Integrand size = 68, antiderivative size = 507 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
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\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \\ \end{align*}
Time = 1.57 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\frac {\left (945 b^3-4224 b^2 c^4-504 a b^2 c^2 x+3072 a b c^6 x-1890 a^2 b^2 x^2+7680 a^2 b c^4 x^2-4096 a^3 c^6 x^3\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (432 b^2 c^3-2048 b c^7+630 a b^2 c x-2304 a b c^5 x+4096 a^2 c^7 x^2+3072 a^3 c^5 x^3\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\sqrt {-b+a^2 x^2} \left (\left (-504 b^2 c^2+1024 b c^6-1890 a b^2 x+7680 a b c^4 x-4096 a^2 c^6 x^2\right ) \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}+\left (630 b^2 c-768 b c^5+4096 a c^7 x+3072 a^2 c^5 x^2\right ) \sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}\right )}{3840 a c^5 \left (a x+\sqrt {-b+a^2 x^2}\right )^{5/2}}+\frac {63 b^2 \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{256 a c^{11/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}}{\sqrt {c}}\right )}{a c^{3/2}} \]
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\[\int \frac {\sqrt {a^{2} x^{2}-b}}{\sqrt {a x +\sqrt {a^{2} x^{2}-b}}\, \sqrt {c +\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}d x\]
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Time = 0.31 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.34 \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\left [\frac {15 \, {\left (256 \, b^{2} c^{4} - 63 \, b^{3}\right )} \sqrt {c} \log \left (-2 \, {\left (a \sqrt {c} x - \sqrt {a^{2} x^{2} - b} \sqrt {c}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} + 2 \, {\left (a c x - \sqrt {a^{2} x^{2} - b} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} + b\right ) + 2 \, {\left (2048 \, b c^{8} + 864 \, a^{2} b c^{4} x^{2} - 432 \, b^{2} c^{4} + 6 \, {\left (128 \, a b c^{6} + 105 \, a b^{2} c^{2}\right )} x + 6 \, {\left (128 \, b c^{6} - 144 \, a b c^{4} x - 105 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1536 \, a^{3} c^{5} x^{3} + 1024 \, b c^{7} + 1008 \, a^{2} b c^{3} x^{2} - 504 \, b^{2} c^{3} - 3 \, {\left (1664 \, a b c^{5} - 315 \, a b^{2} c\right )} x - 3 \, {\left (512 \, a^{2} c^{5} x^{2} - 1408 \, b c^{5} + 336 \, a b c^{3} x + 315 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{7680 \, a b c^{6}}, \frac {15 \, {\left (256 \, b^{2} c^{4} - 63 \, b^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{c}\right ) + {\left (2048 \, b c^{8} + 864 \, a^{2} b c^{4} x^{2} - 432 \, b^{2} c^{4} + 6 \, {\left (128 \, a b c^{6} + 105 \, a b^{2} c^{2}\right )} x + 6 \, {\left (128 \, b c^{6} - 144 \, a b c^{4} x - 105 \, b^{2} c^{2}\right )} \sqrt {a^{2} x^{2} - b} - {\left (1536 \, a^{3} c^{5} x^{3} + 1024 \, b c^{7} + 1008 \, a^{2} b c^{3} x^{2} - 504 \, b^{2} c^{3} - 3 \, {\left (1664 \, a b c^{5} - 315 \, a b^{2} c\right )} x - 3 \, {\left (512 \, a^{2} c^{5} x^{2} - 1408 \, b c^{5} + 336 \, a b c^{3} x + 315 \, b^{2} c\right )} \sqrt {a^{2} x^{2} - b}\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}\right )} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}}{3840 \, a b c^{6}}\right ] \]
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\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}} \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}\, dx \]
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\[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} - b}}{\sqrt {a x + \sqrt {a^{2} x^{2} - b}} \sqrt {c + \sqrt {a x + \sqrt {a^{2} x^{2} - b}}}} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {-b+a^2 x^2}}{\sqrt {a x+\sqrt {-b+a^2 x^2}} \sqrt {c+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx=\int \frac {\sqrt {a^2\,x^2-b}}{\sqrt {a\,x+\sqrt {a^2\,x^2-b}}\,\sqrt {c+\sqrt {a\,x+\sqrt {a^2\,x^2-b}}}} \,d x \]
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