Optimal. Leaf size=59 \[ \frac {1}{16} \left (x+4 x^3\right ) \sqrt [4]{x^2+x^4}+\frac {3}{32} \text {ArcTan}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right )-\frac {3}{32} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^2+x^4}}\right ) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(125\) vs. \(2(59)=118\).
time = 0.08, antiderivative size = 125, normalized size of antiderivative = 2.12, number of steps
used = 8, number of rules used = 8, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {2046, 2049,
2057, 335, 338, 304, 209, 212} \begin {gather*} \frac {3 \left (x^2+1\right )^{3/4} x^{3/2} \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{32 \left (x^4+x^2\right )^{3/4}}+\frac {1}{16} \sqrt [4]{x^4+x^2} x-\frac {3 \left (x^2+1\right )^{3/4} x^{3/2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{32 \left (x^4+x^2\right )^{3/4}}+\frac {1}{4} \sqrt [4]{x^4+x^2} x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 304
Rule 335
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps
\begin {align*} \int x^2 \sqrt [4]{x^2+x^4} \, dx &=\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}+\frac {1}{8} \int \frac {x^4}{\left (x^2+x^4\right )^{3/4}} \, dx\\ &=\frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {3}{32} \int \frac {x^2}{\left (x^2+x^4\right )^{3/4}} \, dx\\ &=\frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right )^{3/4}} \, dx}{32 \left (x^2+x^4\right )^{3/4}}\\ &=\frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{16 \left (x^2+x^4\right )^{3/4}}\\ &=\frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{16 \left (x^2+x^4\right )^{3/4}}\\ &=\frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}-\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}}+\frac {\left (3 x^{3/2} \left (1+x^2\right )^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}}\\ &=\frac {1}{16} x \sqrt [4]{x^2+x^4}+\frac {1}{4} x^3 \sqrt [4]{x^2+x^4}+\frac {3 x^{3/2} \left (1+x^2\right )^{3/4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}}-\frac {3 x^{3/2} \left (1+x^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{32 \left (x^2+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 89, normalized size = 1.51 \begin {gather*} \frac {\sqrt [4]{x^2+x^4} \left (2 x^{3/2} \sqrt [4]{1+x^2} \left (1+4 x^2\right )+3 \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )-3 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )\right )}{32 \sqrt {x} \sqrt [4]{1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{2} \left (x^{4}+x^{2}\right )^{\frac {1}{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 102 vs.
\(2 (47) = 94\).
time = 0.89, size = 102, normalized size = 1.73 \begin {gather*} \frac {1}{16} \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (4 \, x^{3} + x\right )} + \frac {3}{64} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {3}{64} \, \log \left (-\frac {2 \, x^{3} - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} + x^{2}} x + x - 2 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 57, normalized size = 0.97 \begin {gather*} \frac {1}{16} \, {\left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right )} x^{4} - \frac {3}{32} \, \arctan \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{64} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{64} \, \log \left ({\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^2\,{\left (x^4+x^2\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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