Optimal. Leaf size=124 \[ -\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{4 \sqrt{x^4+1}}+\frac{x^3}{2 \sqrt{x^4+1}}-\frac{\sqrt{x^4+1} x}{2 \left (x^2+1\right )}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
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Rubi [A] time = 0.0207975, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {290, 305, 220, 1196} \[ \frac{x^3}{2 \sqrt{x^4+1}}-\frac{\sqrt{x^4+1} x}{2 \left (x^2+1\right )}-\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{x^4+1}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 290
Rule 305
Rule 220
Rule 1196
Rubi steps
\begin{align*} \int \frac{x^2}{\left (1+x^4\right )^{3/2}} \, dx &=\frac{x^3}{2 \sqrt{1+x^4}}-\frac{1}{2} \int \frac{x^2}{\sqrt{1+x^4}} \, dx\\ &=\frac{x^3}{2 \sqrt{1+x^4}}-\frac{1}{2} \int \frac{1}{\sqrt{1+x^4}} \, dx+\frac{1}{2} \int \frac{1-x^2}{\sqrt{1+x^4}} \, dx\\ &=\frac{x^3}{2 \sqrt{1+x^4}}-\frac{x \sqrt{1+x^4}}{2 \left (1+x^2\right )}+\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{2 \sqrt{1+x^4}}-\frac{\left (1+x^2\right ) \sqrt{\frac{1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{4 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0029229, size = 22, normalized size = 0.18 \[ \frac{1}{3} x^3 \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-x^4\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.007, size = 95, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}-{\frac{{\frac{i}{2}} \left ({\it EllipticF} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) -{\it EllipticE} \left ( x \left ({\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) ,i \right ) \right ) }{{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2}}\sqrt{1-i{x}^{2}}\sqrt{1+i{x}^{2}}{\frac{1}{\sqrt{{x}^{4}+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{x^{2}}{{\left (x^{4} + 1\right )}^{\frac{3}{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 (\frac{\sqrt{x^{4} + 1} x^{2}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.743067, size = 29, normalized size = 0.23 \begin{align*} \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{2} \\ \frac{7}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \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{x^{2}}{{\left (x^{4} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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