Integrand size = 33, antiderivative size = 398 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\frac {\sqrt {a} \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} c}-\frac {\sqrt {a} \text {RootSum}\left [b^2 c-4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2+4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {a b d \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right )+2 i b c \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}-a d \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{a b d+i b c \text {$\#$1}-3 a d \text {$\#$1}^2+i c \text {$\#$1}^3}\&\right ]}{\sqrt {2} c} \]
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\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x\right )}+\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x\right )}\right ) \, dx \\ & = \frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}-\sqrt {-c} x} \, dx}{2 \sqrt {d}}+\frac {\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {d}+\sqrt {-c} x} \, dx}{2 \sqrt {d}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 367, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\frac {\sqrt {a} \left (\log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )\right )-\text {RootSum}\left [b^2 c-4 i a b d \text {$\#$1}+2 b c \text {$\#$1}^2+4 i a d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {a b d \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right )+2 i b c \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right ) \text {$\#$1}-a d \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+i \text {$\#$1}\right )\right ) \text {$\#$1}^2}{a b d+i b c \text {$\#$1}-3 a d \text {$\#$1}^2+i c \text {$\#$1}^3}\&\right ]\right )}{\sqrt {2} c} \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.07
\[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{c \,x^{2}+d}d x\]
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Timed out. \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d} \,d x } \]
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Not integrable
Time = 0.53 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{c x^{2} + d} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{d+c x^2} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{c\,x^2+d} \,d x \]
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