Optimal. Leaf size=46 \[ \frac {1}{72} \sqrt {x^4+x} \left (8 x^7+2 x^4-3 x\right )+\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4+x}}\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 65, normalized size of antiderivative = 1.41, number of steps used = 5, 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}{36} \sqrt {x^4+x} x^4-\frac {1}{24} \sqrt {x^4+x} x+\frac {1}{9} \sqrt {x^4+x} x^7+\frac {1}{24} \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^6 \sqrt {x+x^4} \, dx &=\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{6} \int \frac {x^7}{\sqrt {x+x^4}} \, dx\\ &=\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}-\frac {1}{8} \int \frac {x^4}{\sqrt {x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {x+x^4}+\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{16} \int \frac {x}{\sqrt {x+x^4}} \, dx\\ &=-\frac {1}{24} x \sqrt {x+x^4}+\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^2}{\sqrt {x+x^4}}\right )\\ &=-\frac {1}{24} x \sqrt {x+x^4}+\frac {1}{36} x^4 \sqrt {x+x^4}+\frac {1}{9} x^7 \sqrt {x+x^4}+\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 54, normalized size = 1.17 \begin {gather*} \frac {\sqrt {x^4+x} \left (\frac {3 \sinh ^{-1}\left (x^{3/2}\right )}{\sqrt {x^3+1}}+\left (8 x^6+2 x^3-3\right ) x^{3/2}\right )}{72 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 46, normalized size = 1.00 \begin {gather*} \frac {1}{72} \sqrt {x+x^4} \left (-3 x+2 x^4+8 x^7\right )+\frac {1}{24} \tanh ^{-1}\left (\frac {x^2}{\sqrt {x+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 44, normalized size = 0.96 \begin {gather*} \frac {1}{72} \, {\left (8 \, x^{7} + 2 \, x^{4} - 3 \, x\right )} \sqrt {x^{4} + x} + \frac {1}{48} \, \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 = 0.20, size = 50, normalized size = 1.09 \begin {gather*} \frac {1}{72} \, {\left (2 \, {\left (4 \, x^{3} + 1\right )} x^{3} - 3\right )} \sqrt {x^{4} + x} x + \frac {1}{48} \, \log \left (\sqrt {\frac {1}{x^{3}} + 1} + 1\right ) - \frac {1}{48} \, \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 [A] time = 0.29, size = 43, normalized size = 0.93
method | result | size |
meijerg | \(-\frac {\frac {\sqrt {\pi }\, x^{\frac {3}{2}} \left (-40 x^{6}-10 x^{3}+15\right ) \sqrt {x^{3}+1}}{60}-\frac {\sqrt {\pi }\, \arcsinh \left (x^{\frac {3}{2}}\right )}{4}}{6 \sqrt {\pi }}\) | \(43\) |
trager | \(\frac {x \left (8 x^{6}+2 x^{3}-3\right ) \sqrt {x^{4}+x}}{72}-\frac {\ln \left (2 x^{3}-2 x \sqrt {x^{4}+x}+1\right )}{48}\) | \(44\) |
risch | \(\frac {x^{2} \left (8 x^{6}+2 x^{3}-3\right ) \left (x^{3}+1\right )}{72 \sqrt {x \left (x^{3}+1\right )}}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(322\) |
default | \(\frac {x^{7} \sqrt {x^{4}+x}}{9}+\frac {x^{4} \sqrt {x^{4}+x}}{36}-\frac {x \sqrt {x^{4}+x}}{24}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(325\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}+x}}{9}+\frac {x^{4} \sqrt {x^{4}+x}}{36}-\frac {x \sqrt {x^{4}+x}}{24}-\frac {\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (1+x \right )^{2} \sqrt {-\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \sqrt {-\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}\, \left (-\EllipticF \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )+\EllipticPi \left (\sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (1+x \right )}}, \frac {\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{8 \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (1+x \right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\) | \(325\) |
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^{6}\,{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 = -0.02 \begin {gather*} \int x^6\,\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^{6} \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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