Optimal. Leaf size=161 \[ \frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{2 x^2}-3 \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^2+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^2+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^2+2 x^3}+\sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \]
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Rubi [F]
time = 1.73, 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 {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (3+2 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3 \left (-1-2 x^2+x^3\right )} \, dx &=\int \left (-\frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{x^3}+\frac {4 \left (1+2 x^2+2 x^3\right )^{2/3}}{x}+\frac {\left (3+8 x-4 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}\right ) \, dx\\ &=-\left (3 \int \frac {\left (1+2 x^2+2 x^3\right )^{2/3}}{x^3} \, dx\right )+4 \int \frac {\left (1+2 x^2+2 x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (3+8 x-4 x^2\right ) \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx\\ &=-\left (3 \text {Subst}\left (\int \frac {\left (\frac {31}{27}-\frac {2 x}{3}+2 x^3\right )^{2/3}}{\left (-\frac {1}{3}+x\right )^3} \, dx,x,\frac {1}{3}+x\right )\right )+4 \text {Subst}\left (\int \frac {\left (\frac {31}{27}-\frac {2 x}{3}+2 x^3\right )^{2/3}}{-\frac {1}{3}+x} \, dx,x,\frac {1}{3}+x\right )+\int \left (\frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}+\frac {8 x \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}-\frac {4 x^2 \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3}\right ) \, dx\\ &=3 \int \frac {\left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx-4 \int \frac {x^2 \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx+8 \int \frac {x \left (1+2 x^2+2 x^3\right )^{2/3}}{-1-2 x^2+x^3} \, dx-\frac {\left (9\ 2^{2/3} \sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{6} \sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+2 x\right )^{2/3} \left (\frac {1}{9} \left (-4+\left (62-6 \sqrt {105}\right )^{2/3}+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}\right )-\frac {2}{3} \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+4 x^2\right )^{2/3}}{\left (-\frac {1}{3}+x\right )^3} \, dx,x,\frac {1}{3}+x\right )}{\left (4+\sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+12 x\right )^{2/3} \left (\sqrt [3]{3 \left (9+\sqrt {105}\right )} \left (2-\sqrt [3]{62-6 \sqrt {105}}\right )+2 \left (4-\sqrt [3]{62-6 \sqrt {105}}-\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+12 x^2\right )^{2/3}}+\frac {\left (12\ 2^{2/3} \sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{6} \sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+2 x\right )^{2/3} \left (\frac {1}{9} \left (-4+\left (62-6 \sqrt {105}\right )^{2/3}+\frac {8 \sqrt [3]{2}}{\left (31-3 \sqrt {105}\right )^{2/3}}\right )-\frac {2}{3} \left (\sqrt [3]{62-6 \sqrt {105}}+\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+4 x^2\right )^{2/3}}{-\frac {1}{3}+x} \, dx,x,\frac {1}{3}+x\right )}{\left (4+\sqrt [3]{31+3 \sqrt {105}} \left (2 \sqrt [3]{2}+\left (31-3 \sqrt {105}\right )^{2/3}\right )+12 x\right )^{2/3} \left (\sqrt [3]{3 \left (9+\sqrt {105}\right )} \left (2-\sqrt [3]{62-6 \sqrt {105}}\right )+2 \left (4-\sqrt [3]{62-6 \sqrt {105}}-\frac {2\ 2^{2/3}}{\sqrt [3]{31-3 \sqrt {105}}}\right ) x+12 x^2\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 161, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+2 x^2+2 x^3\right )^{2/3}}{2 x^2}-3 \sqrt [6]{3} \text {ArcTan}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+2 x^2+2 x^3}}\right )+3^{2/3} \log \left (-3 x+3^{2/3} \sqrt [3]{1+2 x^2+2 x^3}\right )-\frac {1}{2} 3^{2/3} \log \left (3 x^2+3^{2/3} x \sqrt [3]{1+2 x^2+2 x^3}+\sqrt [3]{3} \left (1+2 x^2+2 x^3\right )^{2/3}\right ) \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 = 11.15, size = 1105, normalized size = 6.86
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1105\) |
trager | \(\text {Expression too large to display}\) | \(1541\) |
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. 438 vs.
\(2 (131) = 262\).
time = 9.29, size = 438, normalized size = 2.72 \begin {gather*} -\frac {2 \cdot 9^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {2 \cdot 9^{\frac {2}{3}} \sqrt {3} {\left (8 \, x^{7} - 14 \, x^{6} - 4 \, x^{5} - 7 \, x^{4} - 4 \, x^{3} - x\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} - 6 \cdot 9^{\frac {1}{3}} \sqrt {3} {\left (55 \, x^{8} + 50 \, x^{7} + 4 \, x^{6} + 25 \, x^{5} + 4 \, x^{4} + x^{2}\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (377 \, x^{9} + 600 \, x^{8} + 204 \, x^{7} + 308 \, x^{6} + 204 \, x^{5} + 12 \, x^{4} + 51 \, x^{3} + 6 \, x^{2} + 1\right )}}{3 \, {\left (487 \, x^{9} + 480 \, x^{8} + 12 \, x^{7} + 232 \, x^{6} + 12 \, x^{5} - 12 \, x^{4} + 3 \, x^{3} - 6 \, x^{2} - 1\right )}}\right ) - 2 \cdot 9^{\frac {1}{3}} x^{2} \log \left (\frac {3 \cdot 9^{\frac {2}{3}} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}} x^{2} - 9 \, {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} x - 9^{\frac {1}{3}} {\left (x^{3} - 2 \, x^{2} - 1\right )}}{x^{3} - 2 \, x^{2} - 1}\right ) + 9^{\frac {1}{3}} x^{2} \log \left (\frac {9 \cdot 9^{\frac {1}{3}} {\left (8 \, x^{4} + 2 \, x^{3} + x\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}} + 9^{\frac {2}{3}} {\left (55 \, x^{6} + 50 \, x^{5} + 4 \, x^{4} + 25 \, x^{3} + 4 \, x^{2} + 1\right )} + 27 \, {\left (7 \, x^{5} + 4 \, x^{4} + 2 \, x^{2}\right )} {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{5} + 4 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1}\right ) - 9 \, {\left (2 \, x^{3} + 2 \, x^{2} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \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 (2 x^{2} + 3\right ) \left (2 x^{3} + 2 x^{2} + 1\right )^{\frac {2}{3}}}{x^{3} \left (x^{3} - 2 x^{2} - 1\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 {\left (2\,x^2+3\right )\,{\left (2\,x^3+2\,x^2+1\right )}^{2/3}}{x^3\,\left (-x^3+2\,x^2+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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