Optimal. Leaf size=67 \[ \frac{5 \sqrt{3} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right ),2\right )}{8 \sqrt [4]{2}}+\frac{5 \sqrt [4]{3 x^2+2}}{8 x}-\frac{\sqrt [4]{3 x^2+2}}{6 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0148175, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {325, 231} \[ \frac{5 \sqrt [4]{3 x^2+2}}{8 x}-\frac{\sqrt [4]{3 x^2+2}}{6 x^3}+\frac{5 \sqrt{3} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8 \sqrt [4]{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 325
Rule 231
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (2+3 x^2\right )^{3/4}} \, dx &=-\frac{\sqrt [4]{2+3 x^2}}{6 x^3}-\frac{5}{4} \int \frac{1}{x^2 \left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{2+3 x^2}}{6 x^3}+\frac{5 \sqrt [4]{2+3 x^2}}{8 x}+\frac{15}{16} \int \frac{1}{\left (2+3 x^2\right )^{3/4}} \, dx\\ &=-\frac{\sqrt [4]{2+3 x^2}}{6 x^3}+\frac{5 \sqrt [4]{2+3 x^2}}{8 x}+\frac{5 \sqrt{3} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\sqrt{\frac{3}{2}} x\right )\right |2\right )}{8 \sqrt [4]{2}}\\ \end{align*}
Mathematica [C] time = 0.0048023, size = 29, normalized size = 0.43 \[ -\frac{\, _2F_1\left (-\frac{3}{2},\frac{3}{4};-\frac{1}{2};-\frac{3 x^2}{2}\right )}{3\ 2^{3/4} x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.021, size = 45, normalized size = 0.7 \begin{align*}{\frac{45\,{x}^{4}+18\,{x}^{2}-8}{24\,{x}^{3}} \left ( 3\,{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}+{\frac{15\,\sqrt [4]{2}x}{32}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{3}{2}};\,-{\frac{3\,{x}^{2}}{2}})}} \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{1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}} x^{4}}\,{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^{2} + 2\right )}^{\frac{1}{4}}}{3 \, x^{6} + 2 \, x^{4}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 0.963091, size = 32, normalized size = 0.48 \begin{align*} - \frac{\sqrt [4]{2}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{3 x^{2} e^{i \pi }}{2}} \right )}}{6 x^{3}} \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{1}{{\left (3 \, x^{2} + 2\right )}^{\frac{3}{4}} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]