Optimal. Leaf size=39 \[ \frac {1}{12} \sqrt {x^4+x} \left (2 x^4+x\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 49, normalized size of antiderivative = 1.26, 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 = {2021, 2024, 2029, 206} \begin {gather*} \frac {1}{6} \sqrt {x^4+x} x^4+\frac {1}{12} \sqrt {x^4+x} x-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 2021
Rule 2024
Rule 2029
Rubi steps
\begin {align*} \int x^3 \sqrt {x+x^4} \, dx &=\frac {1}{6} x^4 \sqrt {x+x^4}+\frac {1}{4} \int \frac {x^4}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{12} x \sqrt {x+x^4}+\frac {1}{6} x^4 \sqrt {x+x^4}-\frac {1}{8} \int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{12} x \sqrt {x+x^4}+\frac {1}{6} x^4 \sqrt {x+x^4}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=\frac {1}{12} x \sqrt {x+x^4}+\frac {1}{6} x^4 \sqrt {x+x^4}-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 48, normalized size = 1.23 \begin {gather*} \frac {\sqrt {x^4+x} \left (2 x^{9/2}+x^{3/2}-\frac {\sinh ^{-1}\left (x^{3/2}\right )}{\sqrt {x^3+1}}\right )}{12 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{12} \sqrt {x^4+x} \left (2 x^4+x\right )-\frac {1}{12} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 37, normalized size = 0.95 \begin {gather*} \frac {1}{12} \, {\left (2 \, x^{4} + x\right )} \sqrt {x^{4} + x} + \frac {1}{24} \, \log \left (2 \, x^{3} - 2 \, \sqrt {x^{4} + x} x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.34, size = 43, normalized size = 1.10 \begin {gather*} \frac {1}{12} \, \sqrt {x^{4} + x} {\left (2 \, x^{3} + 1\right )} x - \frac {1}{24} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) + \frac {1}{24} \, \log \left ({\left | \sqrt {\frac {1}{x^{3}} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.04, size = 313, normalized size = 8.03 \begin {gather*} \frac {x^{4} \sqrt {x^{4}+x}}{6}+\frac {x \sqrt {x^{4}+x}}{12}+\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{4 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}} \end {gather*}
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*} \int \sqrt {x^{4} + x} x^{3}\,{d x} \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 = -0.03 \begin {gather*} \int x^3\,\sqrt {x^4+x} \,d x \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^{3} \sqrt {x \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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