Integrand size = 21, antiderivative size = 175 \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {d (9 b c-4 a d) x \left (a+b x^3\right )^{2/3}}{18 b^2}+\frac {d x \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )}{6 b}+\frac {\left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3}}-\frac {\left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{7/3}} \]
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Time = 0.08 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {427, 396, 245} \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right ) \left (2 a^2 d^2-6 a b c d+9 b^2 c^2\right )}{9 \sqrt {3} b^{7/3}}-\frac {\left (2 a^2 d^2-6 a b c d+9 b^2 c^2\right ) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{18 b^{7/3}}+\frac {d x \left (a+b x^3\right )^{2/3} (9 b c-4 a d)}{18 b^2}+\frac {d x \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )}{6 b} \]
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Rule 245
Rule 396
Rule 427
Rubi steps \begin{align*} \text {integral}& = \frac {d x \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )}{6 b}+\frac {\int \frac {c (6 b c-a d)+d (9 b c-4 a d) x^3}{\sqrt [3]{a+b x^3}} \, dx}{6 b} \\ & = \frac {d (9 b c-4 a d) x \left (a+b x^3\right )^{2/3}}{18 b^2}+\frac {d x \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )}{6 b}+\frac {\left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \int \frac {1}{\sqrt [3]{a+b x^3}} \, dx}{9 b^2} \\ & = \frac {d (9 b c-4 a d) x \left (a+b x^3\right )^{2/3}}{18 b^2}+\frac {d x \left (a+b x^3\right )^{2/3} \left (c+d x^3\right )}{6 b}+\frac {\left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} b^{7/3}}-\frac {\left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )}{18 b^{7/3}} \\ \end{align*}
Time = 0.99 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.27 \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {3 \sqrt [3]{b} d x \left (a+b x^3\right )^{2/3} \left (-4 a d+3 b \left (4 c+d x^3\right )\right )+2 \sqrt {3} \left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2 \sqrt [3]{a+b x^3}}\right )-2 \left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log \left (-\sqrt [3]{b} x+\sqrt [3]{a+b x^3}\right )+\left (9 b^2 c^2-6 a b c d+2 a^2 d^2\right ) \log \left (b^{2/3} x^2+\sqrt [3]{b} x \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )}{54 b^{7/3}} \]
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Time = 4.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.95
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-9 x \left (\frac {d \,x^{3}}{4}+c \right ) d \left (b \,x^{3}+a \right )^{\frac {2}{3}} b^{\frac {4}{3}}+3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} a \,d^{2} x \,b^{\frac {1}{3}}+\left (\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (b^{\frac {1}{3}} x +2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 b^{\frac {1}{3}} x}\right )+\ln \left (\frac {-b^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (\frac {b^{\frac {2}{3}} x^{2}+b^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}\right ) \left (a^{2} d^{2}-3 a b c d +\frac {9}{2} b^{2} c^{2}\right )\right )}{27 b^{\frac {7}{3}}}\) | \(166\) |
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none
Time = 0.33 (sec) , antiderivative size = 554, normalized size of antiderivative = 3.17 \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (9 \, b^{3} c^{2} - 6 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} \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 ) - 2 \, {\left (9 \, b^{2} c^{2} - 6 \, a b c d + 2 \, a^{2} d^{2}\right )} \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 ) + {\left (9 \, b^{2} c^{2} - 6 \, a b c d + 2 \, a^{2} d^{2}\right )} \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 (3 \, b^{2} d^{2} x^{4} + 4 \, {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (9 \, b^{3} c^{2} - 6 \, a b^{2} c d + 2 \, a^{2} b d^{2}\right )} \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 ) + 2 \, {\left (9 \, b^{2} c^{2} - 6 \, a b c d + 2 \, a^{2} d^{2}\right )} \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 ) - {\left (9 \, b^{2} c^{2} - 6 \, a b c d + 2 \, a^{2} d^{2}\right )} \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 (3 \, b^{2} d^{2} x^{4} + 4 \, {\left (3 \, b^{2} c d - a b d^{2}\right )} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, b^{3}}\right ] \]
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Result contains complex when optimal does not.
Time = 2.81 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.72 \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\frac {c^{2} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {2 c d x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {d^{2} x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (148) = 296\).
Time = 0.28 (sec) , antiderivative size = 436, normalized size of antiderivative = 2.49 \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=-\frac {1}{6} \, {\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 {1}{3}}} - \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 {1}{3}}} + \frac {2 \, \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {1}{3}}}\right )} c^{2} + \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 {4}{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 {4}{3}}} + \frac {2 \, a \log \left (-b^{\frac {1}{3}} + \frac {{\left (b x^{3} + a\right )}^{\frac {1}{3}}}{x}\right )}{b^{\frac {4}{3}}} - \frac {6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{{\left (b^{2} - \frac {{\left (b x^{3} + a\right )} b}{x^{3}}\right )} x^{2}}\right )} c d - \frac {1}{54} \, {\left (\frac {4 \, \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 {7}{3}}} - \frac {2 \, 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 {7}{3}}} + \frac {4 \, a^{2} \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 {7 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{2} b}{x^{2}} - \frac {4 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}\right )}}{b^{4} - \frac {2 \, {\left (b x^{3} + a\right )} b^{3}}{x^{3}} + \frac {{\left (b x^{3} + a\right )}^{2} b^{2}}{x^{6}}}\right )} d^{2} \]
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\[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\int { \frac {{\left (d x^{3} + c\right )}^{2}}{{\left (b x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (c+d x^3\right )^2}{\sqrt [3]{a+b x^3}} \, dx=\int \frac {{\left (d\,x^3+c\right )}^2}{{\left (b\,x^3+a\right )}^{1/3}} \,d x \]
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