Integrand size = 36, antiderivative size = 388 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {e x}{c}-\frac {2 e \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c^2}+\frac {(d e-c f) \log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{c^2}+\frac {2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {-b^2 c e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a d^2 e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+a c d f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+a c f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3-c d e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3+c^2 f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 a d \text {$\#$1}+3 a \text {$\#$1}^2+2 c \text {$\#$1}^3}\&\right ]}{c^2} \]
[Out]
\[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {f}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}}\right ) \, dx \\ & = e \int \frac {x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx+f \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \\ & = e \int \left (\frac {1}{c}+\frac {-b^2+d^4-d^2 (2 a-3 c d) x-3 c d (a-c d) x^2-c^2 (a-c d) x^3}{c \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 )}+\frac {d 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 x^2 \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 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 {2 c \left (1-\frac {a}{2 c d}\right ) d 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 {c^2 x^3 \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 {x \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+f \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 \\ & = \frac {e x}{c}+e \int \frac {x \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 {e \int \frac {-b^2+d^4-d^2 (2 a-3 c d) x-3 c d (a-c d) x^2-c^2 (a-c d) x^3}{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}+(c e) \int \frac {x^2 \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+\left (c^2 e\right ) \int \frac {x^3 \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 e) \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+\left (d^2 e\right ) \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-((a-2 c d) e) \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+f \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+f \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+(c f) \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+\left (c^2 f\right ) \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 f) \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+\left (d^2 f\right ) \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) f) \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 \\ & = \frac {e x}{c}+\frac {(a-c d) e \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^3}+\frac {f \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}+e \int \frac {x \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 {e \int \frac {2 c^2 \left (2 b^2 c^2+a d^2 (a-3 c d)\right )+4 a c^3 d (2 a-3 c d) x+6 a c^4 (a-c d) 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^5}+(c e) \int \frac {x^2 \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+\left (c^2 e\right ) \int \frac {x^3 \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 e) \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+\left (d^2 e\right ) \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-((a-2 c d) e) \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+f \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 {f \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 f) \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+\left (c^2 f\right ) \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 f) \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+\left (d^2 f\right ) \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) f) \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 \\ & = \frac {e x}{c}+\frac {(a-c d) e \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^3}+\frac {f \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}+e \int \frac {x \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 {e \int \left (\frac {2 c^2 \left (2 b^2 c^2+a d^2 (a-3 c d)\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 {4 a c^3 d (2 a-3 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 {6 a c^4 (a-c d) 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}{4 c^5}+(c e) \int \frac {x^2 \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+\left (c^2 e\right ) \int \frac {x^3 \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 e) \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+\left (d^2 e\right ) \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-((a-2 c d) e) \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+f \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 {f \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 f) \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+\left (c^2 f\right ) \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 f) \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+\left (d^2 f\right ) \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) f) \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 \\ & = \frac {e x}{c}+\frac {(a-c d) e \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^3}+\frac {f \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}+e \int \frac {x \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+(c e) \int \frac {x^2 \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+\left (c^2 e\right ) \int \frac {x^3 \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 e) \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+\left (d^2 e\right ) \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-\frac {(a d (2 a-3 c d) e) \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}{c^2}-((a-2 c d) e) \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-\frac {(3 a (a-c d) e) \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}{2 c}-\frac {\left (\left (2 b^2 c^2+a d^2 (a-3 c d)\right ) e\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^3}+f \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 {(a f) \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 f) \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+\left (c^2 f\right ) \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 f) \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+\left (d^2 f\right ) \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) f) \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 \\ & = \frac {e x}{c}+\frac {(a-c d) e \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^3}+\frac {f \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}+e \int \frac {x \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+(c e) \int \frac {x^2 \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+\left (c^2 e\right ) \int \frac {x^3 \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 e) \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+\left (d^2 e\right ) \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-\frac {(a d (2 a-3 c d) e) \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}{c^2}-((a-2 c d) e) \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-\frac {(3 a (a-c d) e) \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}{2 c}-\frac {\left (\left (2 b^2 c^2+a d^2 (a-3 c d)\right ) e\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^3}+f \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 {(a f) \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 f) \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+\left (c^2 f\right ) \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 f) \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+\left (d^2 f\right ) \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) f) \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 \\ & = \frac {e x}{c}+\frac {(a-c d) e \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^3}+\frac {f \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}+e \int \frac {x \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+(c e) \int \frac {x^2 \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+\left (c^2 e\right ) \int \frac {x^3 \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 e) \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+\left (d^2 e\right ) \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-\frac {(a d (2 a-3 c d) e) \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}{c^2}-((a-2 c d) e) \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-\frac {(3 a (a-c d) e) \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}{2 c}-\frac {\left (\left (2 b^2 c^2+a d^2 (a-3 c d)\right ) e\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^3}+f \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+(c f) \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 f) \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+\left (c^2 f\right ) \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 f) \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 f) \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+\left (d^2 f\right ) \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 f\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) f) \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 \\ \end{align*}
Time = 1.00 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.98 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {c e x-2 e \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(d e-c f) \log \left (a x+\sqrt {b^2+a^2 x^2}\right )+2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {-b^2 c e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a d^2 e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+a c d f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+a c f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3-c d e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3+c^2 f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 a d \text {$\#$1}+3 a \text {$\#$1}^2+2 c \text {$\#$1}^3}\&\right ]}{c^2} \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08
\[\int \frac {e x +f}{d +c x +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]
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Timed out. \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\text {Timed out} \]
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Not integrable
Time = 1.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {e x + f}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.51 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {e x + f}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Not integrable
Time = 4.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {e x + f}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]
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Not integrable
Time = 7.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {f+e\,x}{d+c\,x+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]
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