Integrand size = 20, antiderivative size = 180 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{18} \left (a+x^3\right )^{2/3} \left (-4 a x+6 b x+6 c x+3 x^4\right )+\frac {1}{27} \left (2 \sqrt {3} a^2-3 \sqrt {3} a b-3 \sqrt {3} a c+9 \sqrt {3} b c\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+\frac {1}{27} \left (-2 a^2+3 a b+3 a c-9 b c\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\frac {1}{54} \left (2 a^2-3 a b-3 a c+9 b c\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.69, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {542, 396, 245} \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{a+x^3}}+1}{\sqrt {3}}\right ) \left (2 a^2-3 a (b+c)+9 b c\right )}{9 \sqrt {3}}-\frac {1}{18} \left (2 a^2-3 a (b+c)+9 b c\right ) \log \left (\sqrt [3]{a+x^3}-x\right )-\frac {1}{18} x \left (a+x^3\right )^{2/3} (4 a-3 b-6 c)+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right ) \]
[In]
[Out]
Rule 245
Rule 396
Rule 542
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{6} \int \frac {-b (a-6 c)-(4 a-3 b-6 c) x^3}{\sqrt [3]{a+x^3}} \, dx \\ & = -\frac {1}{18} (4 a-3 b-6 c) x \left (a+x^3\right )^{2/3}+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {1}{9} \left (2 a^2+9 b c-3 a (b+c)\right ) \int \frac {1}{\sqrt [3]{a+x^3}} \, dx \\ & = -\frac {1}{18} (4 a-3 b-6 c) x \left (a+x^3\right )^{2/3}+\frac {1}{6} x \left (a+x^3\right )^{2/3} \left (b+x^3\right )+\frac {\left (2 a^2+9 b c-3 a (b+c)\right ) \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{a+x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3}}-\frac {1}{18} \left (2 a^2+9 b c-3 a (b+c)\right ) \log \left (-x+\sqrt [3]{a+x^3}\right ) \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {1}{54} \left (3 x \left (a+x^3\right )^{2/3} \left (-4 a+6 b+6 c+3 x^3\right )+2 \sqrt {3} \left (2 a^2+9 b c-3 a (b+c)\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a+x^3}}\right )+2 \left (-2 a^2-9 b c+3 a (b+c)\right ) \log \left (-x+\sqrt [3]{a+x^3}\right )+\left (2 a^2+9 b c-3 a (b+c)\right ) \log \left (x^2+x \sqrt [3]{a+x^3}+\left (a+x^3\right )^{2/3}\right )\right ) \]
[In]
[Out]
Time = 0.30 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\frac {\sqrt {3}\, \left (a^{2}+\frac {3 \left (-b -c \right ) a}{2}+\frac {9 b c}{2}\right ) \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}+a \right )^{\frac {1}{3}}\right )}{3 x}\right )}{3}-\frac {3 x^{4} \left (x^{3}+a \right )^{\frac {2}{3}}}{4}+\left (a -\frac {3 b}{2}-\frac {3 c}{2}\right ) \left (x^{3}+a \right )^{\frac {2}{3}} x -\frac {\left (\ln \left (\frac {x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (x^{3}+a \right )^{\frac {1}{3}}}{x}\right )\right ) \left (a^{2}+\frac {3 \left (-b -c \right ) a}{2}+\frac {9 b c}{2}\right )}{6}\right ) a^{2}}{9 \left (x^{2}+x \left (x^{3}+a \right )^{\frac {1}{3}}+\left (x^{3}+a \right )^{\frac {2}{3}}\right )^{2} {\left (x -\left (x^{3}+a \right )^{\frac {1}{3}}\right )}^{2}}\) | \(178\) |
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.88 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=-\frac {1}{27} \, \sqrt {3} {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + a\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{27} \, {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \log \left (-\frac {x - {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{54} \, {\left (2 \, a^{2} - 3 \, a b - 3 \, {\left (a - 3 \, b\right )} c\right )} \log \left (\frac {x^{2} + {\left (x^{3} + a\right )}^{\frac {1}{3}} x + {\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}}\right ) + \frac {1}{18} \, {\left (3 \, x^{4} - 2 \, {\left (2 \, a - 3 \, b - 3 \, c\right )} x\right )} {\left (x^{3} + a\right )}^{\frac {2}{3}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 3.47 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\frac {b c x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {4}{3}\right )} + \frac {b x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {c x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {7}{3}\right )} + \frac {x^{7} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac {10}{3}\right )} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (152) = 304\).
Time = 0.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 2.29 \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=-\frac {2}{27} \, \sqrt {3} a^{2} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {1}{6} \, {\left (2 \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right )\right )} b c + \frac {1}{27} \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {2}{27} \, a^{2} \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {1}{18} \, {\left (2 \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {6 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a}{x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}}\right )} b + \frac {1}{18} \, {\left (2 \, \sqrt {3} a \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + a\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + 2 \, a \log \left (\frac {{\left (x^{3} + a\right )}^{\frac {1}{3}}}{x} - 1\right ) + \frac {6 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a}{x^{2} {\left (\frac {x^{3} + a}{x^{3}} - 1\right )}}\right )} c - \frac {\frac {7 \, {\left (x^{3} + a\right )}^{\frac {2}{3}} a^{2}}{x^{2}} - \frac {4 \, {\left (x^{3} + a\right )}^{\frac {5}{3}} a^{2}}{x^{5}}}{18 \, {\left (\frac {2 \, {\left (x^{3} + a\right )}}{x^{3}} - \frac {{\left (x^{3} + a\right )}^{2}}{x^{6}} - 1\right )}} \]
[In]
[Out]
\[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\int { \frac {{\left (x^{3} + b\right )} {\left (x^{3} + c\right )}}{{\left (x^{3} + a\right )}^{\frac {1}{3}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (b+x^3\right ) \left (c+x^3\right )}{\sqrt [3]{a+x^3}} \, dx=\int \frac {\left (x^3+b\right )\,\left (x^3+c\right )}{{\left (x^3+a\right )}^{1/3}} \,d x \]
[In]
[Out]