\(\int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2986]

Optimal result

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]

Unintegrable

Rubi [F]

\[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]

[In]

Int[(f + e*x)/(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

(e*x)/c + ((a - c*d)*e*Log[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])/(4*c^3) + (f*Log[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])/(4*c) - ((2*b^2*c^2 + a*d^2*(a - 3*c*d))*e*Defer[Int][(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)^(-1), x])/(2*c^3) - (a*d^2*f*Defer[Int][(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)^(-1), x])/(2*c) - (a*d*(2*a - 3*c*d)
*e*Defer[Int][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),
x])/c^2 - a*d*f*Defer[Int][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), x] - (3*a*(a - c*d)*e*Defer[Int][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), x])/(2*c) - (a*c*f*Defer[Int][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), x])/2 + d*f*Defer[Int][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), x] + d*e*Defer[Int][(x*Sq
rt[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), x] + c*f*Defer[Int][(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), x] + c*e*Defer[Int][(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), x] + d^2*f*Defer[Int][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), x] +
d^2*e*Defer[Int][(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), x] - (a - 2*c*d)*f*Defer[Int][(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), x] - (a - 2*c*d)*e*D
efer[Int][(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), x] + c^2*f*Defer[Int][(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), x] + c^2*e*Defer[Int][(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), x] + f*Defer[Int][(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), x] + e*Defer[Int][(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), x]

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*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.98 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\frac {c e x-2 e \sqrt {a x+\sqrt {b^2+a^2 x^2}}+(d e-c f) \log \left (a x+\sqrt {b^2+a^2 x^2}\right )+2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {-b^2 c e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )-a d^2 e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+a c d f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+a c f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2+a e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3-c d e \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3+c^2 f \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3}{2 a d \text {$\#$1}+3 a \text {$\#$1}^2+2 c \text {$\#$1}^3}\&\right ]}{c^2} \]

[In]

Integrate[(f + e*x)/(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

(c*e*x - 2*e*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] + (d*e - c*f)*Log[a*x + Sqrt[b^2 + a^2*x^2]] + 2*RootSum[b^2*c -
2*a*d*#1^2 - 2*a*#1^3 - c*#1^4 & , (-(b^2*c*e*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]) - a*d^2*e*Log[Sqrt[a*
x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + a*c*d*f*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + a*c*f*Log[Sqrt[a*x
+ Sqrt[b^2 + a^2*x^2]] - #1]*#1^2 + a*e*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^3 - c*d*e*Log[Sqrt[a*x +
Sqrt[b^2 + a^2*x^2]] - #1]*#1^3 + c^2*f*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1^3)/(2*a*d*#1 + 3*a*#1^2 +
2*c*#1^3) & ])/c^2

Maple [N/A] (verified)

Not integrable

Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.08

\[\int \frac {e x +f}{d +c x +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

[In]

int((e*x+f)/(d+c*x+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x)

[Out]

int((e*x+f)/(d+c*x+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\text {Timed out} \]

[In]

integrate((e*x+f)/(d+c*x+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.39 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.08 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {e x + f}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]

[In]

integrate((e*x+f)/(d+c*x+(a*x+(a**2*x**2+b**2)**(1/2))**(1/2)),x)

[Out]

Integral((e*x + f)/(c*x + d + sqrt(a*x + sqrt(a**2*x**2 + b**2))), x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.51 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {e x + f}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

[In]

integrate((e*x+f)/(d+c*x+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x, algorithm="maxima")

[Out]

1/2*e*x/c - 1/2*(d*e - c*f)*log(c*x + d)/c^2 - integrate(-1/2*(c^2*e*x^3 + d^2*f + (2*c*d*e + c^2*f - a*e)*x^2
+ (d^2*e + 2*c*d*f - a*f)*x - sqrt(a^2*x^2 + b^2)*(e*x + f))/(c^3*x^3 + d^3 + (3*c^2*d + a*c)*x^2 + (3*c*d^2
+ a*d)*x + 2*(c^2*x^2 + 2*c*d*x + d^2)*sqrt(a*x + sqrt(a^2*x^2 + b^2)) + sqrt(a^2*x^2 + b^2)*(c*x + d)), x)

Giac [N/A]

Not integrable

Time = 4.77 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {e x + f}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

[In]

integrate((e*x+f)/(d+c*x+(a*x+(a^2*x^2+b^2)^(1/2))^(1/2)),x, algorithm="giac")

[Out]

integrate((e*x + f)/(c*x + d + sqrt(a*x + sqrt(a^2*x^2 + b^2))), x)

Mupad [N/A]

Not integrable

Time = 7.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.09 \[ \int \frac {f+e x}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {f+e\,x}{d+c\,x+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]

[In]

int((f + e*x)/(d + c*x + (a*x + (b^2 + a^2*x^2)^(1/2))^(1/2)),x)

[Out]

int((f + e*x)/(d + c*x + (a*x + (b^2 + a^2*x^2)^(1/2))^(1/2)), x)