Integrand size = 22, antiderivative size = 139 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\frac {38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac {8}{27} a x \left (a+b x^3\right )^{5/3}-\frac {1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac {76 a^3 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3} \sqrt [3]{b}}-\frac {38 a^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{81 \sqrt [3]{b}} \]
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Time = 0.04 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {427, 396, 201, 245} \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\frac {76 a^3 \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{81 \sqrt {3} \sqrt [3]{b}}-\frac {38 a^3 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{81 \sqrt [3]{b}}+\frac {38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac {8}{27} a x \left (a+b x^3\right )^{5/3}-\frac {1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3} \]
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Rule 201
Rule 245
Rule 396
Rule 427
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac {\int \left (a+b x^3\right )^{2/3} \left (10 a^2 b-16 a b^2 x^3\right ) \, dx}{9 b} \\ & = -\frac {8}{27} a x \left (a+b x^3\right )^{5/3}-\frac {1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac {1}{27} \left (38 a^2\right ) \int \left (a+b x^3\right )^{2/3} \, dx \\ & = \frac {38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac {8}{27} a x \left (a+b x^3\right )^{5/3}-\frac {1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac {1}{81} \left (76 a^3\right ) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx \\ & = \frac {38}{81} a^2 x \left (a+b x^3\right )^{2/3}-\frac {8}{27} a x \left (a+b x^3\right )^{5/3}-\frac {1}{9} x \left (a-b x^3\right ) \left (a+b x^3\right )^{5/3}+\frac {76 a^3 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3} \sqrt [3]{b}}-\frac {38 a^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{81 \sqrt [3]{b}} \\ \end{align*}
Time = 0.76 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.20 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\frac {3 \sqrt [3]{b} \left (a+b x^3\right )^{2/3} \left (5 a^2 x-24 a b x^4+9 b^2 x^7\right )+76 \sqrt {3} a^3 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-76 a^3 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+38 a^3 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{243 \sqrt [3]{b}} \]
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Time = 4.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.18
method | result | size |
pseudoelliptic | \(\frac {27 b^{\frac {7}{3}} \left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{7}-72 a \,b^{\frac {4}{3}} x^{4} \left (b \,x^{3}+a \right )^{\frac {2}{3}}+15 a^{2} x \,b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {2}{3}}-76 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a^{3}-76 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a^{3}+38 \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a^{3}}{243 b^{\frac {1}{3}}}\) | \(164\) |
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none
Time = 0.44 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.03 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\left [\frac {114 \, \sqrt {\frac {1}{3}} a^{3} b \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} b x^{3} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b\right )^{\frac {2}{3}} x\right )} \sqrt {\frac {\left (-b\right )^{\frac {1}{3}}}{b}} + 2 \, a\right ) - 76 \, a^{3} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + 38 \, a^{3} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (9 \, b^{3} x^{7} - 24 \, a b^{2} x^{4} + 5 \, a^{2} b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{243 \, b}, -\frac {228 \, \sqrt {\frac {1}{3}} a^{3} b \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (\left (-b\right )^{\frac {1}{3}} x - 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-b\right )^{\frac {1}{3}}}{b}}}{x}\right ) + 76 \, a^{3} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 38 \, a^{3} \left (-b\right )^{\frac {2}{3}} \log \left (\frac {\left (-b\right )^{\frac {2}{3}} x^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 3 \, {\left (9 \, b^{3} x^{7} - 24 \, a b^{2} x^{4} + 5 \, a^{2} b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{243 \, b}\right ] \]
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Result contains complex when optimal does not.
Time = 5.34 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.91 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\frac {a^{\frac {8}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {2 a^{\frac {5}{3}} b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {a^{\frac {2}{3}} b^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (109) = 218\).
Time = 0.29 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.97 \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=-\frac {1}{9} \, {\left (\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {1}{3}}} - \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {1}{3}}} + \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {1}{3}}} + \frac {3 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{{\left (b - \frac {b x^{3} + a}{x^{3}}\right )} x^{2}}\right )} a^{2} - \frac {1}{27} \, {\left (\frac {2 \, \sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} - \frac {a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}} + \frac {3 \, {\left (\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{3} - \frac {2 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b}{x^{6}}}\right )} a b - \frac {1}{243} \, {\left (\frac {4 \, \sqrt {3} a^{3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} - \frac {2 \, a^{3} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, a^{3} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (\frac {2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{3} b^{2}}{x^{2}} + \frac {11 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{3} b}{x^{5}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} a^{3}}{x^{8}}\right )}}{b^{5} - \frac {3 \, {\left (b x^{3} + a\right )} b^{4}}{x^{3}} + \frac {3 \, {\left (b x^{3} + a\right )}^{2} b^{3}}{x^{6}} - \frac {{\left (b x^{3} + a\right )}^{3} b^{2}}{x^{9}}}\right )} b^{2} \]
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\[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b x^{3} - a\right )}^{2} \,d x } \]
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Timed out. \[ \int \left (a-b x^3\right )^2 \left (a+b x^3\right )^{2/3} \, dx=\int {\left (b\,x^3+a\right )}^{2/3}\,{\left (a-b\,x^3\right )}^2 \,d x \]
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