Integrand size = 45, antiderivative size = 117 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=-\frac {4 \left (3 b c+32 a c x-8 c^3 x\right ) \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{105 b^2 x^2}+\frac {4 \left (15 b+20 a x+4 c^2 x\right ) \sqrt {-b x+a x^2} \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{105 b^2 x^3} \]
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\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-b+a x}\right ) \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^{7/2} \sqrt {-b+a x}} \, dx}{\sqrt {-b x+a x^2}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-b+a x}\right ) \text {Subst}\left (\int \frac {\sqrt {c x^4-x^2 \sqrt {-b x^2+a x^4}}}{x^6 \sqrt {-b+a x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-b x+a x^2}} \\ \end{align*}
\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx \]
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\[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3} \sqrt {a \,x^{2}-b x}}d x\]
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Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=-\frac {4 \, {\left (3 \, b c x - 8 \, {\left (c^{3} - 4 \, a c\right )} x^{2} - \sqrt {a x^{2} - b x} {\left (4 \, {\left (c^{2} + 5 \, a\right )} x + 15 \, b\right )}\right )} \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{105 \, b^{2} x^{3}} \]
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\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {x \left (c x - \sqrt {a x^{2} - b x}\right )}}{x^{3} \sqrt {x \left (a x - b\right )}}\, dx \]
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\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int { \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{\sqrt {a x^{2} - b x} x^{3}} \,d x } \]
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\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int { \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{\sqrt {a x^{2} - b x} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3 \sqrt {-b x+a x^2}} \, dx=\int \frac {\sqrt {c\,x^2-x\,\sqrt {a\,x^2-b\,x}}}{x^3\,\sqrt {a\,x^2-b\,x}} \,d x \]
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