Integrand size = 45, antiderivative size = 220 \[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {-b x+a^2 x^2} \left (115 b^2-88 a^2 b x+32 a^4 x^2\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{40 b^2 x}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (\frac {-145 a b^2+104 a^3 b x-32 a^5 x^2}{40 b^2}+\frac {9 \sqrt {b} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{8 \sqrt {2} \sqrt {a} x}\right ) \]
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\[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-b x+a^2 x^2} \int \frac {x^{3/2} \left (-b+a^2 x\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, 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^4 \left (-b+a^2 x^2\right )^{3/2}}{\left (a x^4+x^2 \sqrt {-b x^2+a^2 x^4}\right )^{3/2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a^2 x}} \\ \end{align*}
Time = 5.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {\sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (-2 \sqrt {a} x \left (115 b^3+8 a^3 b x \left (15 a x-13 \sqrt {x \left (-b+a^2 x\right )}\right )+32 a^5 x^2 \left (-a x+\sqrt {x \left (-b+a^2 x\right )}\right )+29 a b^2 \left (-7 a x+5 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+45 \sqrt {2} b^{5/2} \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {2} \sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{80 \sqrt {a} b^2 x \sqrt {x \left (-b+a^2 x\right )}} \]
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\[\int \frac {\left (a^{2} x^{2}-b x \right )^{\frac {3}{2}}}{\left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]
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Time = 0.33 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.67 \[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {45 \, \sqrt {2} \sqrt {a} b^{3} x \log \left (-\frac {4 \, a^{2} x^{2} + 4 \, \sqrt {a^{2} x^{2} - b x} a x - b x - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a^{2} x^{2} - b x} \sqrt {a}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{x}\right ) - 4 \, {\left (32 \, a^{6} x^{3} - 104 \, a^{4} b x^{2} + 145 \, a^{2} b^{2} x - {\left (32 \, a^{5} x^{2} - 88 \, a^{3} b x + 115 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{160 \, a b^{2} x}, \frac {45 \, \sqrt {2} \sqrt {-a} b^{3} x \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (32 \, a^{6} x^{3} - 104 \, a^{4} b x^{2} + 145 \, a^{2} b^{2} x - {\left (32 \, a^{5} x^{2} - 88 \, a^{3} b x + 115 \, a b^{2}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{80 \, a b^{2} x}\right ] \]
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\[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {\left (x \left (a^{2} x - b\right )\right )^{\frac {3}{2}}}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}}}{{\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}}}{{\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (-b x+a^2 x^2\right )^{3/2}}{\left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {{\left (a^2\,x^2-b\,x\right )}^{3/2}}{{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]
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