Optimal. Leaf size=198 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac{3 b \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]
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Rubi [A] time = 0.21155, antiderivative size = 198, 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 \[ \frac{a \tan ^{-1}\left (\frac{x}{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}\right )}{2\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} \sqrt [3]{x^2+1}\right )}{x}\right )}{2\ 2^{2/3} \sqrt{3}}-\frac{a \tan ^{-1}(x)}{6\ 2^{2/3}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{3}}{x}\right )}{2\ 2^{2/3} \sqrt{3}}+\frac{b \log \left (3-x^2\right )}{4\ 2^{2/3}}-\frac{3 b \log \left (2^{2/3}-\sqrt [3]{x^2+1}\right )}{4\ 2^{2/3}}-\frac{\sqrt{3} b \tan ^{-1}\left (\frac{\sqrt [3]{2} \sqrt [3]{x^2+1}+1}{\sqrt{3}}\right )}{2\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x)/((3 - x^2)*(1 + x^2)^(1/3)),x]
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Rubi in Sympy [A] time = 13.1202, size = 282, normalized size = 1.42 \[ \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (- x + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (x + \sqrt{3} \right )}}{24} + \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (- x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} \sqrt{3} a \log{\left (x - \sqrt [3]{2} \sqrt{3} \sqrt [3]{x^{2} + 1} + \sqrt{3} \right )}}{24} - \frac{\sqrt [3]{2} a \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (- x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt [3]{2} a \operatorname{atan}{\left (\frac{2^{\frac{2}{3}} \left (x + \sqrt{3}\right )}{3 \sqrt [3]{x^{2} + 1}} + \frac{\sqrt{3}}{3} \right )}}{12} + \frac{\sqrt [3]{2} b \log{\left (- x^{2} + 3 \right )}}{8} - \frac{3 \sqrt [3]{2} b \log{\left (- \sqrt [3]{x^{2} + 1} + 2^{\frac{2}{3}} \right )}}{8} - \frac{\sqrt [3]{2} \sqrt{3} b \operatorname{atan}{\left (\sqrt{3} \left (\frac{\sqrt [3]{2} \sqrt [3]{x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)/(-x**2+3)/(x**2+1)**(1/3),x)
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Mathematica [C] time = 0.441686, size = 220, normalized size = 1.11 \[ \frac{3 x \left (-\frac{3 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}{2 x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-x^2,\frac{x^2}{3}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-x^2,\frac{x^2}{3}\right )\right )+9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-x^2,\frac{x^2}{3}\right )}-\frac{b x F_1\left (1;\frac{1}{3},1;2;-x^2,\frac{x^2}{3}\right )}{x^2 \left (F_1\left (2;\frac{1}{3},2;3;-x^2,\frac{x^2}{3}\right )-F_1\left (2;\frac{4}{3},1;3;-x^2,\frac{x^2}{3}\right )\right )+6 F_1\left (1;\frac{1}{3},1;2;-x^2,\frac{x^2}{3}\right )}\right )}{\left (x^2-3\right ) \sqrt [3]{x^2+1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(a + b*x)/((3 - x^2)*(1 + x^2)^(1/3)),x]
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Maple [F] time = 0.073, size = 0, normalized size = 0. \[ \int{\frac{bx+a}{-{x}^{2}+3}{\frac{1}{\sqrt [3]{{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)/(-x^2+3)/(x^2+1)^(1/3),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{b x + a}{{\left (x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)/((x^2 + 1)^(1/3)*(x^2 - 3)),x, algorithm="maxima")
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)/((x^2 + 1)^(1/3)*(x^2 - 3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{a}{x^{2} \sqrt [3]{x^{2} + 1} - 3 \sqrt [3]{x^{2} + 1}}\, dx - \int \frac{b x}{x^{2} \sqrt [3]{x^{2} + 1} - 3 \sqrt [3]{x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x+a)/(-x**2+3)/(x**2+1)**(1/3),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{b x + a}{{\left (x^{2} + 1\right )}^{\frac{1}{3}}{\left (x^{2} - 3\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(b*x + a)/((x^2 + 1)^(1/3)*(x^2 - 3)),x, algorithm="giac")
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