Optimal. Leaf size=132 \[ \frac{173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac{161 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 \sqrt{3}} \]
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Rubi [A] time = 0.108624, antiderivative size = 132, 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, 834, 806, 720, 724, 206} \[ \frac{173 \left (x^4+5 x^2+3\right )^{3/2}}{3240 x^6}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{36 x^8}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{15 x^{10}}-\frac{161 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{5184 x^4}+\frac{2093 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 834
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^{11}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \sqrt{3+5 x+x^2}}{x^6} \, dx,x,x^2\right )\\ &=-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{1}{30} \operatorname{Subst}\left (\int \frac{(-10+4 x) \sqrt{3+5 x+x^2}}{x^5} \, dx,x,x^2\right )\\ &=-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{1}{360} \operatorname{Subst}\left (\int \frac{(-173-10 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}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac{161}{432} \operatorname{Subst}\left (\int \frac{\sqrt{3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{161 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{5184 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}-\frac{2093 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )}{10368}\\ &=-\frac{161 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{5184 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac{2093 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{6+5 x^2}{\sqrt{3+5 x^2+x^4}}\right )}{5184}\\ &=-\frac{161 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{5184 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{15 x^{10}}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{36 x^8}+\frac{173 \left (3+5 x^2+x^4\right )^{3/2}}{3240 x^6}+\frac{2093 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{10368 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0370601, size = 84, normalized size = 0.64 \[ \frac{10465 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )-\frac{6 \sqrt{x^4+5 x^2+3} \left (2641 x^8-1370 x^6+1176 x^4+10800 x^2+5184\right )}{x^{10}}}{155520} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 152, normalized size = 1.2 \begin{align*} -{\frac{1}{15\,{x}^{10}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{1}{36\,{x}^{8}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{173}{3240\,{x}^{6}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{161}{2592\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{805}{15552\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{2093}{31104}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{2093\,\sqrt{3}}{31104}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{1610\,{x}^{2}+4025}{31104}\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 = 1.50393, size = 180, normalized size = 1.36 \begin{align*} \frac{2093}{31104} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{161}{2592} \, \sqrt{x^{4} + 5 \, x^{2} + 3} + \frac{805 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{5184 \, x^{2}} - \frac{161 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{2592 \, x^{4}} + \frac{173 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{3240 \, x^{6}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{36 \, x^{8}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{15 \, x^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33106, size = 292, normalized size = 2.21 \begin{align*} \frac{10465 \, \sqrt{3} x^{10} \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 ) - 15846 \, x^{10} - 6 \,{\left (2641 \, x^{8} - 1370 \, x^{6} + 1176 \, x^{4} + 10800 \, x^{2} + 5184\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{155520 \, x^{10}} \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^{11}}\, 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^{11}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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