Optimal. Leaf size=16 \[ \frac {4 \left (x^5+x\right )^{7/4}}{7 x^7} \]
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Rubi [B] time = 0.23, antiderivative size = 33, normalized size of antiderivative = 2.06, number of steps used = 11, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2052, 2025, 2032, 364} \begin {gather*} \frac {4 \left (x^5+x\right )^{3/4}}{7 x^6}+\frac {4 \left (x^5+x\right )^{3/4}}{7 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )}{x^6 \sqrt [4]{x+x^5}} \, dx &=\int \left (-\frac {3}{x^6 \sqrt [4]{x+x^5}}-\frac {2}{x^2 \sqrt [4]{x+x^5}}+\frac {x^2}{\sqrt [4]{x+x^5}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{x^2 \sqrt [4]{x+x^5}} \, dx\right )-3 \int \frac {1}{x^6 \sqrt [4]{x+x^5}} \, dx+\int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {8 \left (x+x^5\right )^{3/4}}{5 x^2}+\frac {9}{7} \int \frac {1}{x^2 \sqrt [4]{x+x^5}} \, dx-\frac {14}{5} \int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx+\frac {\left (\sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{\sqrt [4]{x+x^5}}\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}+\frac {4 x^3 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};-x^4\right )}{11 \sqrt [4]{x+x^5}}+\frac {9}{5} \int \frac {x^2}{\sqrt [4]{x+x^5}} \, dx-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{5 \sqrt [4]{x+x^5}}\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}-\frac {36 x^3 \sqrt [4]{1+x^4} \, _2F_1\left (\frac {1}{4},\frac {11}{16};\frac {27}{16};-x^4\right )}{55 \sqrt [4]{x+x^5}}+\frac {\left (9 \sqrt [4]{x} \sqrt [4]{1+x^4}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^4}} \, dx}{5 \sqrt [4]{x+x^5}}\\ &=\frac {4 \left (x+x^5\right )^{3/4}}{7 x^6}+\frac {4 \left (x+x^5\right )^{3/4}}{7 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 1.31 \begin {gather*} \frac {4 \left (x^4+1\right ) \left (x^5+x\right )^{3/4}}{7 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.21, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \left (x+x^5\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 17, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (x^{5} + x\right )}^{\frac {3}{4}} {\left (x^{4} + 1\right )}}{7 \, x^{6}} \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 (x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 18, normalized size = 1.12
method | result | size |
trager | \(\frac {4 \left (x^{4}+1\right ) \left (x^{5}+x \right )^{\frac {3}{4}}}{7 x^{6}}\) | \(18\) |
gosper | \(\frac {4 \left (x^{4}+1\right )^{2}}{7 x^{5} \left (x^{5}+x \right )^{\frac {1}{4}}}\) | \(20\) |
risch | \(\frac {\frac {4}{7} x^{8}+\frac {8}{7} x^{4}+\frac {4}{7}}{x^{5} \left (x \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\) | \(25\) |
meijerg | \(\frac {4 \hypergeom \left (\left [-\frac {21}{16}, \frac {1}{4}\right ], \left [-\frac {5}{16}\right ], -x^{4}\right )}{7 x^{\frac {21}{4}}}+\frac {8 \hypergeom \left (\left [-\frac {5}{16}, \frac {1}{4}\right ], \left [\frac {11}{16}\right ], -x^{4}\right )}{5 x^{\frac {5}{4}}}+\frac {4 \hypergeom \left (\left [\frac {1}{4}, \frac {11}{16}\right ], \left [\frac {27}{16}\right ], -x^{4}\right ) x^{\frac {11}{4}}}{11}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} + 1\right )} {\left (x^{4} - 3\right )}}{{\left (x^{5} + x\right )}^{\frac {1}{4}} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 27, normalized size = 1.69 \begin {gather*} \frac {4\,{\left (x^5+x\right )}^{3/4}+4\,x^4\,{\left (x^5+x\right )}^{3/4}}{7\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} - 3\right ) \left (x^{4} + 1\right )}{x^{6} \sqrt [4]{x \left (x^{4} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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