Optimal. Leaf size=28 \[ \frac {4 \left (3 x^5-7 x^3+3\right ) \left (x^6+x\right )^{3/4}}{21 x^6} \]
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Rubi [A] time = 0.32, antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 16, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2052, 2025, 2032, 364} \begin {gather*} \frac {4 \left (x^6+x\right )^{3/4}}{7 x}+\frac {4 \left (x^6+x\right )^{3/4}}{7 x^6}-\frac {4 \left (x^6+x\right )^{3/4}}{3 x^3} \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 (1-x^3+x^5\right ) \left (-3+2 x^5\right )}{x^6 \sqrt [4]{x+x^6}} \, dx &=\int \left (-\frac {3}{x^6 \sqrt [4]{x+x^6}}+\frac {3}{x^3 \sqrt [4]{x+x^6}}-\frac {1}{x \sqrt [4]{x+x^6}}-\frac {2 x^2}{\sqrt [4]{x+x^6}}+\frac {2 x^4}{\sqrt [4]{x+x^6}}\right ) \, dx\\ &=-\left (2 \int \frac {x^2}{\sqrt [4]{x+x^6}} \, dx\right )+2 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx-3 \int \frac {1}{x^6 \sqrt [4]{x+x^6}} \, dx+3 \int \frac {1}{x^3 \sqrt [4]{x+x^6}} \, dx-\int \frac {1}{x \sqrt [4]{x+x^6}} \, dx\\ &=\frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{x}+\frac {6}{7} \int \frac {1}{x \sqrt [4]{x+x^6}} \, dx+2 \int \frac {x^2}{\sqrt [4]{x+x^6}} \, dx-14 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx-\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}-\frac {8 x^3 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {11}{20};\frac {31}{20};-x^5\right )}{11 \sqrt [4]{x+x^6}}+\frac {8 x^5 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {19}{20};\frac {39}{20};-x^5\right )}{19 \sqrt [4]{x+x^6}}+12 \int \frac {x^4}{\sqrt [4]{x+x^6}} \, dx+\frac {\left (2 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{7/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}-\frac {\left (14 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}-\frac {48 x^5 \sqrt [4]{1+x^5} \, _2F_1\left (\frac {1}{4},\frac {19}{20};\frac {39}{20};-x^5\right )}{19 \sqrt [4]{x+x^6}}+\frac {\left (12 \sqrt [4]{x} \sqrt [4]{1+x^5}\right ) \int \frac {x^{15/4}}{\sqrt [4]{1+x^5}} \, dx}{\sqrt [4]{x+x^6}}\\ &=\frac {4 \left (x+x^6\right )^{3/4}}{7 x^6}-\frac {4 \left (x+x^6\right )^{3/4}}{3 x^3}+\frac {4 \left (x+x^6\right )^{3/4}}{7 x}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 126, normalized size = 4.50 \begin {gather*} \frac {4 \sqrt [4]{x^5+1} \left (627 \, _2F_1\left (-\frac {21}{20},\frac {1}{4};-\frac {1}{20};-x^5\right )+7 x^3 \left (-114 x^5 \, _2F_1\left (\frac {1}{4},\frac {11}{20};\frac {31}{20};-x^5\right )-209 \, _2F_1\left (-\frac {9}{20},\frac {1}{4};\frac {11}{20};-x^5\right )+66 x^7 \, _2F_1\left (\frac {1}{4},\frac {19}{20};\frac {39}{20};-x^5\right )+627 x^2 \, _2F_1\left (-\frac {1}{20},\frac {1}{4};\frac {19}{20};-x^5\right )\right )\right )}{4389 x^5 \sqrt [4]{x^6+x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 2.64, size = 28, normalized size = 1.00 \begin {gather*} \frac {4 \left (3 x^5-7 x^3+3\right ) \left (x^6+x\right )^{3/4}}{21 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 24, normalized size = 0.86 \begin {gather*} \frac {4 \, {\left (x^{6} + x\right )}^{\frac {3}{4}} {\left (3 \, x^{5} - 7 \, x^{3} + 3\right )}}{21 \, 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 (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + 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 = 0.01, size = 44, normalized size = 1.57 \begin {gather*} \frac {4 \left (x^{4}-x^{3}+x^{2}-x +1\right ) \left (1+x \right ) \left (3 x^{5}-7 x^{3}+3\right )}{21 x^{5} \left (x^{6}+x \right )^{\frac {1}{4}}} \end {gather*}
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 (2 \, x^{5} - 3\right )} {\left (x^{5} - x^{3} + 1\right )}}{{\left (x^{6} + 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 = 0.32, size = 39, normalized size = 1.39 \begin {gather*} \frac {12\,{\left (x^6+x\right )}^{3/4}-28\,x^3\,{\left (x^6+x\right )}^{3/4}+12\,x^5\,{\left (x^6+x\right )}^{3/4}}{21\,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 (2 x^{5} - 3\right ) \left (x^{5} - x^{3} + 1\right )}{x^{6} \sqrt [4]{x \left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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