Integrand size = 17, antiderivative size = 212 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{162} \left (a+x^3\right )^{2/3} \left (28 a^2 x-108 a b x+162 b^2 x-21 a x^4+81 b x^4+18 x^7\right )+\frac {1}{243} \left (-14 \sqrt {3} a^3+54 \sqrt {3} a^2 b-81 \sqrt {3} a b^2+81 \sqrt {3} b^3\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \]
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Time = 0.08 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {427, 542, 396, 245} \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{162} x \left (28 a^2-87 a b+99 b^2\right ) \left (a+x^3\right )^{2/3}-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \arctan \left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (\sqrt [3]{a+x^3}-x\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {1}{54} x (7 a-15 b) \left (a+x^3\right )^{2/3} \left (b+x^3\right ) \]
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Rule 245
Rule 396
Rule 427
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{9} \int \frac {\left (b+x^3\right ) \left (-((a-9 b) b)+(-7 a+15 b) x^3\right )}{\sqrt [3]{a+x^3}} \, dx \\ & = -\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{54} \int \frac {b \left (7 a^2-21 a b+54 b^2\right )+\left (28 a^2-87 a b+99 b^2\right ) x^3}{\sqrt [3]{a+x^3}} \, dx \\ & = \frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2+\frac {1}{81} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx \\ & = \frac {1}{162} \left (28 a^2-87 a b+99 b^2\right ) x \left (a+x^3\right )^{2/3}-\frac {1}{54} (7 a-15 b) x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )^2-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{162} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right ) \\ \end{align*}
Time = 0.98 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.92 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{162} x \left (a+x^3\right )^{2/3} \left (28 a^2-108 a b+162 b^2-21 a x^3+81 b x^3+18 x^6\right )-\frac {\left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )}{81 \sqrt {3}}+\frac {1}{243} \left (14 a^3-54 a^2 b+81 a b^2-81 b^3\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{486} \left (-14 a^3+54 a^2 b-81 a b^2+81 b^3\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \]
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Time = 0.58 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.97
method | result | size |
pseudoelliptic | \(-\frac {14 \left (\sqrt {3}\, \left (a^{3}-\frac {27}{7} a^{2} b +\frac {81}{14} a \,b^{2}-\frac {81}{14} b^{3}\right ) \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+a \right )^{\frac {1}{3}}\right )}{3 x}\right )+\frac {27 x^{7} \left (x^{3}+a \right )^{\frac {2}{3}}}{14}-\frac {9 \left (a -\frac {27 b}{7}\right ) \left (x^{3}+a \right )^{\frac {2}{3}} x^{4}}{4}+3 \left (a^{2}-\frac {27}{7} a b +\frac {81}{14} b^{2}\right ) \left (x^{3}+a \right )^{\frac {2}{3}} x +\left (\ln \left (\frac {-x +\left (x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a^{3}-\frac {27}{7} a^{2} b +\frac {81}{14} a \,b^{2}-\frac {81}{14} b^{3}\right )\right ) a^{3}}{243 \left (x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}\right )^{3} {\left (x -\left (x^{3}+a \right )^{\frac {1}{3}}\right )}^{3}}\) | \(205\) |
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none
Time = 0.29 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{243} \, \sqrt {3} {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{243} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{486} \, {\left (14 \, a^{3} - 54 \, a^{2} b + 81 \, a b^{2} - 81 \, b^{3}\right )} \log \left (\frac {x^{2} + {\left (x^{3} + a\right )}^{\frac {1}{3}} x + {\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{162} \, {\left (18 \, x^{7} - 3 \, {\left (7 \, a - 27 \, b\right )} x^{4} + 2 \, {\left (14 \, a^{2} - 54 \, a b + 81 \, b^{2}\right )} x\right )} {\left (x^{3} + a\right )}^{\frac {2}{3}} \]
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Result contains complex when optimal does not.
Time = 8.94 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.71 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {b^{3} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {b^{2} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {b x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{\sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} + \frac {x^{10} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {13}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 480 vs. \(2 (184) = 368\).
Time = 0.30 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.26 \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\frac {14}{243} \, \sqrt {3} a^{3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {1}{6} \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right )\right )} b^{3} - \frac {7}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {14}{243} \, a^{3} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{6} \, {\left (2 \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {6 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a}{x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}}\right )} b^{2} - \frac {1}{18} \, {\left (4 \, \sqrt {3} a^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - 2 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 4 \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {3 \, {\left (\frac {7 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{2}}{x^{2}} - \frac {4 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1}\right )} b + \frac {\frac {67 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{3}}{x^{2}} - \frac {77 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{3}}{x^{5}} + \frac {28 \, {\left (x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}}{162 \, {\left (\frac {3 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} + a\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} + a\right )}^{3}}{x^{9}} - 1\right )}} \]
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\[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\int { \frac {{\left (x^{3} + b\right )}^{3}}{{\left (x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (b+x^3\right )^3}{\sqrt [3]{a+x^3}} \, dx=\int \frac {{\left (x^3+b\right )}^3}{{\left (x^3+a\right )}^{1/3}} \,d x \]
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