Integrand size = 59, antiderivative size = 233 \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \left (-255-136 a x^2+384 a^2 x^4\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a^2}+\frac {\left (85 x+568 a x^3-384 a^2 x^5\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{1920 a b}+\frac {17 \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{128 \sqrt {2} a^{3/2} b} \]
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\[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \\ \end{align*}
Time = 9.29 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.86 \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {\sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (-2 x \left (384 a^3 x^5+255 b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}+17 a x \left (-5+8 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )-8 a^2 x^3 \left (71+48 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )\right )+255 \sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )\right )}{3840 a^2 b x} \]
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\[\int \frac {x^{3} \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}{\sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]
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Time = 21.79 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.54 \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\left [\frac {255 \, \sqrt {2} \sqrt {a} \log \left (-4 \, a^{2} x^{2} - 4 \, a b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x + \sqrt {2} \sqrt {a} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} + a\right ) - 4 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{7680 \, a^{2} b}, \frac {255 \, \sqrt {2} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (384 \, a^{3} x^{5} - 568 \, a^{2} x^{3} - 85 \, a x - {\left (384 \, a^{2} b x^{4} - 136 \, a b x^{2} - 255 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{3840 \, a^{2} b}\right ] \]
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\[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {x^{3} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}}{\sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]
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\[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {\sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} x^{3}}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}{\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {x^3\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}{\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]
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