Optimal. Leaf size=38 \[ \frac {4 \left (x^5+1\right )^{3/4} \left (14 x^{10}-11 x^9+28 x^5-11 x^4+14\right )}{77 x^{11}} \]
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Rubi [B] time = 0.19, antiderivative size = 88, normalized size of antiderivative = 2.32, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1820, 1825, 1835, 1586, 449} \begin {gather*} \frac {30 \left (x^5+1\right )^{3/4}}{11 x}+\frac {5 \left (x^5+1\right )^{3/4}}{11 x^6}-\frac {15 \left (x^5+1\right )^{3/4}}{14 x^2}+\frac {1}{154} \left (x^5+1\right )^{3/4} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 1586
Rule 1820
Rule 1825
Rule 1835
Rubi steps
\begin {align*} \int \frac {\left (-4+x^5\right ) \left (1+x^5\right )^{3/4} \left (2-x^4+2 x^5\right )}{x^{12}} \, dx &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}-\frac {15}{4} \int \frac {\frac {8}{11}-\frac {4 x^4}{7}+x^5+\frac {x^9}{2}-2 x^{10}}{x^7 \sqrt [4]{1+x^5}} \, dx\\ &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}+\frac {5 \left (1+x^5\right )^{3/4}}{11 x^6}+\frac {5}{16} \int \frac {\frac {48 x^3}{7}-\frac {96 x^4}{11}-6 x^8+24 x^9}{x^6 \sqrt [4]{1+x^5}} \, dx\\ &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}+\frac {5 \left (1+x^5\right )^{3/4}}{11 x^6}+\frac {5}{16} \int \frac {\frac {48 x^2}{7}-\frac {96 x^3}{11}-6 x^7+24 x^8}{x^5 \sqrt [4]{1+x^5}} \, dx\\ &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}+\frac {5 \left (1+x^5\right )^{3/4}}{11 x^6}+\frac {5}{16} \int \frac {\frac {48 x}{7}-\frac {96 x^2}{11}-6 x^6+24 x^7}{x^4 \sqrt [4]{1+x^5}} \, dx\\ &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}+\frac {5 \left (1+x^5\right )^{3/4}}{11 x^6}+\frac {5}{16} \int \frac {\frac {48}{7}-\frac {96 x}{11}-6 x^5+24 x^6}{x^3 \sqrt [4]{1+x^5}} \, dx\\ &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}+\frac {5 \left (1+x^5\right )^{3/4}}{11 x^6}-\frac {15 \left (1+x^5\right )^{3/4}}{14 x^2}-\frac {5}{64} \int \frac {\frac {384}{11}-96 x^5}{x^2 \sqrt [4]{1+x^5}} \, dx\\ &=\frac {1}{154} \left (\frac {112}{x^{11}}-\frac {88}{x^7}+\frac {154}{x^6}+\frac {77}{x^2}-\frac {308}{x}\right ) \left (1+x^5\right )^{3/4}+\frac {5 \left (1+x^5\right )^{3/4}}{11 x^6}-\frac {15 \left (1+x^5\right )^{3/4}}{14 x^2}+\frac {30 \left (1+x^5\right )^{3/4}}{11 x}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 106, normalized size = 2.79 \begin {gather*} -\frac {2 \, _2F_1\left (-\frac {3}{4},-\frac {1}{5};\frac {4}{5};-x^5\right )}{x}+\frac {8 \, _2F_1\left (-\frac {11}{5},-\frac {3}{4};-\frac {6}{5};-x^5\right )}{11 x^{11}}-\frac {4 \, _2F_1\left (-\frac {7}{5},-\frac {3}{4};-\frac {2}{5};-x^5\right )}{7 x^7}+\frac {\, _2F_1\left (-\frac {6}{5},-\frac {3}{4};-\frac {1}{5};-x^5\right )}{x^6}+\frac {\, _2F_1\left (-\frac {3}{4},-\frac {2}{5};\frac {3}{5};-x^5\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.60, size = 28, normalized size = 0.74 \begin {gather*} \frac {4 \left (1+x^5\right )^{7/4} \left (14-11 x^4+14 x^5\right )}{77 x^{11}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 34, normalized size = 0.89 \begin {gather*} \frac {4 \, {\left (14 \, x^{10} - 11 \, x^{9} + 28 \, x^{5} - 11 \, x^{4} + 14\right )} {\left (x^{5} + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{5} - x^{4} + 2\right )} {\left (x^{5} + 1\right )}^{\frac {3}{4}} {\left (x^{5} - 4\right )}}{x^{12}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 35, normalized size = 0.92
method | result | size |
trager | \(\frac {4 \left (x^{5}+1\right )^{\frac {3}{4}} \left (14 x^{10}-11 x^{9}+28 x^{5}-11 x^{4}+14\right )}{77 x^{11}}\) | \(35\) |
gosper | \(\frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (14 x^{5}-11 x^{4}+14\right ) \left (x^{5}+1\right )^{\frac {3}{4}}}{77 x^{11}}\) | \(44\) |
risch | \(\frac {-\frac {4}{7} x^{14}-\frac {8}{7} x^{9}+\frac {8}{11} x^{15}+\frac {24}{11} x^{10}+\frac {24}{11} x^{5}+\frac {8}{11}-\frac {4}{7} x^{4}}{x^{11} \left (x^{5}+1\right )^{\frac {1}{4}}}\) | \(45\) |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {6}{5}, -\frac {3}{4}\right ], \left [-\frac {1}{5}\right ], -x^{5}\right )}{x^{6}}-\frac {4 \hypergeom \left (\left [-\frac {7}{5}, -\frac {3}{4}\right ], \left [-\frac {2}{5}\right ], -x^{5}\right )}{7 x^{7}}+\frac {8 \hypergeom \left (\left [-\frac {11}{5}, -\frac {3}{4}\right ], \left [-\frac {6}{5}\right ], -x^{5}\right )}{11 x^{11}}-\frac {2 \hypergeom \left (\left [-\frac {3}{4}, -\frac {1}{5}\right ], \left [\frac {4}{5}\right ], -x^{5}\right )}{x}+\frac {\hypergeom \left (\left [-\frac {3}{4}, -\frac {2}{5}\right ], \left [\frac {3}{5}\right ], -x^{5}\right )}{2 x^{2}}\) | \(81\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 50, normalized size = 1.32 \begin {gather*} \frac {4 \, {\left (14 \, x^{10} - 11 \, x^{9} + 28 \, x^{5} - 11 \, x^{4} + 14\right )} {\left (x^{4} - x^{3} + x^{2} - x + 1\right )}^{\frac {3}{4}} {\left (x + 1\right )}^{\frac {3}{4}}}{77 \, x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.70, size = 61, normalized size = 1.61 \begin {gather*} \frac {8\,{\left (x^5+1\right )}^{3/4}}{11\,x}-\frac {4\,{\left (x^5+1\right )}^{3/4}}{7\,x^2}+\frac {16\,{\left (x^5+1\right )}^{3/4}}{11\,x^6}-\frac {4\,{\left (x^5+1\right )}^{3/4}}{7\,x^7}+\frac {8\,{\left (x^5+1\right )}^{3/4}}{11\,x^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 6.57, size = 192, normalized size = 5.05 \begin {gather*} \frac {2 \Gamma \left (- \frac {1}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {1}{5} \\ \frac {4}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x \Gamma \left (\frac {4}{5}\right )} - \frac {\Gamma \left (- \frac {2}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, - \frac {2}{5} \\ \frac {3}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{2} \Gamma \left (\frac {3}{5}\right )} - \frac {6 \Gamma \left (- \frac {6}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {6}{5}, - \frac {3}{4} \\ - \frac {1}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{6} \Gamma \left (- \frac {1}{5}\right )} + \frac {4 \Gamma \left (- \frac {7}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{5}, - \frac {3}{4} \\ - \frac {2}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{7} \Gamma \left (- \frac {2}{5}\right )} - \frac {8 \Gamma \left (- \frac {11}{5}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {11}{5}, - \frac {3}{4} \\ - \frac {6}{5} \end {matrix}\middle | {x^{5} e^{i \pi }} \right )}}{5 x^{11} \Gamma \left (- \frac {6}{5}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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