Optimal. Leaf size=97 \[ -\frac{\sqrt{x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{7 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0830542, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1251, 812, 843, 621, 206, 724} \[ -\frac{\sqrt{x^4+5 x^2+3} \left (2-3 x^2\right )}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{7 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 812
Rule 843
Rule 621
Rule 206
Rule 724
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \sqrt{3+5 x+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\left (2-3 x^2\right ) \sqrt{3+5 x^2+x^4}}{2 x^2}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{-28-19 x}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\left (2-3 x^2\right ) \sqrt{3+5 x^2+x^4}}{2 x^2}+\frac{19}{4} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )+7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\left (2-3 x^2\right ) \sqrt{3+5 x^2+x^4}}{2 x^2}+\frac{19}{2} \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{5+2 x^2}{\sqrt{3+5 x^2+x^4}}\right )-14 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{6+5 x^2}{\sqrt{3+5 x^2+x^4}}\right )\\ &=-\frac{\left (2-3 x^2\right ) \sqrt{3+5 x^2+x^4}}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )-\frac{7 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0401172, size = 97, normalized size = 1. \[ \frac{\sqrt{x^4+5 x^2+3} \left (3 x^2-2\right )}{2 x^2}+\frac{19}{4} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{7 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 104, normalized size = 1.1 \begin{align*}{\frac{7}{3}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{19}{4}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-{\frac{7\,\sqrt{3}}{3}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1}{3\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{x}^{2}+5}{6}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46671, size = 120, normalized size = 1.24 \begin{align*} -\frac{7}{3} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{3}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{19}{4} \, \log \left (2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52972, size = 292, normalized size = 3.01 \begin{align*} \frac{56 \, \sqrt{3} x^{2} \log \left (\frac{25 \, x^{2} - 2 \, \sqrt{3}{\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (5 \, \sqrt{3} - 6\right )} + 30}{x^{2}}\right ) - 114 \, x^{2} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 21 \, x^{2} + 12 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} - 2\right )}}{24 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x^{2} + 2\right ) \sqrt{x^{4} + 5 x^{2} + 3}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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