Optimal. Leaf size=93 \[ \frac{1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac{15}{32} \sqrt [4]{4 x^4+3} x^3-\frac{45 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}} \]
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Rubi [A] time = 0.0293518, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {279, 331, 298, 203, 206} \[ \frac{1}{8} \left (4 x^4+3\right )^{5/4} x^3+\frac{15}{32} \sqrt [4]{4 x^4+3} x^3-\frac{45 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{4 x^4+3}}\right )}{128 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 279
Rule 331
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int x^2 \left (3+4 x^4\right )^{5/4} \, dx &=\frac{1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac{15}{8} \int x^2 \sqrt [4]{3+4 x^4} \, dx\\ &=\frac{15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac{45}{32} \int \frac{x^2}{\left (3+4 x^4\right )^{3/4}} \, dx\\ &=\frac{15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac{45}{32} \operatorname{Subst}\left (\int \frac{x^2}{1-4 x^4} \, dx,x,\frac{x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac{15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^3 \left (3+4 x^4\right )^{5/4}+\frac{45}{128} \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{x}{\sqrt [4]{3+4 x^4}}\right )-\frac{45}{128} \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{x}{\sqrt [4]{3+4 x^4}}\right )\\ &=\frac{15}{32} x^3 \sqrt [4]{3+4 x^4}+\frac{1}{8} x^3 \left (3+4 x^4\right )^{5/4}-\frac{45 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt{2}}+\frac{45 \tanh ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3+4 x^4}}\right )}{128 \sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0046128, size = 26, normalized size = 0.28 \[ \sqrt [4]{3} x^3 \, _2F_1\left (-\frac{5}{4},\frac{3}{4};\frac{7}{4};-\frac{4 x^4}{3}\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 0.021, size = 42, normalized size = 0.5 \begin{align*}{\frac{{x}^{3} \left ( 16\,{x}^{4}+27 \right ) }{32}\sqrt [4]{4\,{x}^{4}+3}}+{\frac{5\,\sqrt [4]{3}{x}^{3}}{32}{\mbox{$_2$F$_1$}({\frac{3}{4}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{\frac{4\,{x}^{4}}{3}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46926, size = 176, normalized size = 1.89 \begin{align*} \frac{45}{256} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{45}{512} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) + \frac{9 \,{\left (\frac{20 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x} - \frac{9 \,{\left (4 \, x^{4} + 3\right )}^{\frac{5}{4}}}{x^{5}}\right )}}{32 \,{\left (\frac{8 \,{\left (4 \, x^{4} + 3\right )}}{x^{4}} - \frac{{\left (4 \, x^{4} + 3\right )}^{2}}{x^{8}} - 16\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.86701, size = 298, normalized size = 3.2 \begin{align*} \frac{45}{256} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{45}{512} \, \sqrt{2} \log \left (8 \, x^{4} + 4 \, \sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} x^{3} + 4 \, \sqrt{4 \, x^{4} + 3} x^{2} + 2 \, \sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{3}{4}} x + 3\right ) + \frac{1}{32} \,{\left (16 \, x^{7} + 27 \, x^{3}\right )}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.09364, size = 41, normalized size = 0.44 \begin{align*} \frac{3 \sqrt [4]{3} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{4 x^{4} e^{i \pi }}{3}} \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 [A] time = 1.08375, size = 149, normalized size = 1.6 \begin{align*} \frac{1}{32} \, x^{8}{\left (\frac{9 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}{\left (\frac{3}{x^{4}} + 4\right )}}{x} - \frac{20 \,{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}\right )} + \frac{45}{256} \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{45}{512} \, \sqrt{2} \log \left (-\frac{\sqrt{2} - \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}{\sqrt{2} + \frac{{\left (4 \, x^{4} + 3\right )}^{\frac{1}{4}}}{x}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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