Integrand size = 30, antiderivative size = 183 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {\log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{c}+\frac {2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {a d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+a \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 a d+3 a \text {$\#$1}+2 c \text {$\#$1}^2}\&\right ]}{c} \]
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\[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(d+c x) \left (-d^2+(a-2 c d) x-c^2 x^2\right )}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}+\frac {d \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4}+\frac {c x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4}+\frac {d^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}+\frac {2 c \left (1-\frac {a}{2 c d}\right ) d x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}+\frac {c^2 x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}+\frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}\right ) \, dx \\ & = c \int \frac {x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+c^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+d^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+(-a+2 c d) \int \frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {(d+c x) \left (-d^2+(a-2 c d) x-c^2 x^2\right )}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx \\ & = \frac {\log \left (b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4\right )}{4 c}-\frac {\int \frac {2 a c^3 d^2+4 a c^4 d x+2 a c^5 x^2}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx}{4 c^4}+c \int \frac {x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+c^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+d^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+(-a+2 c d) \int \frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx \\ & = \frac {\log \left (b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4\right )}{4 c}-\frac {\int \frac {2 a c^3 (d+c x)^2}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx}{4 c^4}+c \int \frac {x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+c^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+d^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+(-a+2 c d) \int \frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx \\ & = \frac {\log \left (b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4\right )}{4 c}-\frac {a \int \frac {(d+c x)^2}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx}{2 c}+c \int \frac {x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+c^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+d^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+(-a+2 c d) \int \frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx \\ & = \frac {\log \left (b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4\right )}{4 c}-\frac {a \int \left (\frac {d^2}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}+\frac {2 c d x}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}+\frac {c^2 x^2}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4}\right ) \, dx}{2 c}+c \int \frac {x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+c^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx+d^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+(-a+2 c d) \int \frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx \\ & = \frac {\log \left (b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4\right )}{4 c}+c \int \frac {x \sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx-\frac {1}{2} (a c) \int \frac {x^2}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+c^2 \int \frac {x^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d \int \frac {\sqrt {b^2+a^2 x^2}}{-b^2+d^4-2 d^2 (a-2 c d) x-2 c d (2 a-3 c d) x^2-2 c^2 (a-2 c d) x^3+c^4 x^4} \, dx-(a d) \int \frac {x}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+d^2 \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx-\frac {\left (a d^2\right ) \int \frac {1}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx}{2 c}+(-a+2 c d) \int \frac {x \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx+\int \frac {\sqrt {b^2+a^2 x^2} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{b^2-d^4+2 d^2 (a-2 c d) x+2 c d (2 a-3 c d) x^2+2 c^2 (a-2 c d) x^3-c^4 x^4} \, dx \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.98 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {\log \left (a x+\sqrt {b^2+a^2 x^2}\right )-2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {a d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+a \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 a d+3 a \text {$\#$1}+2 c \text {$\#$1}^2}\&\right ]}{c} \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.14
\[\int \frac {1}{d +c x +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
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Timed out. \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Not integrable
Time = 6.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{d+c\,x+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]
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