Optimal. Leaf size=33 \[ \frac {4 \sqrt [4]{(-1+x)^3} \left (731-549 x-1029 x^2+154 x^3+693 x^4\right )}{4389} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(33)=66\).
time = 0.12, antiderivative size = 72, normalized size of antiderivative = 2.18, number of steps
used = 17, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6874, 2092,
15, 30, 2106, 45} \begin {gather*} \frac {12}{19} \sqrt [4]{(x-1)^3} (1-x)^4+\frac {36}{11} \sqrt [4]{(x-1)^3} (1-x)^2-\frac {4}{7} \sqrt [4]{(x-1)^3} (1-x)+\frac {8}{3} \left ((x-1)^3\right )^{5/4} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 45
Rule 2092
Rule 2106
Rule 6874
Rubi steps
\begin {align*} \int \sqrt [4]{-1+3 x-3 x^2+x^3} \left (-1-2 x+x^2+3 x^3\right ) \, dx &=\int \left (-\sqrt [4]{-1+3 x-3 x^2+x^3}-2 x \sqrt [4]{-1+3 x-3 x^2+x^3}+x^2 \sqrt [4]{-1+3 x-3 x^2+x^3}+3 x^3 \sqrt [4]{-1+3 x-3 x^2+x^3}\right ) \, dx\\ &=-\left (2 \int x \sqrt [4]{-1+3 x-3 x^2+x^3} \, dx\right )+3 \int x^3 \sqrt [4]{-1+3 x-3 x^2+x^3} \, dx-\int \sqrt [4]{-1+3 x-3 x^2+x^3} \, dx+\int x^2 \sqrt [4]{-1+3 x-3 x^2+x^3} \, dx\\ &=-\left (2 \text {Subst}\left (\int \sqrt [4]{x^3} (1+x) \, dx,x,-1+x\right )\right )+3 \text {Subst}\left (\int \sqrt [4]{x^3} (1+x)^3 \, dx,x,-1+x\right )-\text {Subst}\left (\int \sqrt [4]{x^3} \, dx,x,-1+x\right )+\text {Subst}\left (\int \sqrt [4]{x^3} (1+x)^2 \, dx,x,-1+x\right )\\ &=-\frac {\sqrt [4]{(-1+x)^3} \text {Subst}\left (\int x^{3/4} \, dx,x,-1+x\right )}{(-1+x)^{3/4}}+\frac {\sqrt [4]{(-1+x)^3} \text {Subst}\left (\int x^{3/4} (1+x)^2 \, dx,x,-1+x\right )}{(-1+x)^{3/4}}-\frac {\left (2 \sqrt [4]{(-1+x)^3}\right ) \text {Subst}\left (\int x^{3/4} (1+x) \, dx,x,-1+x\right )}{(-1+x)^{3/4}}+\frac {\left (3 \sqrt [4]{(-1+x)^3}\right ) \text {Subst}\left (\int x^{3/4} (1+x)^3 \, dx,x,-1+x\right )}{(-1+x)^{3/4}}\\ &=\frac {4}{7} (1-x) \sqrt [4]{(-1+x)^3}+\frac {\sqrt [4]{(-1+x)^3} \text {Subst}\left (\int \left (x^{3/4}+2 x^{7/4}+x^{11/4}\right ) \, dx,x,-1+x\right )}{(-1+x)^{3/4}}-\frac {\left (2 \sqrt [4]{(-1+x)^3}\right ) \text {Subst}\left (\int \left (x^{3/4}+x^{7/4}\right ) \, dx,x,-1+x\right )}{(-1+x)^{3/4}}+\frac {\left (3 \sqrt [4]{(-1+x)^3}\right ) \text {Subst}\left (\int \left (x^{3/4}+3 x^{7/4}+3 x^{11/4}+x^{15/4}\right ) \, dx,x,-1+x\right )}{(-1+x)^{3/4}}\\ &=-\frac {4}{7} (1-x) \sqrt [4]{(-1+x)^3}+\frac {36}{11} (1-x)^2 \sqrt [4]{(-1+x)^3}+\frac {12}{19} (1-x)^4 \sqrt [4]{(-1+x)^3}+\frac {8}{3} \left ((-1+x)^3\right )^{5/4}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 31, normalized size = 0.94 \begin {gather*} \frac {4 (-1+x) \sqrt [4]{(-1+x)^3} \left (-731-182 x+847 x^2+693 x^3\right )}{4389} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 30, normalized size = 0.91
method | result | size |
risch | \(\frac {4 \left (\left (-1+x \right )^{3}\right )^{\frac {1}{4}} \left (693 x^{4}+154 x^{3}-1029 x^{2}-549 x +731\right )}{4389}\) | \(30\) |
gosper | \(\frac {4 \left (-1+x \right ) \left (693 x^{3}+847 x^{2}-182 x -731\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}}{4389}\) | \(36\) |
trager | \(\left (\frac {12}{19} x^{4}+\frac {8}{57} x^{3}-\frac {196}{209} x^{2}-\frac {732}{1463} x +\frac {2924}{4389}\right ) \left (x^{3}-3 x^{2}+3 x -1\right )^{\frac {1}{4}}\) | \(37\) |
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. 74 vs.
\(2 (29) = 58\).
time = 0.26, size = 74, normalized size = 2.24 \begin {gather*} \frac {12}{7315} \, {\left (385 \, x^{4} - 77 \, x^{3} - 84 \, x^{2} - 96 \, x - 128\right )} {\left (x - 1\right )}^{\frac {3}{4}} + \frac {4}{1155} \, {\left (77 \, x^{3} - 21 \, x^{2} - 24 \, x - 32\right )} {\left (x - 1\right )}^{\frac {3}{4}} - \frac {8}{77} \, {\left (7 \, x^{2} - 3 \, x - 4\right )} {\left (x - 1\right )}^{\frac {3}{4}} - \frac {4}{7} \, {\left (x - 1\right )}^{\frac {7}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 37, normalized size = 1.12 \begin {gather*} \frac {4}{4389} \, {\left (693 \, x^{4} + 154 \, x^{3} - 1029 \, x^{2} - 549 \, x + 731\right )} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}^{\frac {1}{4}} \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 \left (3 x^{3} + x^{2} - 2 x - 1\right ) \sqrt [4]{\left (x - 1\right )^{3}}\, dx \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 = 0.20, size = 35, normalized size = 1.06 \begin {gather*} -\frac {4\,\left (x-1\right )\,{\left (x^3-3\,x^2+3\,x-1\right )}^{1/4}\,\left (-693\,x^3-847\,x^2+182\,x+731\right )}{4389} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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