Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \left (-1+x^3\right )^{3/4} \left (3-3 x^3+7 x^4\right )}{21 x^7} \]
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Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1849, 1600, 457, 75} \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \left (x^3-1\right )^{3/4}}{3 x^3}-\frac {4 \left (x^3-1\right )^{3/4}}{7 x^7}+\frac {4 \left (x^3-1\right )^{3/4}}{7 x^4} \]
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Rule 75
Rule 457
Rule 1600
Rule 1849
Rubi steps \begin{align*} \text {integral}& = -\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^7}+\frac {1}{14} \int \frac {32 x^2-56 x^3-14 x^5+14 x^6}{x^7 \sqrt [4]{-1+x^3}} \, dx \\ & = -\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^7}+\frac {1}{14} \int \frac {32 x-56 x^2-14 x^4+14 x^5}{x^6 \sqrt [4]{-1+x^3}} \, dx \\ & = -\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^7}+\frac {1}{14} \int \frac {32-56 x-14 x^3+14 x^4}{x^5 \sqrt [4]{-1+x^3}} \, dx \\ & = -\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^7}+\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^4}+\frac {1}{112} \int \frac {-448+112 x^3}{x^4 \sqrt [4]{-1+x^3}} \, dx \\ & = -\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^7}+\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^4}+\frac {1}{336} \text {Subst}\left (\int \frac {-448+112 x}{\sqrt [4]{-1+x} x^2} \, dx,x,x^3\right ) \\ & = -\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^7}+\frac {4 \left (-1+x^3\right )^{3/4}}{7 x^4}-\frac {4 \left (-1+x^3\right )^{3/4}}{3 x^3} \\ \end{align*}
Time = 0.65 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \left (-1+x^3\right )^{3/4} \left (3-3 x^3+7 x^4\right )}{21 x^7} \]
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Time = 0.98 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
method | result | size |
trager | \(-\frac {4 \left (x^{3}-1\right )^{\frac {3}{4}} \left (7 x^{4}-3 x^{3}+3\right )}{21 x^{7}}\) | \(25\) |
gosper | \(-\frac {4 \left (x -1\right ) \left (x^{2}+x +1\right ) \left (7 x^{4}-3 x^{3}+3\right )}{21 x^{7} \left (x^{3}-1\right )^{\frac {1}{4}}}\) | \(34\) |
risch | \(-\frac {4 \left (7 x^{7}-3 x^{6}-7 x^{4}+6 x^{3}-3\right )}{21 x^{7} \left (x^{3}-1\right )^{\frac {1}{4}}}\) | \(35\) |
meijerg | \(\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {7}{3}, \frac {1}{4}\right ], \left [-\frac {4}{3}\right ], x^{3}\right )}{7 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{7}}+\frac {2 \sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (-\frac {5 \pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {9}{4}\right ], \left [2, 3\right ], x^{3}\right )}{32 \Gamma \left (\frac {3}{4}\right )}-\frac {\left (3-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right ) x^{3}}\right )}{3 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}}}-\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {4}{3}, \frac {1}{4}\right ], \left [-\frac {1}{3}\right ], x^{3}\right )}{4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{4}}+\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{3}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \left (2\right )-\frac {\pi }{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{6 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {1}{4}\right ], \left [\frac {2}{3}\right ], x^{3}\right )}{\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x}\) | \(267\) |
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Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \, {\left (7 \, x^{4} - 3 \, x^{3} + 3\right )} {\left (x^{3} - 1\right )}^{\frac {3}{4}}}{21 \, x^{7}} \]
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Result contains complex when optimal does not.
Time = 2.54 (sec) , antiderivative size = 189, normalized size of antiderivative = 6.75 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=\frac {e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {1}{4} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} - \frac {5 e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {4}{3}, \frac {1}{4} \\ - \frac {1}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{4} \Gamma \left (- \frac {1}{3}\right )} + \frac {4 e^{\frac {3 i \pi }{4}} \Gamma \left (- \frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{3}, \frac {1}{4} \\ - \frac {4}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x^{7} \Gamma \left (- \frac {4}{3}\right )} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{\frac {3}{4}} \Gamma \left (\frac {5}{4}\right )} + \frac {4 \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{\frac {15}{4}} \Gamma \left (\frac {9}{4}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=-\frac {4 \, {\left (7 \, x^{7} - 3 \, x^{6} - 7 \, x^{4} + 6 \, x^{3} - 3\right )}}{21 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{7}} \]
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\[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=\int { \frac {{\left (x^{4} - x^{3} + 1\right )} {\left (x^{3} - 4\right )}}{{\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{8}} \,d x } \]
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Time = 5.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^8 \sqrt [4]{-1+x^3}} \, dx=-\frac {12\,{\left (x^3-1\right )}^{3/4}-12\,x^3\,{\left (x^3-1\right )}^{3/4}+28\,x^4\,{\left (x^3-1\right )}^{3/4}}{21\,x^7} \]
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