Optimal. Leaf size=98 \[ -\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{256} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1265, 846, 793,
635, 212} \begin {gather*} -\frac {89}{48} \sqrt {x^4+5 x^2+3} x^4-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {x^4+5 x^2+3}+\frac {32801}{256} \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )+\frac {3}{8} \sqrt {x^4+5 x^2+3} x^6 \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 635
Rule 793
Rule 846
Rule 1265
Rubi steps
\begin {align*} \int \frac {x^7 \left (2+3 x^2\right )}{\sqrt {3+5 x^2+x^4}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^3 (2+3 x)}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}+\frac {1}{8} \text {Subst}\left (\int \frac {\left (-27-\frac {89 x}{2}\right ) x^2}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}+\frac {1}{24} \text {Subst}\left (\int \frac {x \left (267+\frac {1901 x}{4}\right )}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{256} \text {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{128} \text {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )\\ &=-\frac {89}{48} x^4 \sqrt {3+5 x^2+x^4}+\frac {3}{8} x^6 \sqrt {3+5 x^2+x^4}-\frac {1}{384} \left (24243-3802 x^2\right ) \sqrt {3+5 x^2+x^4}+\frac {32801}{256} \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 64, normalized size = 0.65 \begin {gather*} \frac {1}{384} \sqrt {3+5 x^2+x^4} \left (-24243+3802 x^2-712 x^4+144 x^6\right )-\frac {32801}{256} \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 87, normalized size = 0.89
method | result | size |
risch | \(\frac {\left (144 x^{6}-712 x^{4}+3802 x^{2}-24243\right ) \sqrt {x^{4}+5 x^{2}+3}}{384}+\frac {32801 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(53\) |
trager | \(\left (\frac {3}{8} x^{6}-\frac {89}{48} x^{4}+\frac {1901}{192} x^{2}-\frac {8081}{128}\right ) \sqrt {x^{4}+5 x^{2}+3}+\frac {32801 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(56\) |
default | \(\frac {3 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{8}-\frac {89 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{48}+\frac {1901 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{192}-\frac {8081 \sqrt {x^{4}+5 x^{2}+3}}{128}+\frac {32801 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(87\) |
elliptic | \(\frac {3 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{8}-\frac {89 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{48}+\frac {1901 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{192}-\frac {8081 \sqrt {x^{4}+5 x^{2}+3}}{128}+\frac {32801 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 90, normalized size = 0.92 \begin {gather*} \frac {3}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{6} - \frac {89}{48} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{4} + \frac {1901}{192} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {8081}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {32801}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 56, normalized size = 0.57 \begin {gather*} \frac {1}{384} \, {\left (144 \, x^{6} - 712 \, x^{4} + 3802 \, x^{2} - 24243\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {32801}{256} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\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 \frac {x^{7} \cdot \left (3 x^{2} + 2\right )}{\sqrt {x^{4} + 5 x^{2} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.97, size = 60, normalized size = 0.61 \begin {gather*} \frac {1}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (18 \, x^{2} - 89\right )} x^{2} + 1901\right )} x^{2} - 24243\right )} - \frac {32801}{256} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\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 = -0.01 \begin {gather*} \int \frac {x^7\,\left (3\,x^2+2\right )}{\sqrt {x^4+5\,x^2+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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