Optimal. Leaf size=90 \[ -\frac{\left (x^4+5 x^2+3\right )^{3/2}}{9 x^6}-\frac{\left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{18 x^4}+\frac{13 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{36 \sqrt{3}} \]
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Rubi [A] time = 0.0681416, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1251, 806, 720, 724, 206} \[ -\frac{\left (x^4+5 x^2+3\right )^{3/2}}{9 x^6}-\frac{\left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{18 x^4}+\frac{13 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{36 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 806
Rule 720
Rule 724
Rule 206
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \sqrt{3+5 x^2+x^4}}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \sqrt{3+5 x+x^2}}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{9 x^6}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sqrt{3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{18 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{9 x^6}-\frac{13}{36} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{\left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{18 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{9 x^6}+\frac{13}{18} \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+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{18 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{9 x^6}+\frac{13 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{36 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0262554, size = 74, normalized size = 0.82 \[ \frac{1}{108} \left (13 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )-\frac{6 \sqrt{x^4+5 x^2+3} \left (7 x^4+16 x^2+6\right )}{x^6}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 118, normalized size = 1.3 \begin{align*} -{\frac{1}{9\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{5}{54\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{13}{108}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{13\,\sqrt{3}}{108}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{10\,{x}^{2}+25}{108}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{1}{9\,{x}^{6}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43131, size = 134, normalized size = 1.49 \begin{align*} \frac{13}{108} \, \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}{9} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{5 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{18 \, x^{2}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{9 \, x^{4}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{9 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.391, size = 234, normalized size = 2.6 \begin{align*} \frac{13 \, \sqrt{3} x^{6} \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 ) - 42 \, x^{6} - 6 \,{\left (7 \, x^{4} + 16 \, x^{2} + 6\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{108 \, x^{6}} \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^{7}}\, 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^{7}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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