Optimal. Leaf size=282 \[ \frac{2 x^2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{2 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{\sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{2 \left (x^3+1\right )}{3 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.194509, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{2 x^2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{2 \sqrt{2} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}+\frac{\sqrt{2-\sqrt{3}} \sqrt{x+1} \sqrt{\frac{x^2-x+1}{\left (x+\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x-\sqrt{3}+1}{x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x+1}{\left (x+\sqrt{3}+1\right )^2}} \sqrt{x^2-x+1}}-\frac{2 \left (x^3+1\right )}{3 \sqrt{x+1} \left (x+\sqrt{3}+1\right ) \sqrt{x^2-x+1}} \]
Antiderivative was successfully verified.
[In] Int[x/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 17.2559, size = 265, normalized size = 0.94 \[ \frac{2 x^{2} \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x^{3} + 1\right )} - \frac{2 \sqrt{x + 1} \sqrt{x^{2} - x + 1}}{3 \left (x + 1 + \sqrt{3}\right )} + \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} E\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} - \frac{2 \sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x + 1\right )^{\frac{3}{2}} \sqrt{x^{2} - x + 1} F\left (\operatorname{asin}{\left (\frac{x - \sqrt{3} + 1}{x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{9 \sqrt{\frac{x + 1}{\left (x + 1 + \sqrt{3}\right )^{2}}} \left (x^{3} + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+x)**(3/2)/(x**2-x+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 1.45613, size = 402, normalized size = 1.43 \[ \frac{2 x^2}{3 \sqrt{x+1} \sqrt{x^2-x+1}}-\frac{(x+1)^{3/2} \left (\frac{12 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \left (x^2-x+1\right )}{(x+1)^2}+\frac{i \sqrt{2} \left (\sqrt{3}+3 i\right ) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}+\frac{3 \sqrt{2} \left (1-i \sqrt{3}\right ) \sqrt{\frac{-\frac{6 i}{x+1}+\sqrt{3}+3 i}{\sqrt{3}+3 i}} \sqrt{\frac{\frac{6 i}{x+1}+\sqrt{3}-3 i}{\sqrt{3}-3 i}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-\frac{6 i}{3 i+\sqrt{3}}}}{\sqrt{x+1}}\right )|\frac{3 i+\sqrt{3}}{3 i-\sqrt{3}}\right )}{\sqrt{x+1}}\right )}{18 \sqrt{-\frac{i}{\sqrt{3}+3 i}} \sqrt{x^2-x+1}} \]
Antiderivative was successfully verified.
[In] Integrate[x/((1 + x)^(3/2)*(1 - x + x^2)^(3/2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.058, size = 356, normalized size = 1.3 \[ -{\frac{1}{3\,{x}^{3}+3}\sqrt{1+x}\sqrt{{x}^{2}-x+1} \left ( i\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) \sqrt{3}+3\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -6\,\sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{i\sqrt{3}+3}}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{-3+i\sqrt{3}}}}{\it EllipticE} \left ( \sqrt{-2\,{\frac{1+x}{-3+i\sqrt{3}}}},\sqrt{-{\frac{-3+i\sqrt{3}}{i\sqrt{3}+3}}} \right ) -2\,{x}^{2} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+x)^(3/2)/(x^2-x+1)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{{\left (x^{3} + 1\right )} \sqrt{x^{2} - x + 1} \sqrt{x + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\left (x + 1\right )^{\frac{3}{2}} \left (x^{2} - x + 1\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+x)**(3/2)/(x**2-x+1)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{{\left (x^{2} - x + 1\right )}^{\frac{3}{2}}{\left (x + 1\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/((x^2 - x + 1)^(3/2)*(x + 1)^(3/2)),x, algorithm="giac")
[Out]