Optimal. Leaf size=28 \[ -\frac {4 \sqrt [4]{1+x^3} \left (-1-x^3+5 x^4\right )}{5 x^5} \]
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Rubi [A]
time = 0.08, antiderivative size = 47, normalized size of antiderivative = 1.68, number of steps
used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {1849, 1600,
460} \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 460
Rule 1600
Rule 1849
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 [A]
time = 3.05, size = 28, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{1+x^3} \left (-1-x^3+5 x^4\right )}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 25, normalized size = 0.89
method | result | size |
trager | \(-\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}} \left (5 x^{4}-x^{3}-1\right )}{5 x^{5}}\) | \(25\) |
risch | \(-\frac {4 \left (5 x^{7}-x^{6}+5 x^{4}-2 x^{3}-1\right )}{5 \left (x^{3}+1\right )^{\frac {3}{4}} x^{5}}\) | \(35\) |
gosper | \(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (5 x^{4}-x^{3}-1\right )}{5 x^{5} \left (x^{3}+1\right )^{\frac {3}{4}}}\) | \(36\) |
meijerg | \(\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {3}{4}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}-x \hypergeom \left (\left [\frac {1}{3}, \frac {3}{4}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )+\frac {5 \hypergeom \left (\left [-\frac {2}{3}, \frac {3}{4}\right ], \left [\frac {1}{3}\right ], -x^{3}\right )}{2 x^{2}}+\frac {4 \hypergeom \left (\left [-\frac {5}{3}, \frac {3}{4}\right ], \left [-\frac {2}{3}\right ], -x^{3}\right )}{5 x^{5}}-\frac {4 \hypergeom \left (\left [-\frac {1}{3}, \frac {3}{4}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 42, normalized size = 1.50 \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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Fricas [A]
time = 0.37, size = 24, normalized size = 0.86 \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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Sympy [C] Result contains complex when optimal does not.
time = 2.48, size = 167, normalized size = 5.96 \begin {gather*} \frac {x^{2} \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} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} - \frac {x \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} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} + \frac {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} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {5 \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} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {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} e^{i \pi }} \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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = 1.39 \begin {gather*} \frac {4\,{\left (x^3+1\right )}^{1/4}+4\,x^3\,{\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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