Integrand size = 29, antiderivative size = 215 \[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\left (-4 b^2+3 a^3 x\right ) \left (b^2 x^2+a^3 x^3\right )^{2/3}}{6 a^5 x}-\frac {\left (9 a^5 b-2 b^4\right ) \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{3 \sqrt {3} a^6}+\frac {\left (9 a^5 b-2 b^4\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{9 a^6}+\frac {\left (-9 a^5 b+2 b^4\right ) \log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{18 a^6} \]
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Leaf count is larger than twice the leaf count of optimal. \(466\) vs. \(2(215)=430\).
Time = 0.19 (sec) , antiderivative size = 466, normalized size of antiderivative = 2.17, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2078, 2036, 61, 2049} \[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} b x^{2/3} \sqrt [3]{a^3 x+b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {b x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {3 b x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^4 x^{2/3} \sqrt [3]{a^3 x+b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3} a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 x^{2/3} \log (x) \sqrt [3]{a^3 x+b^2}}{9 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {b^4 x^{2/3} \sqrt [3]{a^3 x+b^2} \log \left (\frac {\sqrt [3]{a^3 x+b^2}}{a \sqrt [3]{x}}-1\right )}{3 a^6 \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {2 b^2 \left (a^3 x^3+b^2 x^2\right )^{2/3}}{3 a^5 x}+\frac {\left (a^3 x^3+b^2 x^2\right )^{2/3}}{2 a^2} \]
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Rule 61
Rule 2036
Rule 2049
Rule 2078
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b}{\sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}}\right ) \, dx \\ & = a \int \frac {x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx-b \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx \\ & = \frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {\left (2 b^2\right ) \int \frac {x}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx}{3 a^2}-\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}+\frac {\sqrt {3} b x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {b x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 b x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2 b^4\right ) \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx}{9 a^5} \\ & = \frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}+\frac {\sqrt {3} b x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {b x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 b x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (2 b^4 x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{9 a^5 \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \frac {\left (b^2 x^2+a^3 x^3\right )^{2/3}}{2 a^2}-\frac {2 b^2 \left (b^2 x^2+a^3 x^3\right )^{2/3}}{3 a^5 x}+\frac {\sqrt {3} b x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {2 b^4 x^{2/3} \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{3 \sqrt {3} a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {b x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b^4 x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{9 a^6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {3 b x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {b^4 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{3 a^6 \sqrt [3]{b^2 x^2+a^3 x^3}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.44 \[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \left (-12 a b^4 \sqrt [3]{x}-3 a^4 b^2 x^{4/3}+9 a^7 x^{7/3}+2 \sqrt {3} b \left (-9 a^5+2 b^3\right ) \sqrt [3]{b^2+a^3 x} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )+2 b \left (9 a^5-2 b^3\right ) \sqrt [3]{b^2+a^3 x} \log \left (-a \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right )-9 a^5 b \sqrt [3]{b^2+a^3 x} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )+2 b^4 \sqrt [3]{b^2+a^3 x} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )\right )}{18 a^6 \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]
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Time = 0.38 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.15
method | result | size |
pseudoelliptic | \(\frac {\left (-\frac {\left (a^{5}-\frac {2 b^{3}}{9}\right ) x b \ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, b x \left (a^{5}-\frac {2 b^{3}}{9}\right ) \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+\left (a^{5}-\frac {2 b^{3}}{9}\right ) x b \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+\frac {a \left (a^{3} x -\frac {4 b^{2}}{3}\right ) \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{2}\right ) x^{3} b^{4}}{{\left (a x -\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )}^{2} a^{6} {\left (\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}+a x \left (a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )\right )}^{2}}\) | \(248\) |
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Time = 0.28 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.96 \[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b - 2 \, b^{4}\right )} x \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + 2 \, {\left (9 \, a^{5} b - 2 \, b^{4}\right )} x \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (9 \, a^{5} b - 2 \, b^{4}\right )} x \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (3 \, a^{4} x - 4 \, a b^{2}\right )}}{18 \, a^{6} x} \]
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\[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a x^{2} - b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )}}\, dx \]
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\[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]
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Time = 0.34 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.91 \[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\frac {2 \, \sqrt {3} {\left (9 \, a^{5} b^{3} - 2 \, b^{6}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a^{6}} - \frac {{\left (9 \, a^{5} b^{3} - 2 \, b^{6}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{a^{6}} + \frac {2 \, {\left (9 \, a^{5} b^{3} - 2 \, b^{6}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{6}} + \frac {3 \, {\left (7 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}} a^{3} b^{6} - 4 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {5}{3}} b^{6}\right )} x^{2}}{a^{5} b^{4}}}{18 \, b^{2}} \]
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Timed out. \[ \int \frac {-b+a x^2}{\sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int -\frac {b-a\,x^2}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}} \,d x \]
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