Integrand size = 45, antiderivative size = 200 \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\frac {\left (-9 b+8 a^2 x\right ) \sqrt {-b x+a^2 x^2} \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{12 a b x}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (\frac {19 b-8 a^2 x}{12 b}+\frac {3 \sqrt {b} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{4 \sqrt {2} a^{3/2} x}\right ) \]
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\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-b x+a^2 x^2} \int \frac {\sqrt {x} \sqrt {-b+a^2 x}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx}{\sqrt {x} \sqrt {-b+a^2 x}} \\ & = \frac {\left (2 \sqrt {-b x+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2 \sqrt {-b+a^2 x^2}}{\sqrt {a x^4+x^2 \sqrt {-b x^2+a^2 x^4}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a^2 x}} \\ \end{align*}
Time = 4.20 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (-2 \sqrt {a} x \left (-9 b^2+a b \left (17 a x-19 \sqrt {x \left (-b+a^2 x\right )}\right )+8 a^3 x \left (-a x+\sqrt {x \left (-b+a^2 x\right )}\right )\right )+9 \sqrt {2} b^{3/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{24 a^{3/2} b x \sqrt {x \left (-b+a^2 x\right )}} \]
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\[\int \frac {\sqrt {a^{2} x^{2}-b x}}{\sqrt {a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}}}d x\]
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Time = 0.27 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.62 \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\left [\frac {9 \, \sqrt {2} \sqrt {a} b^{2} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (8 \, a^{4} x^{2} - 19 \, a^{2} b x - {\left (8 \, a^{3} x - 9 \, a b\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{48 \, a^{2} b x}, \frac {9 \, \sqrt {2} \sqrt {-a} b^{2} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (8 \, a^{4} x^{2} - 19 \, a^{2} b x - {\left (8 \, a^{3} x - 9 \, a b\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{24 \, a^{2} b x}\right ] \]
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\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int \frac {\sqrt {x \left (a^{2} x - b\right )}}{\sqrt {x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )}}\, dx \]
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\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b x}}{\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}} \,d x } \]
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\[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b x}}{\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {-b x+a^2 x^2}}{\sqrt {a x^2+x \sqrt {-b x+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b\,x}}{\sqrt {a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}}} \,d x \]
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