Integrand size = 30, antiderivative size = 319 \[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {-3+3 i \sqrt {3}} \arctan \left (\frac {\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} x}{\sqrt [3]{a} \sqrt [3]{a^2-b} x-2 \sqrt [3]{-1} \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )}{\sqrt {2} \sqrt [3]{a} \sqrt [3]{a^2-b} b}-\frac {i \left (-i+\sqrt {3}\right ) \log \left (\sqrt [3]{a} \sqrt [3]{a^2-b} x+\sqrt [3]{-1} \sqrt [3]{-b^2 x^2+a^3 x^3}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b} b}+\frac {\left (1+i \sqrt {3}\right ) \log \left (a^{2/3} \left (a^2-b\right )^{2/3} x^2-\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{a^2-b} x \sqrt [3]{-b^2 x^2+a^3 x^3}+(-1)^{2/3} \left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 \sqrt [3]{a} \sqrt [3]{a^2-b} b} \]
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Time = 0.09 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.91, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2081, 93} \[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b^2} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b^2}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{x} \sqrt [3]{a^2-b}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b^2} \log (a x-b)}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b^2} \log \left (\frac {\sqrt [3]{a^3 x-b^2}}{\sqrt [3]{a} \sqrt [3]{a^2-b}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} b \sqrt [3]{a^2-b} \sqrt [3]{a^3 x^3-b^2 x^2}} \]
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Rule 93
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} (-b+a x) \sqrt [3]{-b^2+a^3 x}} \, dx}{\sqrt [3]{-b^2 x^2+a^3 x^3}} \\ & = \frac {\sqrt {3} 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} \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}}\right )}{\sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{-b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b^2+a^3 x} \log (-b+a x)}{2 \sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{-b^2 x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b^2+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-b}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{-b^2 x^2+a^3 x^3}} \\ \end{align*}
Time = 1.45 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{-b^2+a^3 x} \left (2 \sqrt {-6+6 i \sqrt {3}} \arctan \left (\frac {3 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{-b^2+a^3 x}}\right )-i \left (-i+\sqrt {3}\right ) \left (2 \log \left (2 \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{-b^2+a^3 x}\right )-\log \left (-2 i a^{2/3} \left (a^2-b\right )^{2/3} x^{2/3}-\left (-i+\sqrt {3}\right ) \sqrt [3]{a} \sqrt [3]{a^2-b} \sqrt [3]{x} \sqrt [3]{-b^2+a^3 x}+\left (i+\sqrt {3}\right ) \left (-b^2+a^3 x\right )^{2/3}\right )\right )\right )}{4 \sqrt [3]{a} \sqrt [3]{a^2-b} b \sqrt [3]{x^2 \left (-b^2+a^3 x\right )}} \]
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Time = 0.75 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.58
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left ({\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x +2 \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}\right )}{3 {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x}\right )+2 \ln \left (\frac {-{\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {{\left (a \left (a^{2}-b \right )\right )}^{\frac {2}{3}} x^{2}+{\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 {\left (a \left (a^{2}-b \right )\right )}^{\frac {1}{3}} b}\) | \(185\) |
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Time = 0.26 (sec) , antiderivative size = 549, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\left [\frac {\sqrt {3} {\left (a^{3} - a b\right )} \sqrt {-\frac {1}{{\left (a^{3} - a b\right )}^{\frac {2}{3}}}} \log \left (-\frac {2 \, b^{2} x - {\left (3 \, a^{3} - a b\right )} x^{2} + 3 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {2}{3}} x + \sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {4}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )} x - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}} {\left (a^{3} - a b\right )}^{\frac {2}{3}}\right )} \sqrt {-\frac {1}{{\left (a^{3} - a b\right )}^{\frac {2}{3}}}}}{a x^{2} - b x}\right ) + 2 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} b - a b^{2}\right )}}, \frac {2 \, \sqrt {3} {\left (a^{3} - a b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} x + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}} x}\right ) + 2 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (-\frac {{\left (a^{3} - a b\right )}^{\frac {1}{3}} x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - {\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left (\frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} - a b\right )}^{\frac {1}{3}} x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right )}{2 \, {\left (a^{3} b - a b^{2}\right )}}\right ] \]
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\[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x - b^{2}\right )} \left (a x - b\right )}\, dx \]
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\[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (a x - b\right )}} \,d x } \]
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Time = 19.94 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=\frac {3 \, {\left (a^{3} - a b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left ({\left (a^{3} - a b\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3} - a b\right )}^{\frac {1}{3}}}\right )}{\sqrt {3} a^{3} b - \sqrt {3} a b^{2}} - \frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left ({\left (a^{3} - a b\right )}^{\frac {2}{3}} + {\left (a^{3} - a b\right )}^{\frac {1}{3}} {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {2}{3}}\right )}{2 \, {\left (a^{3} b - a b^{2}\right )}} + \frac {{\left (a^{3} - a b\right )}^{\frac {2}{3}} \log \left ({\left | -{\left (a^{3} - a b\right )}^{\frac {1}{3}} + {\left (a^{3} - \frac {b^{2}}{x}\right )}^{\frac {1}{3}} \right |}\right )}{a^{3} b - a b^{2}} \]
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Timed out. \[ \int \frac {1}{(-b+a x) \sqrt [3]{-b^2 x^2+a^3 x^3}} \, dx=-\int \frac {1}{{\left (a^3\,x^3-b^2\,x^2\right )}^{1/3}\,\left (b-a\,x\right )} \,d x \]
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