Optimal. Leaf size=159 \[ \frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}}+\frac{1}{5} \sqrt{x^4+x^2+1} x^3+\frac{14 \sqrt{x^4+x^2+1} x}{15 \left (x^2+1\right )}+\frac{11}{15} \sqrt{x^4+x^2+1} x-\frac{14 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}} \]
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Rubi [A] time = 0.0714685, antiderivative size = 159, 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, 1679, 1197, 1103, 1195} \[ \frac{1}{5} \sqrt{x^4+x^2+1} x^3+\frac{14 \sqrt{x^4+x^2+1} x}{15 \left (x^2+1\right )}+\frac{11}{15} \sqrt{x^4+x^2+1} x+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}}-\frac{14 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1206
Rule 1679
Rule 1197
Rule 1103
Rule 1195
Rubi steps
\begin{align*} \int \frac{\left (1+x^2\right )^3}{\sqrt{1+x^2+x^4}} \, dx &=\frac{1}{5} x^3 \sqrt{1+x^2+x^4}+\frac{1}{5} \int \frac{5+12 x^2+11 x^4}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{11}{15} x \sqrt{1+x^2+x^4}+\frac{1}{5} x^3 \sqrt{1+x^2+x^4}+\frac{1}{15} \int \frac{4+14 x^2}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{11}{15} x \sqrt{1+x^2+x^4}+\frac{1}{5} x^3 \sqrt{1+x^2+x^4}-\frac{14}{15} \int \frac{1-x^2}{\sqrt{1+x^2+x^4}} \, dx+\frac{6}{5} \int \frac{1}{\sqrt{1+x^2+x^4}} \, dx\\ &=\frac{11}{15} x \sqrt{1+x^2+x^4}+\frac{1}{5} x^3 \sqrt{1+x^2+x^4}+\frac{14 x \sqrt{1+x^2+x^4}}{15 \left (1+x^2\right )}-\frac{14 \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 )}{15 \sqrt{1+x^2+x^4}}+\frac{3 \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 )}{5 \sqrt{1+x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.173071, size = 157, normalized size = 0.99 \[ \frac{2 \sqrt [3]{-1} \left (2 \sqrt [3]{-1}-7\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} \text{EllipticF}\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right ),(-1)^{2/3}\right )+x \left (3 x^6+14 x^4+14 x^2+11\right )+14 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )}{15 \sqrt{x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.027, size = 233, normalized size = 1.5 \begin{align*}{\frac{{x}^{3}}{5}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{11\,x}{15}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{8}{15\,\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{56}{15\,\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 \frac{{\left (x^{2} + 1\right )}^{3}}{\sqrt{x^{4} + x^{2} + 1}}\,{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 (\frac{x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1}{\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 \frac{\left (x^{2} + 1\right )^{3}}{\sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, 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 \frac{{\left (x^{2} + 1\right )}^{3}}{\sqrt{x^{4} + x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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