Integrand size = 29, antiderivative size = 506 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}-\frac {\left (3 a^2 b+b^2\right ) \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {3} a^3}-\frac {\sqrt {-3-3 i \sqrt {3}} b \sqrt [3]{a^2+b} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b} x}{\sqrt [3]{a} \sqrt [3]{a^2+b} x-2 \sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{\sqrt {2} a^{5/3}}+\frac {\left (-3 a^2 b-b^2\right ) \log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{3 a^3}+\frac {i \left (i b \sqrt [3]{a^2+b}+\sqrt {3} b \sqrt [3]{a^2+b}\right ) \log \left (\sqrt [3]{a} \sqrt [3]{a^2+b} x+\sqrt [3]{-1} \sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a^{5/3}}+\frac {\left (3 a^2 b+b^2\right ) \log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{6 a^3}+\frac {\left (b \sqrt [3]{a^2+b}-i \sqrt {3} b \sqrt [3]{a^2+b}\right ) \log \left (a^{2/3} \left (a^2+b\right )^{2/3} x^2-\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{a^2+b} x \sqrt [3]{b^2 x^2+a^3 x^3}+(-1)^{2/3} \left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a^{5/3}} \]
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Time = 0.17 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.01, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2081, 103, 163, 61, 93} \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {\sqrt [3]{a^3 x^3+b^2 x^2}}{a}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \arctan \left (\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (a^3 x+b^2\right )}{6 a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}-1\right )}{2 a^3 x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {\sqrt {3} b \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3+b^2 x^2} \arctan \left (\frac {2 \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2+b}}{\sqrt {3} \sqrt [3]{a^3 x+b^2}}+\frac {1}{\sqrt {3}}\right )}{a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}}-\frac {b \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3+b^2 x^2} \log (a x-b)}{2 a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}}+\frac {3 b \sqrt [3]{a^2+b} \sqrt [3]{a^3 x^3+b^2 x^2} \log \left (\sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2+b}-\sqrt [3]{a^3 x+b^2}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{a^3 x+b^2}} \]
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Rule 61
Rule 93
Rule 103
Rule 163
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [3]{b^2 x^2+a^3 x^3} \int \frac {x^{2/3} \sqrt [3]{b^2+a^3 x}}{-b+a x} \, dx}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \\ & = \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}-\frac {\sqrt [3]{b^2 x^2+a^3 x^3} \int \frac {-\frac {2 b^3}{3}-\frac {1}{3} a b \left (3 a^2+b\right ) x}{\sqrt [3]{x} (-b+a x) \left (b^2+a^3 x\right )^{2/3}} \, dx}{a x^{2/3} \sqrt [3]{b^2+a^3 x}} \\ & = \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}+\frac {\left (b^2 \left (a^2+b\right ) \sqrt [3]{b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} (-b+a x) \left (b^2+a^3 x\right )^{2/3}} \, dx}{a x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\left (b \left (3 a^2+b\right ) \sqrt [3]{b^2 x^2+a^3 x^3}\right ) \int \frac {1}{\sqrt [3]{x} \left (b^2+a^3 x\right )^{2/3}} \, dx}{3 a x^{2/3} \sqrt [3]{b^2+a^3 x}} \\ & = \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{a}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 a \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{b^2+a^3 x}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\sqrt {3} b \sqrt [3]{a^2+b} \sqrt [3]{b^2 x^2+a^3 x^3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{b^2+a^3 x}}\right )}{a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b \sqrt [3]{a^2+b} \sqrt [3]{b^2 x^2+a^3 x^3} \log (-b+a x)}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{b^2 x^2+a^3 x^3} \log \left (b^2+a^3 x\right )}{6 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{b^2 x^2+a^3 x^3} \log \left (-1+\frac {a \sqrt [3]{x}}{\sqrt [3]{b^2+a^3 x}}\right )}{2 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {3 b \sqrt [3]{a^2+b} \sqrt [3]{b^2 x^2+a^3 x^3} \log \left (\sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}} \\ \end{align*}
Time = 1.95 (sec) , antiderivative size = 694, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {\sqrt [3]{x^2 \left (b^2+a^3 x\right )}}{a}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )}{\sqrt {3} a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {i \left (-3 i+\sqrt {3}\right ) b \sqrt [3]{a^2+b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \text {arctanh}\left (\frac {i \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}+\left (-i+\sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b \left (3 a^2+b\right ) \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (-a \sqrt [3]{x}+\sqrt [3]{b^2+a^3 x}\right )}{3 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {i \left (i+\sqrt {3}\right ) b \sqrt [3]{a^2+b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2+b} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{b^2+a^3 x}\right )}{2 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {b \left (3 a^2+b\right ) \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )}{6 a^3 x^{2/3} \sqrt [3]{b^2+a^3 x}}+\frac {\left (1-i \sqrt {3}\right ) b \sqrt [3]{a^2+b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )} \log \left (-2 i a^{2/3} \left (a^2+b\right )^{2/3} x^{2/3}+\sqrt [3]{a} \sqrt [3]{a^2+b} \left (i \sqrt [3]{x}-\sqrt {3} \sqrt [3]{x}\right ) \sqrt [3]{b^2+a^3 x}+\left (i+\sqrt {3}\right ) \left (b^2+a^3 x\right )^{2/3}\right )}{4 a^{5/3} x^{2/3} \sqrt [3]{b^2+a^3 x}} \]
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Time = 0.77 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(\frac {\left (-2 {\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a^{2}+\frac {b \left (-\left (3 a^{2}+b \right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+\ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )\right ) {\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}}+3 a^{2} \left (a^{2}+b \right ) \left (2 \arctan \left (\frac {\sqrt {3}\, \left ({\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right )}{3 {\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} x}\right ) \sqrt {3}+\ln \left (\frac {{\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}} x^{2}+{\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-{\left (a \left (a^{2}+b \right )\right )}^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )\right )\right )}{3}\right ) b^{2} x^{2}}{2 {\left (a \left (a^{2}+b \right )\right )}^{\frac {2}{3}} \left (a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}\right ) \left (a x -\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) a^{3}}\) | \(409\) |
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Time = 0.26 (sec) , antiderivative size = 408, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=-\frac {6 \, \sqrt {3} a^{2} b \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (a^{2} + b\right )} x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {2}{3}}}{3 \, {\left (a^{2} + b\right )} x}\right ) - 6 \, a^{2} b \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} \log \left (-\frac {a x \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + 3 \, a^{2} b \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} \log \left (\frac {a^{2} x^{2} \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {2}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x \left (\frac {a^{2} + b}{a^{2}}\right )^{\frac {1}{3}} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, a^{2} b + b^{2}\right )} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) - 6 \, {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a^{2} + 2 \, {\left (3 \, a^{2} b + b^{2}\right )} \log \left (-\frac {a x - {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (3 \, a^{2} b + b^{2}\right )} \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{6 \, a^{3}} \]
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\[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\int \frac {\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )}}{a x - b}\, dx \]
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\[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\int { \frac {{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x - b} \,d x } \]
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Time = 23.25 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=\frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{2} b + b^{2}\right )} \log \left ({\left | -{\left (a^{3} + a b\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{4} + a^{2} b} - \frac {\sqrt {3} {\left (a^{3} + a b\right )}^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} + a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} + a b\right )}^{\frac {1}{3}}}\right )}{a^{2}} + \frac {{\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} x}{a} - \frac {{\left (a^{3} + a b\right )}^{\frac {1}{3}} b \log \left ({\left (a^{3} + a b\right )}^{\frac {2}{3}} + {\left (a^{3} + a b\right )}^{\frac {1}{3}} {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, a^{2}} + \frac {\sqrt {3} {\left (3 \, a^{2} b + b^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{3 \, a^{3}} + \frac {{\left (3 \, a^{2} b + b^{2}\right )} \log \left (a^{2} + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{6 \, a^{3}} - \frac {{\left (3 \, a^{2} b + b^{2}\right )} \log \left ({\left | -a + {\left (a^{3} + \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{3 \, a^{3}} \]
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Timed out. \[ \int \frac {\sqrt [3]{b^2 x^2+a^3 x^3}}{-b+a x} \, dx=-\int \frac {{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}}{b-a\,x} \,d x \]
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