Optimal. Leaf size=38 \[ \frac {3 \left (x^5-1\right )^{2/3} \left (5 x^{10}+8 x^8-10 x^5-8 x^3+5\right )}{40 x^8} \]
________________________________________________________________________________________
Rubi [B] time = 0.17, antiderivative size = 89, normalized size of antiderivative = 2.34, number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1826, 1835, 1586, 1584, 449} \begin {gather*} \frac {6 \left (x^5-1\right )^{2/3}}{5 x^5}-\frac {15 \left (x^5-1\right )^{2/3}}{56 x^8}-\frac {15 \left (x^5-1\right )^{2/3}}{4 x^3}+\frac {3 \left (x^5-1\right )^{2/3} \left (35 x^{11}+56 x^9+280 x^6-168 x^4+60 x\right )}{280 x^9} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 449
Rule 1584
Rule 1586
Rule 1826
Rule 1835
Rubi steps
\begin {align*} \int \frac {\left (-1+x^5\right )^{2/3} \left (-1+x^3+x^5\right ) \left (3+2 x^5\right )}{x^9} \, dx &=\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}-\frac {10}{3} \int \frac {\frac {9}{14}-\frac {9 x^3}{5}+3 x^5+\frac {3 x^8}{5}+\frac {3 x^{10}}{8}}{x^9 \sqrt [3]{-1+x^5}} \, dx\\ &=-\frac {15 \left (-1+x^5\right )^{2/3}}{56 x^8}+\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}-\frac {5}{24} \int \frac {-\frac {144 x^2}{5}+54 x^4+\frac {48 x^7}{5}+6 x^9}{x^8 \sqrt [3]{-1+x^5}} \, dx\\ &=-\frac {15 \left (-1+x^5\right )^{2/3}}{56 x^8}+\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}-\frac {5}{24} \int \frac {-\frac {144 x}{5}+54 x^3+\frac {48 x^6}{5}+6 x^8}{x^7 \sqrt [3]{-1+x^5}} \, dx\\ &=-\frac {15 \left (-1+x^5\right )^{2/3}}{56 x^8}+\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}-\frac {5}{24} \int \frac {-\frac {144}{5}+54 x^2+\frac {48 x^5}{5}+6 x^7}{x^6 \sqrt [3]{-1+x^5}} \, dx\\ &=-\frac {15 \left (-1+x^5\right )^{2/3}}{56 x^8}+\frac {6 \left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}-\frac {1}{48} \int \frac {540 x+60 x^6}{x^5 \sqrt [3]{-1+x^5}} \, dx\\ &=-\frac {15 \left (-1+x^5\right )^{2/3}}{56 x^8}+\frac {6 \left (-1+x^5\right )^{2/3}}{5 x^5}+\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}-\frac {1}{48} \int \frac {540+60 x^5}{x^4 \sqrt [3]{-1+x^5}} \, dx\\ &=-\frac {15 \left (-1+x^5\right )^{2/3}}{56 x^8}+\frac {6 \left (-1+x^5\right )^{2/3}}{5 x^5}-\frac {15 \left (-1+x^5\right )^{2/3}}{4 x^3}+\frac {3 \left (-1+x^5\right )^{2/3} \left (60 x-168 x^4+280 x^6+56 x^9+35 x^{11}\right )}{280 x^9}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.12, size = 139, normalized size = 3.66 \begin {gather*} \frac {\left (x^5-1\right )^{2/3} \left (225 \, _2F_1\left (-\frac {8}{5},-\frac {2}{3};-\frac {3}{5};x^5\right )+8 x^5 \left (75 x^5 \, _2F_1\left (-\frac {2}{3},\frac {2}{5};\frac {7}{5};x^5\right )-25 \, _2F_1\left (-\frac {2}{3},-\frac {3}{5};\frac {2}{5};x^5\right )+9 \left (1-x^5\right )^{2/3} x^3 \left (-5 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};1-x^5\right )+3 \left (x^5-1\right ) \, _2F_1\left (\frac {5}{3},2;\frac {8}{3};1-x^5\right )+5\right )\right )\right )}{600 x^8 \left (1-x^5\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.83, size = 28, normalized size = 0.74 \begin {gather*} \frac {3 \left (-1+x^5\right )^{5/3} \left (-5+8 x^3+5 x^5\right )}{40 x^8} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 34, normalized size = 0.89 \begin {gather*} \frac {3 \, {\left (5 \, x^{10} + 8 \, x^{8} - 10 \, x^{5} - 8 \, x^{3} + 5\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{40 \, x^{8}} \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*} \int \frac {{\left (2 \, x^{5} + 3\right )} {\left (x^{5} + x^{3} - 1\right )} {\left (x^{5} - 1\right )}^{\frac {2}{3}}}{x^{9}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.15, size = 35, normalized size = 0.92
method | result | size |
trager | \(\frac {3 \left (x^{5}-1\right )^{\frac {2}{3}} \left (5 x^{10}+8 x^{8}-10 x^{5}-8 x^{3}+5\right )}{40 x^{8}}\) | \(35\) |
gosper | \(\frac {3 \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (-1+x \right ) \left (5 x^{5}+8 x^{3}-5\right ) \left (x^{5}-1\right )^{\frac {2}{3}}}{40 x^{8}}\) | \(40\) |
risch | \(\frac {-\frac {6}{5} x^{8}+\frac {3}{5} x^{3}-\frac {9}{8} x^{10}+\frac {9}{8} x^{5}-\frac {3}{8}+\frac {3}{8} x^{15}+\frac {3}{5} x^{13}}{x^{8} \left (x^{5}-1\right )^{\frac {1}{3}}}\) | \(45\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, \frac {2}{5}\right ], \left [\frac {7}{5}\right ], x^{5}\right ) x^{2}}{\left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}}}-\frac {\mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {3}{5}\right ], \left [\frac {2}{5}\right ], x^{5}\right )}{3 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}} x^{3}}-\frac {2 \mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \sqrt {3}\, \left (-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+5 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}+\frac {2 \hypergeom \left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{5}\right ) \pi \sqrt {3}\, x^{5}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{15 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}} \pi }+\frac {\mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \Gamma \left (\frac {2}{3}\right ) \sqrt {3}\, \left (-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{5}}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}-1+5 \ln \relax (x )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{5}\right ) \pi \sqrt {3}\, x^{5}}{9 \Gamma \left (\frac {2}{3}\right )}\right )}{5 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}} \pi }+\frac {3 \mathrm {signum}\left (x^{5}-1\right )^{\frac {2}{3}} \hypergeom \left (\left [-\frac {8}{5}, -\frac {2}{3}\right ], \left [-\frac {3}{5}\right ], x^{5}\right )}{8 \left (-\mathrm {signum}\left (x^{5}-1\right )\right )^{\frac {2}{3}} x^{8}}\) | \(276\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 46, normalized size = 1.21 \begin {gather*} \frac {3 \, {\left (5 \, x^{10} + 8 \, x^{8} - 10 \, x^{5} - 8 \, x^{3} + 5\right )} {\left (x^{4} + x^{3} + x^{2} + x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )}^{\frac {2}{3}}}{40 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.49, size = 52, normalized size = 1.37 \begin {gather*} {\left (x^5-1\right )}^{2/3}\,\left (\frac {3\,x^2}{8}+\frac {3}{5}\right )-\frac {3\,{\left (x^5-1\right )}^{2/3}}{4\,x^3}-\frac {3\,{\left (x^5-1\right )}^{2/3}}{5\,x^5}+\frac {3\,{\left (x^5-1\right )}^{2/3}}{8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 5.71, size = 194, normalized size = 5.11 \begin {gather*} - \frac {2 x^{\frac {10}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 \Gamma \left (\frac {1}{3}\right )} + \frac {2 x^{2} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {2}{5} \\ \frac {7}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 \Gamma \left (\frac {7}{5}\right )} - \frac {e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {3}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {3}{5} \\ \frac {2}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{3} \Gamma \left (\frac {2}{5}\right )} + \frac {3 e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {8}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {8}{5}, - \frac {2}{3} \\ - \frac {3}{5} \end {matrix}\middle | {x^{5}} \right )}}{5 x^{8} \Gamma \left (- \frac {3}{5}\right )} - \frac {3 \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{5}}} \right )}}{5 x^{\frac {5}{3}} \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________