Optimal. Leaf size=312 \[ -\frac {\left (14-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{7 x}+\frac {1}{35} x \left (129 x^2+655\right ) \sqrt {x^4+5 x^2+3}+\frac {412 x \left (2 x^2+\sqrt {13}+5\right )}{35 \sqrt {x^4+5 x^2+3}}+\frac {19 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {206 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{35 \sqrt {x^4+5 x^2+3}} \]
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Rubi [A] time = 0.15, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1271, 1176, 1189, 1099, 1135} \[ -\frac {\left (14-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{7 x}+\frac {1}{35} x \left (129 x^2+655\right ) \sqrt {x^4+5 x^2+3}+\frac {412 x \left (2 x^2+\sqrt {13}+5\right )}{35 \sqrt {x^4+5 x^2+3}}+\frac {19 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {x^4+5 x^2+3}}-\frac {206 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {\left (5-\sqrt {13}\right ) x^2+6}{\left (5+\sqrt {13}\right ) x^2+6}} \left (\left (5+\sqrt {13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{35 \sqrt {x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
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Rule 1099
Rule 1135
Rule 1176
Rule 1189
Rule 1271
Rubi steps
\begin {align*} \int \frac {\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^2} \, dx &=-\frac {\left (14-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{7 x}-\frac {3}{7} \int \left (-88-43 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx\\ &=\frac {1}{35} x \left (655+129 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {\left (14-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{7 x}-\frac {1}{35} \int \frac {-1995-824 x^2}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {1}{35} x \left (655+129 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {\left (14-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{7 x}+\frac {824}{35} \int \frac {x^2}{\sqrt {3+5 x^2+x^4}} \, dx+57 \int \frac {1}{\sqrt {3+5 x^2+x^4}} \, dx\\ &=\frac {412 x \left (5+\sqrt {13}+2 x^2\right )}{35 \sqrt {3+5 x^2+x^4}}+\frac {1}{35} x \left (655+129 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {\left (14-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{7 x}-\frac {206 \sqrt {\frac {2}{3} \left (5+\sqrt {13}\right )} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{35 \sqrt {3+5 x^2+x^4}}+\frac {19 \sqrt {\frac {3}{2 \left (5+\sqrt {13}\right )}} \sqrt {\frac {6+\left (5-\sqrt {13}\right ) x^2}{6+\left (5+\sqrt {13}\right ) x^2}} \left (6+\left (5+\sqrt {13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt {\frac {1}{6} \left (5+\sqrt {13}\right )} x\right )|\frac {1}{6} \left (-13+5 \sqrt {13}\right )\right )}{\sqrt {3+5 x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 235, normalized size = 0.75 \[ \frac {30 x^{10}+418 x^8+2130 x^6+3884 x^4-i \sqrt {2} \left (412 \sqrt {13}-65\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} x F\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )+412 i \sqrt {2} \left (\sqrt {13}-5\right ) \sqrt {\frac {-2 x^2+\sqrt {13}-5}{\sqrt {13}-5}} \sqrt {2 x^2+\sqrt {13}+5} x E\left (i \sinh ^{-1}\left (\sqrt {\frac {2}{5+\sqrt {13}}} x\right )|\frac {19}{6}+\frac {5 \sqrt {13}}{6}\right )-1260}{70 x \sqrt {x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{6} + 17 \, x^{4} + 19 \, x^{2} + 6\right )} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 260, normalized size = 0.83 \[ \frac {3 \sqrt {x^{4}+5 x^{2}+3}\, x^{5}}{7}+\frac {134 \sqrt {x^{4}+5 x^{2}+3}\, x^{3}}{35}+10 \sqrt {x^{4}+5 x^{2}+3}\, x +\frac {342 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )}{\sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}}-\frac {6 \sqrt {x^{4}+5 x^{2}+3}}{x}-\frac {29664 \sqrt {-\left (-\frac {5}{6}+\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {5}{6}-\frac {\sqrt {13}}{6}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )+\EllipticF \left (\frac {\sqrt {-30+6 \sqrt {13}}\, x}{6}, \frac {5 \sqrt {3}}{6}+\frac {\sqrt {39}}{6}\right )\right )}{35 \sqrt {-30+6 \sqrt {13}}\, \sqrt {x^{4}+5 x^{2}+3}\, \left (\sqrt {13}+5\right )} \]
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 \[ \int \frac {{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} {\left (3 \, x^{2} + 2\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{x^2} \,d x \]
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 \[ \int \frac {\left (3 x^{2} + 2\right ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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