Optimal. Leaf size=26 \[ -\frac {4 \sqrt [4]{x^3-1} \left (5 x^4+x^3-1\right )}{5 x^5} \]
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Rubi [A] time = 0.13, antiderivative size = 47, normalized size of antiderivative = 1.81, number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1835, 1586, 449} \begin {gather*} -\frac {4 \sqrt [4]{x^3-1}}{x}+\frac {4 \sqrt [4]{x^3-1}}{5 x^5}-\frac {4 \sqrt [4]{x^3-1}}{5 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 1586
Rule 1835
Rubi steps
\begin {align*} \int \frac {\left (-4+x^3\right ) \left (-1+x^3+x^4\right )}{x^6 \left (-1+x^3\right )^{3/4}} \, dx &=\frac {4 \sqrt [4]{-1+x^3}}{5 x^5}+\frac {1}{10} \int \frac {-16 x^2-40 x^3+10 x^5+10 x^6}{x^5 \left (-1+x^3\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-1+x^3}}{5 x^5}+\frac {1}{10} \int \frac {-16 x-40 x^2+10 x^4+10 x^5}{x^4 \left (-1+x^3\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-1+x^3}}{5 x^5}+\frac {1}{10} \int \frac {-16-40 x+10 x^3+10 x^4}{x^3 \left (-1+x^3\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-1+x^3}}{5 x^5}-\frac {4 \sqrt [4]{-1+x^3}}{5 x^2}+\frac {1}{40} \int \frac {-160+40 x^3}{x^2 \left (-1+x^3\right )^{3/4}} \, dx\\ &=\frac {4 \sqrt [4]{-1+x^3}}{5 x^5}-\frac {4 \sqrt [4]{-1+x^3}}{5 x^2}-\frac {4 \sqrt [4]{-1+x^3}}{x}\\ \end {align*}
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Mathematica [C] time = 0.07, size = 115, normalized size = 4.42 \begin {gather*} \frac {\left (1-x^3\right )^{3/4} \left (5 x^3 \left (2 x^3 \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};x^3\right )+8 x \, _2F_1\left (-\frac {1}{3},\frac {3}{4};\frac {2}{3};x^3\right )+5 \, _2F_1\left (-\frac {2}{3},\frac {3}{4};\frac {1}{3};x^3\right )+x^4 \, _2F_1\left (\frac {2}{3},\frac {3}{4};\frac {5}{3};x^3\right )\right )-8 \, _2F_1\left (-\frac {5}{3},\frac {3}{4};-\frac {2}{3};x^3\right )\right )}{10 x^5 \left (x^3-1\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.02, size = 26, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{x^3-1} \left (5 x^4+x^3-1\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 22, normalized size = 0.85 \begin {gather*} -\frac {4 \, {\left (5 \, x^{4} + x^{3} - 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{5 \, x^{5}} \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} + x^{3} - 1\right )} {\left (x^{3} - 4\right )}}{{\left (x^{3} - 1\right )}^{\frac {3}{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 = 32, normalized size = 1.23 \begin {gather*} -\frac {4 \left (-1+x \right ) \left (x^{2}+x +1\right ) \left (5 x^{4}+x^{3}-1\right )}{5 x^{5} \left (x^{3}-1\right )^{\frac {3}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 38, normalized size = 1.46 \begin {gather*} -\frac {4 \, {\left (5 \, x^{7} + x^{6} - 5 \, x^{4} - 2 \, x^{3} + 1\right )}}{5 \, {\left (x^{2} + x + 1\right )}^{\frac {3}{4}} {\left (x - 1\right )}^{\frac {3}{4}} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.24, size = 39, normalized size = 1.50 \begin {gather*} -\frac {4\,x^3\,{\left (x^3-1\right )}^{1/4}-4\,{\left (x^3-1\right )}^{1/4}+20\,x^4\,{\left (x^3-1\right )}^{1/4}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.05, size = 178, normalized size = 6.85 \begin {gather*} \frac {x^{2} e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{4} \\ \frac {5}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {x e^{- \frac {3 i \pi }{4}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {3}{4} \\ \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{4} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} + \frac {5 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{4} \\ \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {4 e^{\frac {i \pi }{4}} \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, \frac {3}{4} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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