Integrand size = 35, antiderivative size = 238 \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\frac {\left (-b-2 a x^3\right ) \sqrt [3]{b+a x^3}}{2 b x^4}-\frac {a^{4/3} \arctan \left (\frac {3^{5/6} \sqrt [3]{a} x}{\sqrt [3]{3} \sqrt [3]{a} x+2 \sqrt [3]{2} \sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}-\frac {a^{4/3} \log \left (-3 \sqrt [3]{a} x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (3 a^{2/3} x^2+\sqrt [3]{2} 3^{2/3} \sqrt [3]{a} x \sqrt [3]{b+a x^3}+2^{2/3} \sqrt [3]{3} \left (b+a x^3\right )^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b} \]
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Time = 0.12 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {594, 597, 12, 503} \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=-\frac {a^{4/3} \arctan \left (\frac {\frac {2^{2/3} \sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}+\frac {a^{4/3} \log \left (a x^3-2 b\right )}{4 \sqrt [3]{2} 3^{2/3} b}-\frac {\sqrt [3]{\frac {3}{2}} a^{4/3} \log \left (\sqrt [3]{\frac {3}{2}} \sqrt [3]{a} x-\sqrt [3]{a x^3+b}\right )}{4 b}-\frac {a \sqrt [3]{a x^3+b}}{b x}-\frac {\sqrt [3]{a x^3+b}}{2 x^4} \]
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Rule 12
Rule 503
Rule 594
Rule 597
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {\int \frac {16 a b^2+4 a^2 b x^3}{x^2 \left (-2 b+a x^3\right ) \left (b+a x^3\right )^{2/3}} \, dx}{8 b} \\ & = -\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {\int \frac {24 a^2 b^3 x}{\left (-2 b+a x^3\right ) \left (b+a x^3\right )^{2/3}} \, dx}{16 b^3} \\ & = -\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {1}{2} \left (3 a^2\right ) \int \frac {x}{\left (-2 b+a x^3\right ) \left (b+a x^3\right )^{2/3}} \, dx \\ & = -\frac {\sqrt [3]{b+a x^3}}{2 x^4}-\frac {a \sqrt [3]{b+a x^3}}{b x}-\frac {a^{4/3} \arctan \left (\frac {1+\frac {2^{2/3} \sqrt [3]{3} \sqrt [3]{a} x}{\sqrt [3]{b+a x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}+\frac {a^{4/3} \log \left (-2 b+a x^3\right )}{4 \sqrt [3]{2} 3^{2/3} b}-\frac {\sqrt [3]{\frac {3}{2}} a^{4/3} \log \left (\sqrt [3]{\frac {3}{2}} \sqrt [3]{a} x-\sqrt [3]{b+a x^3}\right )}{4 b} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\frac {\left (-b-2 a x^3\right ) \sqrt [3]{b+a x^3}}{2 b x^4}-\frac {a^{4/3} \arctan \left (\frac {3^{5/6} \sqrt [3]{a} x}{\sqrt [3]{3} \sqrt [3]{a} x+2 \sqrt [3]{2} \sqrt [3]{b+a x^3}}\right )}{2 \sqrt [3]{2} \sqrt [6]{3} b}-\frac {a^{4/3} \log \left (-3 \sqrt [3]{a} x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{b+a x^3}\right )}{2 \sqrt [3]{2} 3^{2/3} b}+\frac {a^{4/3} \log \left (3 a^{2/3} x^2+\sqrt [3]{2} 3^{2/3} \sqrt [3]{a} x \sqrt [3]{b+a x^3}+2^{2/3} \sqrt [3]{3} \left (b+a x^3\right )^{2/3}\right )}{4 \sqrt [3]{2} 3^{2/3} b} \]
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Time = 0.73 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.73
method | result | size |
pseudoelliptic | \(\frac {6 \left (-2 a \,x^{3}-b \right ) \left (a \,x^{3}+b \right )^{\frac {1}{3}}+x^{4} 2^{\frac {2}{3}} \left (\left (-\ln \left (\frac {-3^{\frac {1}{3}} 2^{\frac {2}{3}} a^{\frac {1}{3}} x +2 \left (a \,x^{3}+b \right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {3^{\frac {2}{3}} 2^{\frac {1}{3}} a^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} 2^{\frac {2}{3}} a^{\frac {1}{3}} \left (a \,x^{3}+b \right )^{\frac {1}{3}} x +2 \left (a \,x^{3}+b \right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\frac {\ln \left (2\right )}{2}\right ) 3^{\frac {1}{3}}+\arctan \left (\frac {\sqrt {3}\, \left (2 \,2^{\frac {1}{3}} 3^{\frac {2}{3}} \left (a \,x^{3}+b \right )^{\frac {1}{3}}+3 a^{\frac {1}{3}} x \right )}{9 a^{\frac {1}{3}} x}\right ) 3^{\frac {5}{6}}\right ) a^{\frac {4}{3}}}{12 b \,x^{4}}\) | \(174\) |
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (172) = 344\).
Time = 9.65 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.76 \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=-\frac {2 \cdot 18^{\frac {2}{3}} \sqrt {3} \left (-a\right )^{\frac {1}{3}} a x^{4} \arctan \left (\frac {4 \cdot 18^{\frac {2}{3}} \sqrt {3} {\left (4 \, a^{2} x^{7} - 7 \, a b x^{4} - 2 \, b^{2} x\right )} {\left (a x^{3} + b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} + 6 \cdot 18^{\frac {1}{3}} \sqrt {3} {\left (55 \, a^{2} x^{8} + 50 \, a b x^{5} + 4 \, b^{2} x^{2}\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} {\left (377 \, a^{3} x^{9} + 600 \, a^{2} b x^{6} + 204 \, a b^{2} x^{3} + 8 \, b^{3}\right )}}{3 \, {\left (487 \, a^{3} x^{9} + 480 \, a^{2} b x^{6} + 12 \, a b^{2} x^{3} - 8 \, b^{3}\right )}}\right ) - 2 \cdot 18^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} a x^{4} \log \left (-\frac {3 \cdot 18^{\frac {2}{3}} {\left (a x^{3} + b\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a x^{2} + 18 \, {\left (a x^{3} + b\right )}^{\frac {2}{3}} a x + 18^{\frac {1}{3}} {\left (a x^{3} - 2 \, b\right )} \left (-a\right )^{\frac {2}{3}}}{18 \, {\left (a x^{3} - 2 \, b\right )}}\right ) + 18^{\frac {2}{3}} \left (-a\right )^{\frac {1}{3}} a x^{4} \log \left (\frac {36 \cdot 18^{\frac {1}{3}} {\left (4 \, a x^{4} + b x\right )} {\left (a x^{3} + b\right )}^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - 18^{\frac {2}{3}} {\left (55 \, a^{2} x^{6} + 50 \, a b x^{3} + 4 \, b^{2}\right )} \left (-a\right )^{\frac {1}{3}} + 54 \, {\left (7 \, a^{2} x^{5} + 4 \, a b x^{2}\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}}}{18 \, {\left (a^{2} x^{6} - 4 \, a b x^{3} + 4 \, b^{2}\right )}}\right ) + 108 \, {\left (2 \, a x^{3} + b\right )} {\left (a x^{3} + b\right )}^{\frac {1}{3}}}{216 \, b x^{4}} \]
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\[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int \frac {\left (a x^{3} - 4 b\right ) \sqrt [3]{a x^{3} + b}}{x^{5} \left (a x^{3} - 2 b\right )}\, dx \]
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\[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int { \frac {{\left (a x^{3} + b\right )}^{\frac {1}{3}} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} - 2 \, b\right )} x^{5}} \,d x } \]
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\[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int { \frac {{\left (a x^{3} + b\right )}^{\frac {1}{3}} {\left (a x^{3} - 4 \, b\right )}}{{\left (a x^{3} - 2 \, b\right )} x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\left (-4 b+a x^3\right ) \sqrt [3]{b+a x^3}}{x^5 \left (-2 b+a x^3\right )} \, dx=\int \frac {{\left (a\,x^3+b\right )}^{1/3}\,\left (4\,b-a\,x^3\right )}{x^5\,\left (2\,b-a\,x^3\right )} \,d x \]
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