Optimal. Leaf size=127 \[ -\frac{\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{4 x^4}-\frac{3 \left (28-19 x^2\right ) \sqrt{x^4+5 x^2+3}}{8 x^2}+\frac{453}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{127}{8} \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]
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Rubi [A] time = 0.108955, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{\left (2-3 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{4 x^4}-\frac{3 \left (28-19 x^2\right ) \sqrt{x^4+5 x^2+3}}{8 x^2}+\frac{453}{16} \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-\frac{127}{8} \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right ) \]
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 ) \left (3+5 x^2+x^4\right )^{3/2}}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 x) \left (3+5 x+x^2\right )^{3/2}}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\left (2-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{4 x^4}-\frac{3}{16} \operatorname{Subst}\left (\int \frac{(-56-38 x) \sqrt{3+5 x+x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{3 \left (28-19 x^2\right ) \sqrt{3+5 x^2+x^4}}{8 x^2}-\frac{\left (2-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{4 x^4}+\frac{3}{32} \operatorname{Subst}\left (\int \frac{508+302 x}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{3 \left (28-19 x^2\right ) \sqrt{3+5 x^2+x^4}}{8 x^2}-\frac{\left (2-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{4 x^4}+\frac{453}{16} \operatorname{Subst}\left (\int \frac{1}{\sqrt{3+5 x+x^2}} \, dx,x,x^2\right )+\frac{381}{8} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )\\ &=-\frac{3 \left (28-19 x^2\right ) \sqrt{3+5 x^2+x^4}}{8 x^2}-\frac{\left (2-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{4 x^4}+\frac{453}{8} \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{381}{4} \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{3 \left (28-19 x^2\right ) \sqrt{3+5 x^2+x^4}}{8 x^2}-\frac{\left (2-3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{4 x^4}+\frac{453}{16} \tanh ^{-1}\left (\frac{5+2 x^2}{2 \sqrt{3+5 x^2+x^4}}\right )-\frac{127}{8} \sqrt{3} \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )\\ \end{align*}
Mathematica [A] time = 0.0650273, size = 107, normalized size = 0.84 \[ \frac{1}{16} \left (\frac{2 \sqrt{x^4+5 x^2+3} \left (6 x^6+83 x^4-86 x^2-12\right )}{x^4}+453 \tanh ^{-1}\left (\frac{2 x^2+5}{2 \sqrt{x^4+5 x^2+3}}\right )-254 \sqrt{3} \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 117, normalized size = 0.9 \begin{align*}{\frac{83}{8}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{453}{16}\ln \left ({\frac{5}{2}}+{x}^{2}+\sqrt{{x}^{4}+5\,{x}^{2}+3} \right ) }-{\frac{43}{4\,{x}^{2}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{127\,\sqrt{3}}{8}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }-{\frac{3}{2\,{x}^{4}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+{\frac{3\,{x}^{2}}{4}\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 = 1.48336, size = 185, normalized size = 1.46 \begin{align*} \frac{7}{2} \, \sqrt{x^{4} + 5 \, x^{2} + 3} x^{2} + \frac{1}{6} \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}} - \frac{127}{8} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) + \frac{197}{8} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{23 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{12 \, x^{2}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{5}{2}}}{6 \, x^{4}} + \frac{453}{16} \, \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 = 1.37994, size = 320, normalized size = 2.52 \begin{align*} \frac{1016 \, \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 ) - 1812 \, x^{4} \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} + 5 \, x^{2} + 3} - 5\right ) + 67 \, x^{4} + 8 \,{\left (6 \, x^{6} + 83 \, x^{4} - 86 \, x^{2} - 12\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{64 \, 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 ) \left (x^{4} + 5 x^{2} + 3\right )^{\frac{3}{2}}}{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{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}{\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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