Optimal. Leaf size=164 \[ \frac{1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac{2}{21} x \left (3 x^2+4\right ) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{7 \sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
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Rubi [A] time = 0.140456, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac{2}{21} x \left (3 x^2+4\right ) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{7 \sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
[In] Int[(1 + x^2)^2*Sqrt[1 + x^2 + x^4],x]
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Rubi in Sympy [A] time = 28.8372, size = 156, normalized size = 0.95 \[ \frac{x \left (\frac{30 x^{2}}{7} + \frac{40}{7}\right ) \sqrt{x^{4} + x^{2} + 1}}{15} + \frac{x \left (x^{4} + x^{2} + 1\right )^{\frac{3}{2}}}{7} + \frac{2 x \sqrt{x^{4} + x^{2} + 1}}{3 \left (x^{2} + 1\right )} - \frac{2 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{3 \sqrt{x^{4} + x^{2} + 1}} + \frac{4 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{7 \sqrt{x^{4} + x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**2+1)**2*(x**4+x**2+1)**(1/2),x)
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Mathematica [C] time = 0.232731, size = 162, normalized size = 0.99 \[ \frac{2 \sqrt [3]{-1} \left (5 \sqrt [3]{-1}-7\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+14 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x \left (3 x^8+12 x^6+23 x^4+20 x^2+11\right )}{21 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x^2)^2*Sqrt[1 + x^2 + x^4],x]
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Maple [C] time = 0.011, size = 248, normalized size = 1.5 \[{\frac{11\,x}{21}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{20}{21\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{8}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{3\,{x}^{3}}{7}\sqrt{{x}^{4}+{x}^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^2+1)^2*(x^4+x^2+1)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + x^2 + 1)*(x^2 + 1)^2,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt{x^{4} + x^{2} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + x^2 + 1)*(x^2 + 1)^2,x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**2+1)**2*(x**4+x**2+1)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^4 + x^2 + 1)*(x^2 + 1)^2,x, algorithm="giac")
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