Integrand size = 48, antiderivative size = 154 \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=-\frac {4 \sqrt {-b x+a^2 x^2} \left (3003 b^3-4032 a^2 b^2 x-5120 a^4 b x^2-8192 a^6 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{45045 b^5 x^5}+\frac {4 \left (6699 a b^3+448 a^3 b^2 x+1024 a^5 b x^2+8192 a^7 x^3\right ) \sqrt {x \left (a x+\sqrt {-b x+a^2 x^2}\right )}}{45045 b^5 x^4} \]
[Out]
\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {-b+a^2 x}\right ) \int \frac {1}{x^{5/2} \sqrt {-b+a^2 x} \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^4 \sqrt {-b+a^2 x^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 = 3.54 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \sqrt {x \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )} \left (3003 b^4+1024 a^5 b x^2 \left (-3 a x+\sqrt {x \left (-b+a^2 x\right )}\right )+8192 a^7 x^3 \left (a x+\sqrt {x \left (-b+a^2 x\right )}\right )+64 a^3 b^2 x \left (-17 a x+7 \sqrt {x \left (-b+a^2 x\right )}\right )+21 a b^3 \left (-335 a x+319 \sqrt {x \left (-b+a^2 x\right )}\right )\right )}{45045 b^5 x^4 \sqrt {x \left (-b+a^2 x\right )}} \]
[In]
[Out]
\[\int \frac {1}{x^{2} \sqrt {a^{2} x^{2}-b x}\, \left (a \,x^{2}+x \sqrt {a^{2} x^{2}-b x}\right )^{\frac {3}{2}}}d x\]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.75 \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\frac {4 \, {\left (8192 \, a^{7} x^{4} + 1024 \, a^{5} b x^{3} + 448 \, a^{3} b^{2} x^{2} + 6699 \, a b^{3} x + {\left (8192 \, a^{6} x^{3} + 5120 \, a^{4} b x^{2} + 4032 \, a^{2} b^{2} x - 3003 \, b^{3}\right )} \sqrt {a^{2} x^{2} - b x}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{2} - b x} x}}{45045 \, b^{5} x^{5}} \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^{2} \left (x \left (a x + \sqrt {a^{2} x^{2} - b x}\right )\right )^{\frac {3}{2}} \sqrt {x \left (a^{2} x - b\right )}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} x^{2} - b x} {\left (a x^{2} + \sqrt {a^{2} x^{2} - b x} x\right )}^{\frac {3}{2}} x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^2 \sqrt {-b x+a^2 x^2} \left (a x^2+x \sqrt {-b x+a^2 x^2}\right )^{3/2}} \, dx=\int \frac {1}{x^2\,\sqrt {a^2\,x^2-b\,x}\,{\left (a\,x^2+x\,\sqrt {a^2\,x^2-b\,x}\right )}^{3/2}} \,d x \]
[In]
[Out]