Optimal. Leaf size=121 \[ -\frac{2}{189} \left (2-3 x^2\right )^{7/4}+\frac{4}{27} \left (2-3 x^2\right )^{3/4}+\frac{8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0645026, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {440, 261, 266, 43, 439} \[ -\frac{2}{189} \left (2-3 x^2\right )^{7/4}+\frac{4}{27} \left (2-3 x^2\right )^{3/4}+\frac{8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2-3 x^2}+\sqrt{2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 440
Rule 261
Rule 266
Rule 43
Rule 439
Rubi steps
\begin{align*} \int \frac{x^5}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx &=\int \left (-\frac{4 x}{9 \sqrt [4]{2-3 x^2}}-\frac{x^3}{3 \sqrt [4]{2-3 x^2}}+\frac{16 x}{9 \sqrt [4]{2-3 x^2} \left (4-3 x^2\right )}\right ) \, dx\\ &=-\left (\frac{1}{3} \int \frac{x^3}{\sqrt [4]{2-3 x^2}} \, dx\right )-\frac{4}{9} \int \frac{x}{\sqrt [4]{2-3 x^2}} \, dx+\frac{16}{9} \int \frac{x}{\sqrt [4]{2-3 x^2} \left (4-3 x^2\right )} \, dx\\ &=\frac{8}{81} \left (2-3 x^2\right )^{3/4}+\frac{8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2}+\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{x}{\sqrt [4]{2-3 x}} \, dx,x,x^2\right )\\ &=\frac{8}{81} \left (2-3 x^2\right )^{3/4}+\frac{8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2}+\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )-\frac{1}{6} \operatorname{Subst}\left (\int \left (\frac{2}{3 \sqrt [4]{2-3 x}}-\frac{1}{3} (2-3 x)^{3/4}\right ) \, dx,x,x^2\right )\\ &=\frac{4}{27} \left (2-3 x^2\right )^{3/4}-\frac{2}{189} \left (2-3 x^2\right )^{7/4}+\frac{8}{27} \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt{2}-\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )+\frac{8}{27} \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt{2}+\sqrt{2-3 x^2}}{2^{3/4} \sqrt [4]{2-3 x^2}}\right )\\ \end{align*}
Mathematica [C] time = 0.0215337, size = 42, normalized size = 0.35 \[ \frac{2}{567} \left (2-3 x^2\right )^{3/4} \left (9 \left (x^2+4\right )-56 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};\frac{3 x^2}{2}-1\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{5}}{-3\,{x}^{2}+4}{\frac{1}{\sqrt [4]{-3\,{x}^{2}+2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.52376, size = 189, normalized size = 1.56 \begin{align*} -\frac{2}{189} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{7}{4}} - \frac{8}{27} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{8}{27} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{4}{27} \cdot 2^{\frac{1}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{4}{27} \cdot 2^{\frac{1}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) + \frac{4}{27} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.34583, size = 784, normalized size = 6.48 \begin{align*} \frac{2}{63} \,{\left (x^{2} + 4\right )}{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} + \frac{8}{27} \cdot 8^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{4} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 4 \, \sqrt{2} + 4 \, \sqrt{-3 \, x^{2} + 2}} - \frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} - 1\right ) + \frac{8}{27} \cdot 8^{\frac{1}{4}} \sqrt{2} \arctan \left (\frac{1}{8} \cdot 8^{\frac{1}{4}} \sqrt{2} \sqrt{-4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}} - \frac{1}{2} \cdot 8^{\frac{1}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 1\right ) + \frac{2}{27} \cdot 8^{\frac{1}{4}} \sqrt{2} \log \left (4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}\right ) - \frac{2}{27} \cdot 8^{\frac{1}{4}} \sqrt{2} \log \left (-4 \cdot 8^{\frac{3}{4}} \sqrt{2}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + 16 \, \sqrt{2} + 16 \, \sqrt{-3 \, x^{2} + 2}\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^{5}}{3 x^{2} \sqrt [4]{2 - 3 x^{2}} - 4 \sqrt [4]{2 - 3 x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.72053, size = 189, normalized size = 1.56 \begin{align*} -\frac{2}{189} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{7}{4}} + \frac{1}{27} \cdot 8^{\frac{3}{4}} \log \left (2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{1}{27} \cdot 8^{\frac{3}{4}} \log \left (-2^{\frac{3}{4}}{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}} + \sqrt{2} + \sqrt{-3 \, x^{2} + 2}\right ) - \frac{8}{27} \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} + 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) - \frac{8}{27} \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{1}{2} \cdot 2^{\frac{1}{4}}{\left (2^{\frac{3}{4}} - 2 \,{\left (-3 \, x^{2} + 2\right )}^{\frac{1}{4}}\right )}\right ) + \frac{4}{27} \,{\left (-3 \, x^{2} + 2\right )}^{\frac{3}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]