Optimal. Leaf size=164 \[ \frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{4}\right )}{7 \sqrt{x^4+x^2+1}}+\frac{1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac{2}{21} x \left (3 x^2+4\right ) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left (x^2+1\right )}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
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Rubi [A] time = 0.0605862, 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, Rules used = {1206, 1176, 1197, 1103, 1195} \[ \frac{1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac{2}{21} x \left (3 x^2+4\right ) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{7 \sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1176
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \left (1+x^2\right )^2 \sqrt{1+x^2+x^4} \, dx &=\frac{1}{7} x \left (1+x^2+x^4\right )^{3/2}+\frac{1}{7} \int \left (6+10 x^2\right ) \sqrt{1+x^2+x^4} \, dx\\ &=\frac{2}{21} x \left (4+3 x^2\right ) \sqrt{1+x^2+x^4}+\frac{1}{7} x \left (1+x^2+x^4\right )^{3/2}+\frac{1}{105} \int \frac{50+70 x^2}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{2}{21} x \left (4+3 x^2\right ) \sqrt{1+x^2+x^4}+\frac{1}{7} x \left (1+x^2+x^4\right )^{3/2}-\frac{2}{3} \int \frac{1-x^2}{\sqrt{1+x^2+x^4}} \, dx+\frac{8}{7} \int \frac{1}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{2 x \sqrt{1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac{2}{21} x \left (4+3 x^2\right ) \sqrt{1+x^2+x^4}+\frac{1}{7} x \left (1+x^2+x^4\right )^{3/2}-\frac{2 \left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{1+x^2+x^4}}+\frac{4 \left (1+x^2\right ) \sqrt{\frac{1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{7 \sqrt{1+x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.154316, size = 162, normalized size = 0.99 \[ \frac{2 \sqrt [3]{-1} \left (5 \sqrt [3]{-1}-7\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+x \left (3 x^8+12 x^6+23 x^4+20 x^2+11\right )+14 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{21 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.006, size = 248, normalized size = 1.5 \begin{align*}{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{3\,{x}^{3}}{7}\sqrt{{x}^{4}+{x}^{2}+1}}+{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt{x^{4} + x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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