Integrand size = 15, antiderivative size = 120 \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=\frac {a x^2 \sqrt [3]{a+b x^3}}{18 b}+\frac {1}{6} x^5 \sqrt [3]{a+b x^3}+\frac {a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{5/3}}+\frac {a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{5/3}} \]
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Time = 0.03 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {285, 327, 337} \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=\frac {a^2 \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{5/3}}+\frac {a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{5/3}}+\frac {1}{6} x^5 \sqrt [3]{a+b x^3}+\frac {a x^2 \sqrt [3]{a+b x^3}}{18 b} \]
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Rule 285
Rule 327
Rule 337
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^5 \sqrt [3]{a+b x^3}+\frac {1}{6} a \int \frac {x^4}{\left (a+b x^3\right )^{2/3}} \, dx \\ & = \frac {a x^2 \sqrt [3]{a+b x^3}}{18 b}+\frac {1}{6} x^5 \sqrt [3]{a+b x^3}-\frac {a^2 \int \frac {x}{\left (a+b x^3\right )^{2/3}} \, dx}{9 b} \\ & = \frac {a x^2 \sqrt [3]{a+b x^3}}{18 b}+\frac {1}{6} x^5 \sqrt [3]{a+b x^3}+\frac {a^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{5/3}}+\frac {a^2 \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{18 b^{5/3}} \\ \end{align*}
Time = 0.36 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.40 \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=\frac {x^2 \sqrt [3]{a+b x^3} \left (a+3 b x^3\right )}{18 b}+\frac {a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{9 \sqrt {3} b^{5/3}}+\frac {a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{27 b^{5/3}}-\frac {a^2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{54 b^{5/3}} \]
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Time = 4.94 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22
method | result | size |
pseudoelliptic | \(\frac {9 x^{5} \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{\frac {5}{3}}+3 a \,x^{2} \left (b \,x^{3}+a \right )^{\frac {1}{3}} b^{\frac {2}{3}}-2 a^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+2 a^{2} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-a^{2} \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{54 b^{\frac {5}{3}}}\) | \(146\) |
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none
Time = 0.26 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.51 \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=-\frac {2 \, \sqrt {3} a^{2} {\left (b^{2}\right )}^{\frac {1}{6}} b \arctan \left (\frac {{\left (\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}}\right )} {\left (b^{2}\right )}^{\frac {1}{6}}}{3 \, b^{2} x}\right ) - 2 \, a^{2} {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (b^{2}\right )}^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) + a^{2} {\left (b^{2}\right )}^{\frac {2}{3}} \log \left (\frac {{\left (b^{2}\right )}^{\frac {1}{3}} b x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right ) - 3 \, {\left (3 \, b^{3} x^{5} + a b^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{54 \, b^{3}} \]
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Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.32 \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=\frac {\sqrt [3]{a} x^{5} \Gamma \left (\frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {5}{3} \\ \frac {8}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {8}{3}\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.51 \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=-\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{27 \, b^{\frac {5}{3}}} - \frac {a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{54 \, b^{\frac {5}{3}}} + \frac {a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{27 \, b^{\frac {5}{3}}} + \frac {\frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{2} b}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2}}{x^{4}}}{18 \, {\left (b^{3} - \frac {2 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b}{x^{6}}\right )}} \]
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\[ \int x^4 \sqrt [3]{a+b x^3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {1}{3}} x^{4} \,d x } \]
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Timed out. \[ \int x^4 \sqrt [3]{a+b x^3} \, dx=\int x^4\,{\left (b\,x^3+a\right )}^{1/3} \,d x \]
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