Integrand size = 48, antiderivative size = 212 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {a \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{c}-\frac {2 a \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+4 a^2 \text {$\#$1}^5+c \text {$\#$1}^8\&,\frac {b^2 c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text {$\#$1}-2 c \text {$\#$1}^4}\&\right ]}{c} \]
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\[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]
[In]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {b^2+a^2 x^2} \left (d+c x^2\right ) \left (-d^2+a x-2 c d x^2-c^2 x^4\right )}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {d \left (b^2+a^2 x^2\right )}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {c x^2 \left (b^2+a^2 x^2\right )}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {d^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {2 c d x^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {c^2 x^4 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {a x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}\right ) \, dx \\ & = a \int \frac {x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+c \int \frac {x^2 \left (b^2+a^2 x^2\right )}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+c^2 \int \frac {x^4 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+d \int \frac {b^2+a^2 x^2}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+(2 c d) \int \frac {x^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+d^2 \int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \left (d+c x^2\right ) \left (-d^2+a x-2 c d x^2-c^2 x^4\right )}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+\int \frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx \\ & = a \int \frac {x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+c \int \frac {x^2 \left (-b^2-a^2 x^2\right )}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+c^2 \int \frac {x^4 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+d \int \frac {-b^2-a^2 x^2}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+(2 c d) \int \frac {x^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+d^2 \int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \left (d+c x^2\right ) \left (-d^2+a x-2 c d x^2-c^2 x^4\right )}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\int \frac {\left (b^2+a^2 x^2\right ) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx \\ & = a \int \frac {x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+c \int \left (\frac {b^2 x^2}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {a^2 x^4}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}\right ) \, dx+c^2 \int \frac {x^4 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+d \int \left (\frac {b^2}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {a^2 x^2}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}\right ) \, dx+(2 c d) \int \frac {x^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+d^2 \int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\int \left (\frac {a d x \sqrt {b^2+a^2 x^2}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {a c x^3 \sqrt {b^2+a^2 x^2}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {d^3 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {3 c d^2 x^2 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {3 c^2 d x^4 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}+\frac {c^3 x^6 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8}\right ) \, dx+\int \left (\frac {b^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}+\frac {a^2 x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8}\right ) \, dx \\ & = a \int \frac {x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+a^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+b^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+(a c) \int \frac {x^3 \sqrt {b^2+a^2 x^2}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+\left (a^2 c\right ) \int \frac {x^4}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+\left (b^2 c\right ) \int \frac {x^2}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+c^2 \int \frac {x^4 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+c^3 \int \frac {x^6 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+(a d) \int \frac {x \sqrt {b^2+a^2 x^2}}{b^2 \left (1-\frac {d^4}{b^2}\right )+2 a d^2 x-4 c d^3 x^2+4 a c d x^3-6 c^2 d^2 x^4+2 a c^2 x^5-4 c^3 d x^6-c^4 x^8} \, dx+\left (a^2 d\right ) \int \frac {x^2}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+\left (b^2 d\right ) \int \frac {1}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+(2 c d) \int \frac {x^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (3 c^2 d\right ) \int \frac {x^4 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+d^2 \int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (3 c d^2\right ) \int \frac {x^2 \sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx+d^3 \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2 \left (1-\frac {d^4}{b^2}\right )-2 a d^2 x+4 c d^3 x^2-4 a c d x^3+6 c^2 d^2 x^4-2 a c^2 x^5+4 c^3 d x^6+c^4 x^8} \, dx \\ & = a \int \frac {x \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+a^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+b^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+(a c) \int \frac {x^3 \sqrt {b^2+a^2 x^2}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (a^2 c\right ) \int \frac {x^4}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (b^2 c\right ) \int \frac {x^2}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+c^2 \int \frac {x^4 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+c^3 \int \frac {x^6 \sqrt {b^2+a^2 x^2}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+(a d) \int \frac {x \sqrt {b^2+a^2 x^2}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (a^2 d\right ) \int \frac {x^2}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (b^2 d\right ) \int \frac {1}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+(2 c d) \int \frac {x^2 \sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (3 c^2 d\right ) \int \frac {x^4 \sqrt {b^2+a^2 x^2}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+d^2 \int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+\left (3 c d^2\right ) \int \frac {x^2 \sqrt {b^2+a^2 x^2}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx+d^3 \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+\left (d+c x^2\right )^2 \left (d^2-2 a x+2 c d x^2+c^2 x^4\right )} \, dx \\ \end{align*}
Time = 3.45 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {a \left (\log \left (a x+\sqrt {b^2+a^2 x^2}\right )-2 \text {RootSum}\left [b^4 c-2 b^2 c \text {$\#$1}^4+4 a^2 d \text {$\#$1}^4+4 a^2 \text {$\#$1}^5+c \text {$\#$1}^8\&,\frac {b^2 c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a^2 \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}}{2 b^2 c-4 a^2 d-5 a^2 \text {$\#$1}-2 c \text {$\#$1}^4}\&\right ]\right )}{c} \]
[In]
[Out]
Not integrable
Time = 0.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.20
\[\int \frac {\sqrt {a^{2} x^{2}+b^{2}}}{d +c \,x^{2}+\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
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[Out]
Timed out. \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\text {Timed out} \]
[In]
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Not integrable
Time = 0.75 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.19 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Not integrable
Time = 1.45 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {\sqrt {a^{2} x^{2} + b^{2}}}{c x^{2} + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Not integrable
Time = 6.70 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.21 \[ \int \frac {\sqrt {b^2+a^2 x^2}}{d+c x^2+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {\sqrt {a^2\,x^2+b^2}}{d+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}+c\,x^2} \,d x \]
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