Optimal. Leaf size=171 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-2-x+x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{-2-x+x^2}}\right )}{2^{2/3}}+\frac {\log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{-2-x+x^2}\right )}{2^{2/3}}-\frac {\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (2^{2/3}+2^{2/3} x\right ) \sqrt [3]{-2-x+x^2}-2 \left (-2-x+x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \]
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Rubi [F]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-5+x}{\sqrt [3]{-2-x+x^2} \left (-3+4 x+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-5+x}{\sqrt [3]{-2-x+x^2} \left (-3+4 x+x^2\right )} \, dx &=\int \frac {-5+x}{\sqrt [3]{-2-x+x^2} \left (-3+4 x+x^2\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 160, normalized size = 0.94 \begin {gather*} \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-2-x+x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{-2-x+x^2}}\right )+2 \log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{-2-x+x^2}\right )-\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (1+x) \sqrt [3]{-2-x+x^2}-2 \left (-2-x+x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 5.78, size = 1404, normalized size = 8.21
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1404\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 300 vs.
\(2 (133) = 266\).
time = 4.55, size = 300, normalized size = 1.75 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 5 \, x^{3} + 4 \, x^{2} + 9 \, x - 9\right )} {\left (x^{2} - x - 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} + 30 \, x^{5} + 3 \, x^{4} + 100 \, x^{3} - 45 \, x^{2} - 306 \, x - 351\right )} + 12 \, {\left (x^{5} - 9 \, x^{4} + 40 \, x^{2} + 75 \, x + 45\right )} {\left (x^{2} - x - 2\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{6} - 42 \, x^{5} - 69 \, x^{4} + 100 \, x^{3} + 315 \, x^{2} + 486 \, x + 81\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{2} + x + 3\right )} {\left (x^{2} - x - 2\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} - 10 \, x^{3} + 10 \, x^{2} + 30 \, x + 45\right )} - 6 \, {\left (x^{3} - x^{2} + 7 \, x + 9\right )} {\left (x^{2} - x - 2\right )}^{\frac {1}{3}}}{x^{4} + 8 \, x^{3} + 10 \, x^{2} - 24 \, x + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 4 \, x - 3\right )} + 6 \cdot 4^{\frac {1}{3}} {\left (x^{2} - x - 2\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 12 \, {\left (x^{2} - x - 2\right )}^{\frac {2}{3}}}{x^{2} + 4 \, x - 3}\right ) \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 {x - 5}{\sqrt [3]{\left (x - 2\right ) \left (x + 1\right )} \left (x^{2} + 4 x - 3\right )}\, 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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x-5}{{\left (x^2-x-2\right )}^{1/3}\,\left (x^2+4\,x-3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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