Integrand size = 15, antiderivative size = 117 \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\frac {a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac {1}{6} x^4 \left (a+b x^3\right )^{2/3}-\frac {a^2 \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{4/3}}+\frac {a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{4/3}} \]
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Time = 0.02 (sec) , antiderivative size = 117, 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, 245} \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=-\frac {a^2 \arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} b^{4/3}}+\frac {a^2 \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{4/3}}+\frac {a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac {1}{6} x^4 \left (a+b x^3\right )^{2/3} \]
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Rule 245
Rule 285
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^4 \left (a+b x^3\right )^{2/3}+\frac {1}{3} a \int \frac {x^3}{\sqrt [3]{a+b x^3}} \, dx \\ & = \frac {a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac {1}{6} x^4 \left (a+b x^3\right )^{2/3}-\frac {a^2 \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{9 b} \\ & = \frac {a x \left (a+b x^3\right )^{2/3}}{9 b}+\frac {1}{6} x^4 \left (a+b x^3\right )^{2/3}-\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^{4/3}}+\frac {a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{4/3}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.44 \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\frac {\left (a+b x^3\right )^{2/3} \left (2 a x+3 b x^4\right )}{18 b}-\frac {a^2 \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )}{9 \sqrt {3} b^{4/3}}+\frac {a^2 \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{27 b^{4/3}}-\frac {a^2 \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{54 b^{4/3}} \]
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Time = 4.06 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.23
method | result | size |
pseudoelliptic | \(\frac {9 x^{4} \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {4}{3}}+6 a x \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {1}{3}}+2 a^{2} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+2 a^{2} \ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-a^{2} \ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{54 b^{\frac {4}{3}}}\) | \(144\) |
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none
Time = 0.27 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.95 \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a^{2} b \sqrt {-\frac {1}{b^{\frac {2}{3}}}} \log \left (3 \, b x^{3} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {2}{3}} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (b^{\frac {4}{3}} x^{3} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b x^{2} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{\frac {2}{3}} x\right )} \sqrt {-\frac {1}{b^{\frac {2}{3}}}} + 2 \, a\right ) + 2 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - a^{2} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (3 \, b^{2} x^{4} + 2 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{2}}, \frac {6 \, \sqrt {\frac {1}{3}} a^{2} b^{\frac {2}{3}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (b^{\frac {1}{3}} x + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{b^{\frac {1}{3}} x}\right ) + 2 \, a^{2} b^{\frac {2}{3}} \log \left (-\frac {b^{\frac {1}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) - a^{2} b^{\frac {2}{3}} \log \left (\frac {b^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (3 \, b^{2} x^{4} + 2 \, a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{2}}\right ] \]
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Result contains complex when optimal does not.
Time = 1.12 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.33 \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\frac {a^{\frac {2}{3}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \Gamma \left (\frac {7}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 181 vs. \(2 (90) = 180\).
Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.55 \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (b^{\frac {1}{3}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}}{3 \, b^{\frac {1}{3}}}\right )}{27 \, b^{\frac {4}{3}}} - \frac {a^{2} \log \left (b^{\frac {2}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{\frac {1}{3}}}{x} + \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right )}{54 \, b^{\frac {4}{3}}} + \frac {a^{2} \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{27 \, b^{\frac {4}{3}}} + \frac {\frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} + \frac {2 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}}{18 \, {\left (b^{3} - \frac {2 \, {\left (b x^{3} + a\right )} b^{2}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b}{x^{6}}\right )}} \]
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\[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\int { {\left (b x^{3} + a\right )}^{\frac {2}{3}} x^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (a+b x^3\right )^{2/3} \, dx=\int x^3\,{\left (b\,x^3+a\right )}^{2/3} \,d x \]
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