Optimal. Leaf size=61 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.016959, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-3 x^2-1}}\right )}{3 \sqrt{6}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 442
Rubi steps
\begin{align*} \int \frac{x^2}{\left (-2-3 x^2\right ) \left (-1-3 x^2\right )^{3/4}} \, dx &=\frac{\tan ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{\frac{3}{2}} x}{\sqrt [4]{-1-3 x^2}}\right )}{3 \sqrt{6}}\\ \end{align*}
Mathematica [C] time = 0.0417319, size = 52, normalized size = 0.85 \[ -\frac{x^3 \left (3 x^2+1\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-3 x^2,-\frac{3 x^2}{2}\right )}{6 \left (-3 x^2-1\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.04, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-3\,{x}^{2}-2} \left ( -3\,{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \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{x^{2}}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 1.66068, size = 343, normalized size = 5.62 \begin{align*} -\frac{1}{36} \, \sqrt{6} \log \left (\frac{\sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{1}{36} \, \sqrt{6} \log \left (-\frac{\sqrt{6} x - 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) - \frac{1}{36} i \, \sqrt{6} \log \left (\frac{i \, \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, x}\right ) + \frac{1}{36} i \, \sqrt{6} \log \left (\frac{-i \, \sqrt{6} x + 2 \,{\left (-3 \, x^{2} - 1\right )}^{\frac{1}{4}}}{2 \, 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{x^{2}}{3 x^{2} \left (- 3 x^{2} - 1\right )^{\frac{3}{4}} + 2 \left (- 3 x^{2} - 1\right )^{\frac{3}{4}}}\, 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{x^{2}}{{\left (3 \, x^{2} + 2\right )}{\left (-3 \, x^{2} - 1\right )}^{\frac{3}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]