Optimal. Leaf size=138 \[ -\frac{\sqrt{x^3+1}}{5 x^5}+\frac{7 \sqrt{x^3+1}}{20 x^2}+\frac{7 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
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Rubi [A] time = 0.081315, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{\sqrt{x^3+1}}{5 x^5}+\frac{7 \sqrt{x^3+1}}{20 x^2}+\frac{7 \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{20 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^3+1}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^6*Sqrt[1 + x^3]),x]
[Out]
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Rubi in Sympy [A] time = 5.68369, size = 124, normalized size = 0.9 \[ \frac{7 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{60 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{x^{3} + 1}} + \frac{7 \sqrt{x^{3} + 1}}{20 x^{2}} - \frac{\sqrt{x^{3} + 1}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**6/(x**3+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.136498, size = 110, normalized size = 0.8 \[ \frac{21 x^6+9 x^3+7 \sqrt [6]{-1} 3^{3/4} \sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )} \sqrt{(-1)^{2/3} x^2+\sqrt [3]{-1} x+1} x^5 F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} (x+1)}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-12}{60 x^5 \sqrt{x^3+1}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[1/(x^6*Sqrt[1 + x^3]),x]
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Maple [A] time = 0.029, size = 141, normalized size = 1. \[ -{\frac{1}{5\,{x}^{5}}\sqrt{{x}^{3}+1}}+{\frac{7}{20\,{x}^{2}}\sqrt{{x}^{3}+1}}+{\frac{{\frac{21}{2}}-{\frac{7\,i}{2}}\sqrt{3}}{20}\sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{1+x}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}+1}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^6/(x^3+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{3} + 1} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 + 1)*x^6),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{x^{3} + 1} x^{6}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 + 1)*x^6),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.85182, size = 36, normalized size = 0.26 \[ \frac{\Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac{2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**6/(x**3+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x^{3} + 1} x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^3 + 1)*x^6),x, algorithm="giac")
[Out]