Optimal. Leaf size=18 \[ \frac {4 \left (x^6-x\right )^{7/4}}{7 x^7} \]
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Rubi [B] time = 0.27, antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps used = 14, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2052, 2025, 2032, 365, 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 2025
Rule 2032
Rule 2052
Rubi steps
\begin {align*} \int \frac {\left (-1+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 {1}{x \sqrt [4]{-x+x^6}}+\frac {2 x^4}{\sqrt [4]{-x+x^6}}\right ) \, dx\\ &=2 \int \frac {x^4}{\sqrt [4]{-x+x^6}} \, dx-3 \int \frac {1}{x^6 \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}}{x}-\frac {6}{7} \int \frac {1}{x \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^{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}}{7 x}+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^{15/4}}{\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}}{7 x}+\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}}-\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 {\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}}{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}}{7 x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x \left (x^5-1\right )\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.31, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^6-x\right )^{7/4}}{7 x^7} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 19, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (x^{6} - x\right )}^{\frac {3}{4}} {\left (x^{5} - 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 (2 \, x^{5} + 3\right )} {\left (x^{5} - 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 [B] time = 0.01, size = 35, normalized size = 1.94 \begin {gather*} \frac {4 \left (-1+x \right ) \left (x^{4}+x^{3}+x^{2}+x +1\right ) \left (x^{5}-1\right )}{7 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} - 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 = 0.28, size = 31, normalized size = 1.72 \begin {gather*} -\frac {4\,{\left (x^6-x\right )}^{3/4}-4\,x^5\,{\left (x^6-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 - 1\right ) \left (2 x^{5} + 3\right ) \left (x^{4} + x^{3} + x^{2} + x + 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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