Optimal. Leaf size=106 \[ -\frac {1}{40} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}+\frac {123}{128} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {4797 \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}}{1024}+\frac {62361 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )}{2048} \]
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Rubi [A] time = 0.07, antiderivative size = 106, 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 = {1251, 779, 612, 621, 206} \[ -\frac {1}{40} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}+\frac {123}{128} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {4797 \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}}{1024}+\frac {62361 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )}{2048} \]
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 621
Rule 779
Rule 1251
Rubi steps
\begin {align*} \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (2+3 x) \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=-\frac {1}{40} \left (27-10 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}+\frac {123}{16} \operatorname {Subst}\left (\int \left (3+5 x+x^2\right )^{3/2} \, dx,x,x^2\right )\\ &=\frac {123}{128} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {1}{40} \left (27-10 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}-\frac {4797}{256} \operatorname {Subst}\left (\int \sqrt {3+5 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {4797 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{1024}+\frac {123}{128} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {1}{40} \left (27-10 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}+\frac {62361 \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+5 x+x^2}} \, dx,x,x^2\right )}{2048}\\ &=-\frac {4797 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{1024}+\frac {123}{128} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {1}{40} \left (27-10 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}+\frac {62361 \operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {5+2 x^2}{\sqrt {3+5 x^2+x^4}}\right )}{1024}\\ &=-\frac {4797 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{1024}+\frac {123}{128} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {1}{40} \left (27-10 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}+\frac {62361 \tanh ^{-1}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )}{2048}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 76, normalized size = 0.72 \[ \frac {311805 \tanh ^{-1}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )+2 \sqrt {x^4+5 x^2+3} \left (1280 x^{10}+9344 x^8+14960 x^6+5064 x^4+12390 x^2-77229\right )}{10240} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 66, normalized size = 0.62 \[ \frac {1}{5120} \, {\left (1280 \, x^{10} + 9344 \, x^{8} + 14960 \, x^{6} + 5064 \, x^{4} + 12390 \, x^{2} - 77229\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {62361}{2048} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 179, normalized size = 1.69 \[ \frac {1}{1024} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, x^{2} + 1\right )} x^{2} - 33\right )} x^{2} + 321\right )} x^{2} - 6837\right )} x^{2} + 87147\right )} + \frac {17}{3840} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + 5\right )} x^{2} - 127\right )} x^{2} + 2635\right )} x^{2} - 33429\right )} + \frac {19}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {1}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} + 5\right )} x^{2} - 51\right )} - \frac {62361}{2048} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 121, normalized size = 1.14 \[ \frac {\sqrt {x^{4}+5 x^{2}+3}\, x^{10}}{4}+\frac {73 \sqrt {x^{4}+5 x^{2}+3}\, x^{8}}{40}+\frac {187 \sqrt {x^{4}+5 x^{2}+3}\, x^{6}}{64}+\frac {633 \sqrt {x^{4}+5 x^{2}+3}\, x^{4}}{640}+\frac {1239 \sqrt {x^{4}+5 x^{2}+3}\, x^{2}}{512}+\frac {62361 \ln \left (x^{2}+\frac {5}{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2048}-\frac {77229 \sqrt {x^{4}+5 x^{2}+3}}{5120} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.88, size = 118, normalized size = 1.11 \[ \frac {1}{4} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} x^{2} + \frac {123}{64} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} - \frac {27}{40} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} - \frac {4797}{512} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} + \frac {615}{128} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} - \frac {23985}{1024} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {62361}{2048} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
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 \[ \int x^3\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]
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 \[ \int x^{3} \left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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