Optimal. Leaf size=99 \[ -\frac{\sqrt{x^4+5 x^2+3} \left (23 x^2+6\right )}{12 x^4}+\frac{3}{2} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{77 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{24 \sqrt{3}} \]
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Rubi [A] time = 0.0834549, antiderivative size = 99, 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, 810, 843, 621, 206, 724} \[ -\frac{\sqrt{x^4+5 x^2+3} \left (23 x^2+6\right )}{12 x^4}+\frac{3}{2} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{77 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{24 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 810
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^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \sqrt{3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (6+23 x^2\right ) \sqrt{3+5 x^2+x^4}}{12 x^4}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{-77-36 x}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\left (6+23 x^2\right ) \sqrt{3+5 x^2+x^4}}{12 x^4}+\frac{3}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )+\frac{77}{24} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\left (6+23 x^2\right ) \sqrt{3+5 x^2+x^4}}{12 x^4}+3 \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{5+2 x^2}{\sqrt{3+5 x^2+x^4}}\right )-\frac{77}{12} \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 (6+23 x^2\right ) \sqrt{3+5 x^2+x^4}}{12 x^4}+\frac{3}{2} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )-\frac{77 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{24 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0397514, size = 97, normalized size = 0.98 \[ \frac{1}{72} \left (-\frac{6 \sqrt{x^4+5 x^2+3} \left (23 x^2+6\right )}{x^4}+108 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-77 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 121, normalized size = 1.2 \begin{align*} -{\frac{1}{6\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{13}{36\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{77}{72}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{77\,\sqrt{3}}{72}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }+{\frac{26\,{x}^{2}+65}{72}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{3}{2}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47378, size = 143, normalized size = 1.44 \begin{align*} -\frac{77}{72} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{1}{6} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{13 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{12 \, x^{2}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{6 \, x^{4}} + \frac{3}{2} \, \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 = 1.63955, size = 293, normalized size = 2.96 \begin{align*} \frac{77 \, \sqrt{3} x^{4} \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 ) - 108 \, x^{4} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) - 138 \, x^{4} - 6 \, \sqrt{x^{4} + 5 \, x^{2} + 3}{\left (23 \, x^{2} + 6\right )}}{72 \, x^{4}} \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^{5}}\, 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^{5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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