Integrand size = 38, antiderivative size = 123 \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {4 \left (-2+a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{15 x^2}-\frac {2 b \left (-3+2 a x^2\right ) \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{15 a x^3} \]
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\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \\ \end{align*}
Time = 5.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \left (-3+a x^2-b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}{15 x^2 \sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}} \]
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\[\int \frac {1}{x^{3} \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]
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Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \, {\left (2 \, a^{2} x^{3} - 4 \, a x - {\left (2 \, a b x^{2} - 3 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{15 \, a x^{3}} \]
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\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {1}{x^{3} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]
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\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{3}} \,d x } \]
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\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {1}{x^3\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]
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