Optimal. Leaf size=90 \[ \frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}(x),\frac{1}{2}\right )}{28 \sqrt{x^4+1}}-\frac{x^9}{2 \sqrt{x^4+1}}+\frac{9}{14} \sqrt{x^4+1} x^5-\frac{15}{14} \sqrt{x^4+1} x \]
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Rubi [A] time = 0.0199829, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {288, 321, 220} \[ -\frac{x^9}{2 \sqrt{x^4+1}}+\frac{9}{14} \sqrt{x^4+1} x^5-\frac{15}{14} \sqrt{x^4+1} x+\frac{15 \left (x^2+1\right ) \sqrt{\frac{x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{2}\right )}{28 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Rule 288
Rule 321
Rule 220
Rubi steps
\begin{align*} \int \frac{x^{12}}{\left (1+x^4\right )^{3/2}} \, dx &=-\frac{x^9}{2 \sqrt{1+x^4}}+\frac{9}{2} \int \frac{x^8}{\sqrt{1+x^4}} \, dx\\ &=-\frac{x^9}{2 \sqrt{1+x^4}}+\frac{9}{14} x^5 \sqrt{1+x^4}-\frac{45}{14} \int \frac{x^4}{\sqrt{1+x^4}} \, dx\\ &=-\frac{x^9}{2 \sqrt{1+x^4}}-\frac{15}{14} x \sqrt{1+x^4}+\frac{9}{14} x^5 \sqrt{1+x^4}+\frac{15}{14} \int \frac{1}{\sqrt{1+x^4}} \, dx\\ &=-\frac{x^9}{2 \sqrt{1+x^4}}-\frac{15}{14} x \sqrt{1+x^4}+\frac{9}{14} x^5 \sqrt{1+x^4}+\frac{15 \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 )}{28 \sqrt{1+x^4}}\\ \end{align*}
Mathematica [C] time = 0.0104249, size = 52, normalized size = 0.58 \[ \frac{x \left (15 \sqrt{x^4+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^4\right )+2 x^8-6 x^4-15\right )}{14 \sqrt{x^4+1}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.02, size = 94, normalized size = 1. \begin{align*} -{\frac{x}{2}{\frac{1}{\sqrt{{x}^{4}+1}}}}+{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+1}}-{\frac{4\,x}{7}\sqrt{{x}^{4}+1}}+{\frac{15\,{\it EllipticF} \left ( x \left ( 1/2\,\sqrt{2}+i/2\sqrt{2} \right ) ,i \right ) }{7\,\sqrt{2}+7\,i\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^{12}}{{\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^{12}}{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 = 1.36499, size = 29, normalized size = 0.32 \begin{align*} \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac{17}{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^{12}}{{\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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