Optimal. Leaf size=50 \[ \frac {4 \sqrt [4]{-x^3+x^4} \left (13923-663 x-21255 x^2+615 x^3+820 x^4+1312 x^5+5248 x^6\right )}{348075 x^7} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(121\) vs. \(2(50)=100\).
time = 0.19, antiderivative size = 121, normalized size of antiderivative = 2.42, number of steps
used = 12, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2077, 2041,
2039} \begin {gather*} -\frac {4 \left (x^4-x^3\right )^{5/4}}{25 x^{10}}-\frac {16 \left (x^4-x^3\right )^{5/4}}{105 x^9}+\frac {164 \left (x^4-x^3\right )^{5/4}}{1785 x^8}+\frac {656 \left (x^4-x^3\right )^{5/4}}{7735 x^7}+\frac {5248 \left (x^4-x^3\right )^{5/4}}{69615 x^6}+\frac {20992 \left (x^4-x^3\right )^{5/4}}{348075 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rule 2077
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{-x^3+x^4}}{x^8} \, dx &=\int \left (-\frac {\sqrt [4]{-x^3+x^4}}{x^8}+\frac {\sqrt [4]{-x^3+x^4}}{x^6}\right ) \, dx\\ &=-\int \frac {\sqrt [4]{-x^3+x^4}}{x^8} \, dx+\int \frac {\sqrt [4]{-x^3+x^4}}{x^6} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{17 x^8}+\frac {12}{17} \int \frac {\sqrt [4]{-x^3+x^4}}{x^5} \, dx-\frac {4}{5} \int \frac {\sqrt [4]{-x^3+x^4}}{x^7} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {4 \left (-x^3+x^4\right )^{5/4}}{17 x^8}+\frac {48 \left (-x^3+x^4\right )^{5/4}}{221 x^7}+\frac {96}{221} \int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx-\frac {64}{105} \int \frac {\sqrt [4]{-x^3+x^4}}{x^6} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {48 \left (-x^3+x^4\right )^{5/4}}{221 x^7}+\frac {128 \left (-x^3+x^4\right )^{5/4}}{663 x^6}+\frac {128}{663} \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx-\frac {256}{595} \int \frac {\sqrt [4]{-x^3+x^4}}{x^5} \, dx\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {656 \left (-x^3+x^4\right )^{5/4}}{7735 x^7}+\frac {128 \left (-x^3+x^4\right )^{5/4}}{663 x^6}+\frac {512 \left (-x^3+x^4\right )^{5/4}}{3315 x^5}-\frac {2048 \int \frac {\sqrt [4]{-x^3+x^4}}{x^4} \, dx}{7735}\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {656 \left (-x^3+x^4\right )^{5/4}}{7735 x^7}+\frac {5248 \left (-x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {512 \left (-x^3+x^4\right )^{5/4}}{3315 x^5}-\frac {8192 \int \frac {\sqrt [4]{-x^3+x^4}}{x^3} \, dx}{69615}\\ &=-\frac {4 \left (-x^3+x^4\right )^{5/4}}{25 x^{10}}-\frac {16 \left (-x^3+x^4\right )^{5/4}}{105 x^9}+\frac {164 \left (-x^3+x^4\right )^{5/4}}{1785 x^8}+\frac {656 \left (-x^3+x^4\right )^{5/4}}{7735 x^7}+\frac {5248 \left (-x^3+x^4\right )^{5/4}}{69615 x^6}+\frac {20992 \left (-x^3+x^4\right )^{5/4}}{348075 x^5}\\ \end {align*}
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Mathematica [A]
time = 1.17, size = 38, normalized size = 0.76 \begin {gather*} \frac {4 \left ((-1+x) x^3\right )^{9/4} \left (13923+27183 x+19188 x^2+11808 x^3+5248 x^4\right )}{348075 x^{13}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 42, normalized size = 0.84
method | result | size |
gosper | \(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (-1+x \right )^{2} \left (5248 x^{4}+11808 x^{3}+19188 x^{2}+27183 x +13923\right )}{348075 x^{7}}\) | \(42\) |
trager | \(\frac {4 \left (x^{4}-x^{3}\right )^{\frac {1}{4}} \left (5248 x^{6}+1312 x^{5}+820 x^{4}+615 x^{3}-21255 x^{2}-663 x +13923\right )}{348075 x^{7}}\) | \(47\) |
risch | \(\frac {4 \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (5248 x^{7}-3936 x^{6}-492 x^{5}-205 x^{4}-21870 x^{3}+20592 x^{2}+14586 x -13923\right )}{348075 x^{7} \left (-1+x \right )}\) | \(55\) |
meijerg | \(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \hypergeom \left (\left [-\frac {25}{4}, -\frac {1}{4}\right ], \left [-\frac {21}{4}\right ], x\right )}{25 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {25}{4}}}-\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} \left (-\frac {128}{195} x^{4}-\frac {32}{195} x^{3}-\frac {4}{39} x^{2}-\frac {1}{13} x +1\right ) \left (1-x \right )^{\frac {1}{4}}}{17 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}} x^{\frac {17}{4}}}\) | \(74\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 46, normalized size = 0.92 \begin {gather*} \frac {4 \, {\left (5248 \, x^{6} + 1312 \, x^{5} + 820 \, x^{4} + 615 \, x^{3} - 21255 \, x^{2} - 663 \, x + 13923\right )} {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{348075 \, x^{7}} \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 {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 91, normalized size = 1.82 \begin {gather*} -\frac {4}{25} \, {\left (\frac {1}{x} - 1\right )}^{6} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {20}{21} \, {\left (\frac {1}{x} - 1\right )}^{5} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {36}{17} \, {\left (\frac {1}{x} - 1\right )}^{4} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {28}{13} \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {8}{9} \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 113, normalized size = 2.26 \begin {gather*} \frac {20992\,{\left (x^4-x^3\right )}^{1/4}}{348075\,x}+\frac {5248\,{\left (x^4-x^3\right )}^{1/4}}{348075\,x^2}+\frac {656\,{\left (x^4-x^3\right )}^{1/4}}{69615\,x^3}+\frac {164\,{\left (x^4-x^3\right )}^{1/4}}{23205\,x^4}-\frac {436\,{\left (x^4-x^3\right )}^{1/4}}{1785\,x^5}-\frac {4\,{\left (x^4-x^3\right )}^{1/4}}{525\,x^6}+\frac {4\,{\left (x^4-x^3\right )}^{1/4}}{25\,x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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