Optimal. Leaf size=111 \[ -\frac{11 \left (x^4+5 x^2+3\right )^{3/2}}{216 x^6}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{12 x^8}+\frac{67 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{1728 x^4}-\frac{871 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{3456 \sqrt{3}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0861399, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ -\frac{11 \left (x^4+5 x^2+3\right )^{3/2}}{216 x^6}-\frac{\left (x^4+5 x^2+3\right )^{3/2}}{12 x^8}+\frac{67 \left (5 x^2+6\right ) \sqrt{x^4+5 x^2+3}}{1728 x^4}-\frac{871 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{3456 \sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
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^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(2+3 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}}{12 x^8}-\frac{1}{24} \operatorname{Subst}\left (\int \frac{(-11+2 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}}{12 x^8}-\frac{11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac{67}{144} \operatorname{Subst}\left (\int \frac{\sqrt{3+5 x+x^2}}{x^3} \, dx,x,x^2\right )\\ &=\frac{67 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{1728 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac{11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}+\frac{871 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{3+5 x+x^2}} \, dx,x,x^2\right )}{3456}\\ &=\frac{67 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{1728 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac{11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac{871 \operatorname{Subst}\left (\int \frac{1}{12-x^2} \, dx,x,\frac{6+5 x^2}{\sqrt{3+5 x^2+x^4}}\right )}{1728}\\ &=\frac{67 \left (6+5 x^2\right ) \sqrt{3+5 x^2+x^4}}{1728 x^4}-\frac{\left (3+5 x^2+x^4\right )^{3/2}}{12 x^8}-\frac{11 \left (3+5 x^2+x^4\right )^{3/2}}{216 x^6}-\frac{871 \tanh ^{-1}\left (\frac{6+5 x^2}{2 \sqrt{3} \sqrt{3+5 x^2+x^4}}\right )}{3456 \sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0282441, size = 82, normalized size = 0.74 \[ \frac{6 \sqrt{x^4+5 x^2+3} \left (247 x^6-182 x^4-984 x^2-432\right )-871 \sqrt{3} x^8 \tanh ^{-1}\left (\frac{5 x^2+6}{2 \sqrt{3} \sqrt{x^4+5 x^2+3}}\right )}{10368 x^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.016, size = 135, normalized size = 1.2 \begin{align*} -{\frac{1}{12\,{x}^{8}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{11}{216\,{x}^{6}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{67}{864\,{x}^{4}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}-{\frac{335}{5184\,{x}^{2}} \left ({x}^{4}+5\,{x}^{2}+3 \right ) ^{{\frac{3}{2}}}}+{\frac{871}{10368}\sqrt{{x}^{4}+5\,{x}^{2}+3}}-{\frac{871\,\sqrt{3}}{10368}{\it Artanh} \left ({\frac{ \left ( 5\,{x}^{2}+6 \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{{x}^{4}+5\,{x}^{2}+3}}}} \right ) }+{\frac{670\,{x}^{2}+1675}{10368}\sqrt{{x}^{4}+5\,{x}^{2}+3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.46421, size = 157, normalized size = 1.41 \begin{align*} -\frac{871}{10368} \, \sqrt{3} \log \left (\frac{2 \, \sqrt{3} \sqrt{x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac{6}{x^{2}} + 5\right ) - \frac{67}{864} \, \sqrt{x^{4} + 5 \, x^{2} + 3} - \frac{335 \, \sqrt{x^{4} + 5 \, x^{2} + 3}}{1728 \, x^{2}} + \frac{67 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{864 \, x^{4}} - \frac{11 \,{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{216 \, x^{6}} - \frac{{\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac{3}{2}}}{12 \, x^{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41226, size = 261, normalized size = 2.35 \begin{align*} \frac{871 \, \sqrt{3} x^{8} \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 ) + 1482 \, x^{8} + 6 \,{\left (247 \, x^{6} - 182 \, x^{4} - 984 \, x^{2} - 432\right )} \sqrt{x^{4} + 5 \, x^{2} + 3}}{10368 \, x^{8}} \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 ) \sqrt{x^{4} + 5 x^{2} + 3}}{x^{9}}\, 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{\sqrt{x^{4} + 5 \, x^{2} + 3}{\left (3 \, x^{2} + 2\right )}}{x^{9}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]