Integrand size = 22, antiderivative size = 113 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}-\frac {10 a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {5 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \]
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Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {424, 396, 245} \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=-\frac {10 a \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}+\frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {5 a \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{3 \sqrt [3]{b}} \]
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Rule 245
Rule 396
Rule 424
Rubi steps \begin{align*} \text {integral}& = \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {\int \frac {-a^2 b+7 a b^2 x^3}{\sqrt [3]{a+b x^3}} \, dx}{a b} \\ & = \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}-\frac {1}{3} (10 a) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx \\ & = \frac {2 x \left (a-b x^3\right )}{\sqrt [3]{a+b x^3}}+\frac {7}{3} x \left (a+b x^3\right )^{2/3}-\frac {10 a \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{b}}+\frac {5 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{3 \sqrt [3]{b}} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{9} \left (\frac {3 \left (13 a x+b x^4\right )}{\sqrt [3]{a+b x^3}}-\frac {10 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{\sqrt [3]{b}}+\frac {10 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{b}}-\frac {5 a \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{\sqrt [3]{b}}\right ) \]
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Time = 4.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.38
method | result | size |
pseudoelliptic | \(\frac {3 b^{\frac {4}{3}} x^{4}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a \left (b \,x^{3}+a \right )^{\frac {1}{3}}+39 x a \,b^{\frac {1}{3}}+10 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a \left (b \,x^{3}+a \right )^{\frac {1}{3}}-5 \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a \left (b \,x^{3}+a \right )^{\frac {1}{3}}}{9 b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}}\) | \(156\) |
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Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (89) = 178\).
Time = 0.36 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.65 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\left [\frac {15 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 10 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (b^{2} x^{4} + 13 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (b^{2} x^{3} + a b\right )}}, \frac {10 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (a b x^{3} + a^{2}\right )} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {30 \, \sqrt {\frac {1}{3}} {\left (a b^{2} x^{3} + a^{2} b\right )} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right )}{b^{\frac {1}{3}}} + 3 \, {\left (b^{2} x^{4} + 13 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{9 \, {\left (b^{2} x^{3} + a b\right )}}\right ] \]
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\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {\left (- a + b x^{3}\right )^{2}}{\left (a + b x^{3}\right )^{\frac {4}{3}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (89) = 178\).
Time = 0.29 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.62 \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\frac {1}{9} \, b^{2} {\left (\frac {4 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {7}{3}}} + \frac {3 \, {\left (3 \, a b - \frac {4 \, {\left (b x^{3} + a\right )} a}{x^{3}}\right )}}{\frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{3}}{x} - \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} b^{2}}{x^{4}}} - \frac {2 \, a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {7}{3}}} + \frac {4 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {7}{3}}}\right )} + \frac {1}{3} \, a b {\left (\frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{b^{\frac {4}{3}}} + \frac {6 \, x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} b} - \frac {\log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{b^{\frac {4}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}}\right )} + \frac {a x}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \]
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\[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{3} - a\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a-b x^3\right )^2}{\left (a+b x^3\right )^{4/3}} \, dx=\int \frac {{\left (a-b\,x^3\right )}^2}{{\left (b\,x^3+a\right )}^{4/3}} \,d x \]
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