Optimal. Leaf size=244 \[ \frac {2}{45} x \left (-1+3 x^2\right )^{3/4}+\frac {8 x \sqrt [4]{-1+3 x^2}}{15 \left (1+\sqrt {-1+3 x^2}\right )}-\frac {1}{9} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{9} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {8 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}+\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.12, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {451, 236, 311,
226, 1210, 327, 407} \begin {gather*} -\frac {1}{9} \sqrt {\frac {2}{3}} \text {ArcTan}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right )+\frac {4 \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) F\left (2 \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}-\frac {8 \sqrt {\frac {x^2}{\left (\sqrt {3 x^2-1}+1\right )^2}} \left (\sqrt {3 x^2-1}+1\right ) E\left (2 \text {ArcTan}\left (\sqrt [4]{3 x^2-1}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}+\frac {2}{45} \left (3 x^2-1\right )^{3/4} x+\frac {8 \sqrt [4]{3 x^2-1} x}{15 \left (\sqrt {3 x^2-1}+1\right )}-\frac {1}{9} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{3 x^2-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 226
Rule 236
Rule 311
Rule 327
Rule 407
Rule 451
Rule 1210
Rubi steps
\begin {align*} \int \frac {x^4}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx &=\int \left (\frac {2}{9 \sqrt [4]{-1+3 x^2}}+\frac {x^2}{3 \sqrt [4]{-1+3 x^2}}+\frac {4}{9 \left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}}\right ) \, dx\\ &=\frac {2}{9} \int \frac {1}{\sqrt [4]{-1+3 x^2}} \, dx+\frac {1}{3} \int \frac {x^2}{\sqrt [4]{-1+3 x^2}} \, dx+\frac {4}{9} \int \frac {1}{\left (-2+3 x^2\right ) \sqrt [4]{-1+3 x^2}} \, dx\\ &=\frac {2}{45} x \left (-1+3 x^2\right )^{3/4}-\frac {1}{9} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{9} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {2}{45} \int \frac {1}{\sqrt [4]{-1+3 x^2}} \, dx+\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{9 \sqrt {3} x}\\ &=\frac {2}{45} x \left (-1+3 x^2\right )^{3/4}-\frac {1}{9} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{9} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )+\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{45 \sqrt {3} x}+\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{9 \sqrt {3} x}-\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{9 \sqrt {3} x}\\ &=\frac {2}{45} x \left (-1+3 x^2\right )^{3/4}+\frac {4 x \sqrt [4]{-1+3 x^2}}{9 \left (1+\sqrt {-1+3 x^2}\right )}-\frac {1}{9} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{9} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x}+\frac {2 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{9 \sqrt {3} x}+\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{45 \sqrt {3} x}-\frac {\left (4 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\sqrt [4]{-1+3 x^2}\right )}{45 \sqrt {3} x}\\ &=\frac {2}{45} x \left (-1+3 x^2\right )^{3/4}+\frac {8 x \sqrt [4]{-1+3 x^2}}{15 \left (1+\sqrt {-1+3 x^2}\right )}-\frac {1}{9} \sqrt {\frac {2}{3}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {1}{9} \sqrt {\frac {2}{3}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt [4]{-1+3 x^2}}\right )-\frac {8 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) E\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}+\frac {4 \sqrt {\frac {x^2}{\left (1+\sqrt {-1+3 x^2}\right )^2}} \left (1+\sqrt {-1+3 x^2}\right ) F\left (2 \tan ^{-1}\left (\sqrt [4]{-1+3 x^2}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 6.14, size = 177, normalized size = 0.73 \begin {gather*} \frac {2 x \left (-1+3 x^2-3 x^2 \sqrt [4]{1-3 x^2} F_1\left (\frac {3}{2};\frac {1}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )-\frac {4 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};3 x^2,\frac {3 x^2}{2}\right )}{\left (-2+3 x^2\right ) \left (2 F_1\left (\frac {1}{2};\frac {1}{4},1;\frac {3}{2};3 x^2,\frac {3 x^2}{2}\right )+x^2 \left (2 F_1\left (\frac {3}{2};\frac {1}{4},2;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )+F_1\left (\frac {3}{2};\frac {5}{4},1;\frac {5}{2};3 x^2,\frac {3 x^2}{2}\right )\right )\right )}\right )}{45 \sqrt [4]{-1+3 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{4}}{\left (3 x^{2}-2\right ) \left (3 x^{2}-1\right )^{\frac {1}{4}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4}}{\left (3 x^{2} - 2\right ) \sqrt [4]{3 x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4}{{\left (3\,x^2-1\right )}^{1/4}\,\left (3\,x^2-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________