 b x F_1\left (1;\frac{1}{4},1;2;-x^2,-\frac{x^2}{2}\right )}{x^2 \left (2 F_1\left (2;\frac{1}{4},2;3;-x^2,-\frac{x^2}{2}\right )+F_1\left (2;\frac{5}{4},1;3;-x^2,-\frac{x^2}{2}\right )\right )-8 F_1\left (1;\frac{1}{4},1;2;-x^2,-\frac{x^2}{2}\right )}\right )}{\sqrt [4]{-x^2-1} \left (x^2+2\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)/((-1 - x^2)^(1/4)*(2 + x^2)),x]
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Maple [F] time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{bx+a}{{x}^{2}+2}{\frac{1}{\sqrt [4]{-{x}^{2}-1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(-x^2-1)^(1/4)/(x^2+2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (x^{2} + 2\right )}{\left (-x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((x^2 + 2)*(-x^2 - 1)^(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((b*x + a)/((x^2 + 2)*(-x^2 - 1)^(1/4)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + b x}{\sqrt [4]{- x^{2} - 1} \left (x^{2} + 2\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(-x**2-1)**(1/4)/(x**2+2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{b x + a}{{\left (x^{2} + 2\right )}{\left (-x^{2} - 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)/((x^2 + 2)*(-x^2 - 1)^(1/4)),x, algorithm="giac")
[Out]