Optimal. Leaf size=18 \[ \frac {4 \left (-x+x^4\right )^{7/4}}{21 x^7} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(37\) vs. \(2(18)=36\).
time = 0.08, antiderivative size = 37, normalized size of antiderivative = 2.06, number of steps
used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2077, 2041,
2039} \begin {gather*} \frac {4 \left (x^4-x\right )^{3/4}}{21 x^3}-\frac {4 \left (x^4-x\right )^{3/4}}{21 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 2039
Rule 2041
Rule 2077
Rubi steps
\begin {align*} \int \frac {-1+x^3}{x^6 \sqrt [4]{-x+x^4}} \, dx &=\int \left (-\frac {1}{x^6 \sqrt [4]{-x+x^4}}+\frac {1}{x^3 \sqrt [4]{-x+x^4}}\right ) \, dx\\ &=-\int \frac {1}{x^6 \sqrt [4]{-x+x^4}} \, dx+\int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx\\ &=-\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {4 \left (-x+x^4\right )^{3/4}}{9 x^3}-\frac {4}{7} \int \frac {1}{x^3 \sqrt [4]{-x+x^4}} \, dx\\ &=-\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^6}+\frac {4 \left (-x+x^4\right )^{3/4}}{21 x^3}\\ \end {align*}
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Mathematica [A]
time = 9.21, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x \left (-1+x^3\right )\right )^{7/4}}{21 x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 20, normalized size = 1.11
method | result | size |
trager | \(\frac {4 \left (x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {3}{4}}}{21 x^{6}}\) | \(20\) |
risch | \(\frac {\frac {4}{21} x^{6}-\frac {8}{21} x^{3}+\frac {4}{21}}{x^{5} \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}\) | \(25\) |
gosper | \(\frac {4 \left (-1+x \right ) \left (x^{2}+x +1\right ) \left (x^{3}-1\right )}{21 x^{5} \left (x^{4}-x \right )^{\frac {1}{4}}}\) | \(29\) |
meijerg | \(\frac {4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (1+\frac {4 x^{3}}{3}\right ) \left (-x^{3}+1\right )^{\frac {3}{4}}}{21 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}}}-\frac {4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (-x^{3}+1\right )^{\frac {3}{4}}}{9 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs.
\(2 (14) = 28\).
time = 0.48, size = 58, normalized size = 3.22 \begin {gather*} \frac {4 \, {\left (x^{4} - x\right )}}{9 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {13}{4}}} - \frac {4 \, {\left (4 \, x^{7} - x^{4} - 3 \, x\right )}}{63 \, {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x - 1\right )}^{\frac {1}{4}} x^{\frac {25}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 19, normalized size = 1.06 \begin {gather*} \frac {4 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} {\left (x^{3} - 1\right )}}{21 \, 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 (x^{2} + x + 1\right )}{x^{6} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 11, normalized size = 0.61 \begin {gather*} -\frac {4}{21} \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 31, normalized size = 1.72 \begin {gather*} -\frac {4\,{\left (x^4-x\right )}^{3/4}-4\,x^3\,{\left (x^4-x\right )}^{3/4}}{21\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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