Integrand size = 26, antiderivative size = 193 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {\left (a x^2+x^3\right )^{2/3} \left (-7280 a^5+5460 a^4 x-4680 a^3 x^2+4212 a^2 x^3-3888 a x^4+3645 x^5\right )}{21870 x}+\frac {\left (728 \sqrt {3} a^6-6561 \sqrt {3} b^2\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a x^2+x^3}}\right )}{6561}+\frac {\left (-728 a^6+6561 b^2\right ) \log \left (-x+\sqrt [3]{a x^2+x^3}\right )}{6561}+\frac {\left (728 a^6-6561 b^2\right ) \log \left (x^2+x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}\right )}{13122} \]
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Leaf count is larger than twice the leaf count of optimal. \(416\) vs. \(2(193)=386\).
Time = 0.26 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.16, number of steps used = 12, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2078, 2036, 61, 2049} \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\frac {728 a^6 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3} \]
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Rule 61
Rule 2036
Rule 2049
Rule 2078
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b^2}{\sqrt [3]{a x^2+x^3}}+\frac {x^6}{\sqrt [3]{a x^2+x^3}}\right ) \, dx \\ & = -\left (b^2 \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx\right )+\int \frac {x^6}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = \frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}-\frac {1}{9} (8 a) \int \frac {x^5}{\sqrt [3]{a x^2+x^3}} \, dx-\frac {\left (b^2 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{\sqrt [3]{a x^2+x^3}} \\ & = -\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {1}{135} \left (104 a^2\right ) \int \frac {x^4}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = \frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {1}{81} \left (52 a^3\right ) \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = -\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {1}{729} \left (364 a^4\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = \frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (728 a^5\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx}{2187} \\ & = \frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (728 a^6\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx}{6561} \\ & = \frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (728 a^6 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{6561 \sqrt [3]{a x^2+x^3}} \\ & = \frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}-\frac {728 a^6 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}} \\ \end{align*}
Time = 0.70 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {-21840 a^6 x-5460 a^5 x^2+2340 a^4 x^3-1404 a^3 x^4+972 a^2 x^5-729 a x^6+10935 x^7+10 \sqrt {3} \left (728 a^6-6561 b^2\right ) x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{a+x}}\right )+10 \left (-728 a^6+6561 b^2\right ) x^{2/3} \sqrt [3]{a+x} \log \left (-\sqrt [3]{x}+\sqrt [3]{a+x}\right )+3640 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{a+x}+(a+x)^{2/3}\right )-32805 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{a+x}+(a+x)^{2/3}\right )}{65610 \sqrt [3]{x^2 (a+x)}} \]
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Time = 0.64 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.06
method | result | size |
pseudoelliptic | \(-\frac {728 \left (-\frac {\left (a^{6}-\frac {6561 b^{2}}{728}\right ) x \ln \left (\frac {\left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, x \left (a^{6}-\frac {6561 b^{2}}{728}\right ) \arctan \left (\frac {\left (2 \left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\left (a^{6}-\frac {6561 b^{2}}{728}\right ) x \ln \left (\frac {\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}-x}{x}\right )+3 \left (a^{5}-\frac {3}{4} a^{4} x +\frac {9}{14} a^{3} x^{2}-\frac {81}{140} a^{2} x^{3}+\frac {243}{455} a \,x^{4}-\frac {729}{1456} x^{5}\right ) \left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}\right ) x^{11} a^{6}}{6561 {\left (-\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}+x \right )}^{6} {\left (\left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}+x \left (x +\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}\right )\right )}^{6}}\) | \(204\) |
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Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\frac {10 \, \sqrt {3} {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 10 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \log \left (-\frac {x - {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \log \left (\frac {x^{2} + {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}} x + {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (7280 \, a^{5} - 5460 \, a^{4} x + 4680 \, a^{3} x^{2} - 4212 \, a^{2} x^{3} + 3888 \, a x^{4} - 3645 \, x^{5}\right )} {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{65610 \, x} \]
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\[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int \frac {\left (- b + x^{3}\right ) \left (b + x^{3}\right )}{\sqrt [3]{x^{2} \left (a + x\right )}}\, dx \]
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\[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int { \frac {{\left (x^{3} + b\right )} {\left (x^{3} - b\right )}}{{\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\frac {10 \, \sqrt {3} {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 5 \, {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \log \left ({\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \log \left ({\left | {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) + \frac {3 \, {\left (7280 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} - 41860 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} + 99320 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} - 123812 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 84592 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 29165 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )} x^{6}}{a^{6}}}{65610 \, a} \]
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Timed out. \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\int \frac {\left (x^3+b\right )\,\left (b-x^3\right )}{{\left (x^3+a\,x^2\right )}^{1/3}} \,d x \]
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