Integrand size = 45, antiderivative size = 232 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {2 \sqrt {-b x+a^2 x^2} \left (210 b^3-200 a^2 b^2 x-456 a^4 b x^2+1601 a^6 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{1155 b^5 x^4 \left (b-a^2 x\right )}+\sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )} \left (-\frac {4 \left (245 a b^2+461 a^3 b x+2533 a^5 x^2\right )}{1155 b^5 x^3}-\frac {6 a^{11/2} \sqrt {-a x+\sqrt {-b x+a^2 x^2}} \arctan \left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {-b x+a^2 x^2}}}{\sqrt {b}}\right )}{b^{11/2} x}\right ) \]
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\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-b+a^2 x}\right ) \int \frac {1}{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 {-b x+a^2 x^2}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {-b+a^2 x}\right ) \text {Subst}\left (\int \frac {1}{x^2 \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 {-b x+a^2 x^2}} \\ \end{align*}
Time = 4.51 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=-\frac {2 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (\sqrt {b} \left (210 b^3+10 a b^2 \left (-20 a x+49 \sqrt {x \left (-b+a^2 x\right )}\right )+2 a^3 b x \left (-228 a x+461 \sqrt {x \left (-b+a^2 x\right )}\right )+a^5 x^2 \left (1601 a x+5066 \sqrt {x \left (-b+a^2 x\right )}\right )\right )+3465 a^{11/2} x^2 \sqrt {x \left (-b+a^2 x\right )} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}} \arctan \left (\frac {\sqrt {a} \sqrt {-a x+\sqrt {x \left (-b+a^2 x\right )}}}{\sqrt {b}}\right )\right )}{1155 b^{11/2} x^3 \sqrt {x \left (-b+a^2 x\right )}} \]
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\[\int \frac {1}{\left (a^{2} x^{2}-b x \right )^{\frac {3}{2}} \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]
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Time = 0.30 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.89 \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\left [\frac {3465 \, {\left (a^{7} x^{5} - a^{5} b x^{4}\right )} \sqrt {a} \log \left (\frac {a^{2} x^{2} + 2 \, \sqrt {a^{2} x^{2} - b x} a x - b x + 2 \, \sqrt {a^{2} x^{2} - b x} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {a}}{a^{2} x^{2} - b x}\right ) - 2 \, {\left (5066 \, a^{7} x^{4} - 4144 \, a^{5} b x^{3} - 432 \, a^{3} b^{2} x^{2} - 490 \, a b^{3} x + {\left (1601 \, a^{6} x^{3} - 456 \, a^{4} b x^{2} - 200 \, a^{2} b^{2} x + 210 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{1155 \, {\left (a^{2} b^{5} x^{5} - b^{6} x^{4}\right )}}, -\frac {2 \, {\left (3465 \, {\left (a^{7} x^{5} - a^{5} b x^{4}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x} \sqrt {-a}}{a x}\right ) + {\left (5066 \, a^{7} x^{4} - 4144 \, a^{5} b x^{3} - 432 \, a^{3} b^{2} x^{2} - 490 \, a b^{3} x + {\left (1601 \, a^{6} x^{3} - 456 \, a^{4} b x^{2} - 200 \, a^{2} b^{2} x + 210 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}\right )}}{1155 \, {\left (a^{2} b^{5} x^{5} - b^{6} x^{4}\right )}}\right ] \]
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\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \left (x \left (a^{2} x - b\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} x^{2} - b x\right )}^{\frac {3}{2}} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (-b x+a^2 x^2\right )^{3/2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{{\left (a^2\,x^2-b\,x\right )}^{3/2}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]
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