Integrand size = 15, antiderivative size = 97 \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac {2 a \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3}}+\frac {a \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{5/3}} \]
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Time = 0.01 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {327, 337} \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {2 a \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3}}+\frac {a \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{5/3}}+\frac {x^2 \sqrt [3]{a+b x^3}}{3 b} \]
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Rule 327
Rule 337
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt [3]{a+b x^3}}{3 b}-\frac {(2 a) \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx}{3 b} \\ & = \frac {x^2 \sqrt [3]{a+b x^3}}{3 b}+\frac {2 a \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3}}+\frac {a \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{3 b^{5/3}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.46 \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {3 b^{2/3} x^2 \sqrt [3]{a+b x^3}+2 \sqrt {3} a \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )+2 a \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )-a \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{9 b^{5/3}} \]
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Time = 4.01 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.26
method | result | size |
pseudoelliptic | \(\frac {3 \left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{2} b^{\frac {2}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right ) a +2 \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) a -\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) a}{9 b^{\frac {5}{3}}}\) | \(122\) |
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (74) = 148\).
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.70 \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{2} x^{2} - 2 \, \sqrt {3} a {\left (b^{2}\right )}^{\frac {1}{6}} b \arctan \left (\frac {{\left (\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}}\right )} {\left (b^{2}\right )}^{\frac {1}{6}}}{3 \, b^{2} x}\right ) + 2 \, a {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2}\right )}^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) - a {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (b^{2}\right )}^{\frac {1}{3}} b x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right )}{9 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.38 \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\frac {x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} \Gamma \left (\frac {8}{3}\right )} \]
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none
Time = 0.34 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.41 \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=-\frac {2 \, \sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{9 \, b^{\frac {5}{3}}} - \frac {a \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{9 \, b^{\frac {5}{3}}} + \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{9 \, b^{\frac {5}{3}}} - \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} a}{3 \, {\left (b^{2} - \frac {{\left (b x^{3} + a\right )} b}{x^{3}}\right )} x} \]
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\[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\int { \frac {x^{4}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx=\int \frac {x^4}{{\left (b\,x^3+a\right )}^{2/3}} \,d x \]
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