Optimal. Leaf size=331 \[ \frac{103 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) \text{EllipticF}\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right ),\frac{1}{6} \left (5 \sqrt{13}-13\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}-\frac{\left (2-5 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{5 x^5}-\frac{\left (40-87 x^2\right ) \sqrt{x^4+5 x^2+3}}{5 x^3}-\frac{722 \sqrt{x^4+5 x^2+3}}{15 x}+\frac{361 x \left (2 x^2+\sqrt{13}+5\right )}{15 \sqrt{x^4+5 x^2+3}}-\frac{361 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{x^4+5 x^2+3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.202751, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1271, 1281, 1189, 1099, 1135} \[ -\frac{\left (2-5 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}}{5 x^5}-\frac{\left (40-87 x^2\right ) \sqrt{x^4+5 x^2+3}}{5 x^3}-\frac{722 \sqrt{x^4+5 x^2+3}}{15 x}+\frac{361 x \left (2 x^2+\sqrt{13}+5\right )}{15 \sqrt{x^4+5 x^2+3}}+\frac{103 \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{x^4+5 x^2+3}}-\frac{361 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{\left (5-\sqrt{13}\right ) x^2+6}{\left (5+\sqrt{13}\right ) x^2+6}} \left (\left (5+\sqrt{13}\right ) x^2+6\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{x^4+5 x^2+3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1271
Rule 1281
Rule 1189
Rule 1099
Rule 1135
Rubi steps
\begin{align*} \int \frac{\left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{x^6} \, dx &=-\frac{\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac{1}{5} \int \frac{\left (-120-87 x^2\right ) \sqrt{3+5 x^2+x^4}}{x^4} \, dx\\ &=-\frac{\left (40-87 x^2\right ) \sqrt{3+5 x^2+x^4}}{5 x^3}-\frac{\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}+\frac{1}{15} \int \frac{2166+1545 x^2}{x^2 \sqrt{3+5 x^2+x^4}} \, dx\\ &=-\frac{722 \sqrt{3+5 x^2+x^4}}{15 x}-\frac{\left (40-87 x^2\right ) \sqrt{3+5 x^2+x^4}}{5 x^3}-\frac{\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac{1}{45} \int \frac{-4635-2166 x^2}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=-\frac{722 \sqrt{3+5 x^2+x^4}}{15 x}-\frac{\left (40-87 x^2\right ) \sqrt{3+5 x^2+x^4}}{5 x^3}-\frac{\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}+\frac{722}{15} \int \frac{x^2}{\sqrt{3+5 x^2+x^4}} \, dx+103 \int \frac{1}{\sqrt{3+5 x^2+x^4}} \, dx\\ &=\frac{361 x \left (5+\sqrt{13}+2 x^2\right )}{15 \sqrt{3+5 x^2+x^4}}-\frac{722 \sqrt{3+5 x^2+x^4}}{15 x}-\frac{\left (40-87 x^2\right ) \sqrt{3+5 x^2+x^4}}{5 x^3}-\frac{\left (2-5 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}}{5 x^5}-\frac{361 \sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} \sqrt{\frac{6+\left (5-\sqrt{13}\right ) x^2}{6+\left (5+\sqrt{13}\right ) x^2}} \left (6+\left (5+\sqrt{13}\right ) x^2\right ) E\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{15 \sqrt{3+5 x^2+x^4}}+\frac{103 \sqrt{\frac{6+\left (5-\sqrt{13}\right ) x^2}{6+\left (5+\sqrt{13}\right ) x^2}} \left (6+\left (5+\sqrt{13}\right ) x^2\right ) F\left (\tan ^{-1}\left (\sqrt{\frac{1}{6} \left (5+\sqrt{13}\right )} x\right )|\frac{1}{6} \left (-13+5 \sqrt{13}\right )\right )}{\sqrt{6 \left (5+\sqrt{13}\right )} \sqrt{3+5 x^2+x^4}}\\ \end{align*}
Mathematica [C] time = 0.316542, size = 244, normalized size = 0.74 \[ \frac{-i \sqrt{2} \left (361 \sqrt{13}-260\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^5 \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right ),\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )+30 x^{10}-634 x^8-4040 x^6-3438 x^4-810 x^2+361 i \sqrt{2} \left (\sqrt{13}-5\right ) \sqrt{\frac{-2 x^2+\sqrt{13}-5}{\sqrt{13}-5}} \sqrt{2 x^2+\sqrt{13}+5} x^5 E\left (i \sinh ^{-1}\left (\sqrt{\frac{2}{5+\sqrt{13}}} x\right )|\frac{19}{6}+\frac{5 \sqrt{13}}{6}\right )-108}{30 x^5 \sqrt{x^4+5 x^2+3}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.02, size = 259, normalized size = 0.8 \begin{align*} -{\frac{6}{5\,{x}^{5}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-7\,{\frac{\sqrt{{x}^{4}+5\,{x}^{2}+3}}{{x}^{3}}}-{\frac{392}{15\,x}\sqrt{{x}^{4}+5\,{x}^{2}+3}}+618\,{\frac{\sqrt{1- \left ( -5/6+1/6\,\sqrt{13} \right ){x}^{2}}\sqrt{1- \left ( -5/6-1/6\,\sqrt{13} \right ){x}^{2}}{\it EllipticF} \left ( 1/6\,x\sqrt{-30+6\,\sqrt{13}},5/6\,\sqrt{3}+1/6\,\sqrt{39} \right ) }{\sqrt{-30+6\,\sqrt{13}}\sqrt{{x}^{4}+5\,{x}^{2}+3}}}-{\frac{8664}{5\,\sqrt{-30+6\,\sqrt{13}} \left ( \sqrt{13}+5 \right ) }\sqrt{1- \left ( -{\frac{5}{6}}+{\frac{\sqrt{13}}{6}} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{5}{6}}-{\frac{\sqrt{13}}{6}} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-30+6\,\sqrt{13}}}{6}},{\frac{5\,\sqrt{3}}{6}}+{\frac{\sqrt{39}}{6}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}}+x\sqrt{{x}^{4}+5\,{x}^{2}+3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (3 \, x^{6} + 17 \, x^{4} + 19 \, x^{2} + 6\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]