\(\int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\) [2332]

Optimal result

Integrand size = 30, antiderivative size = 183 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {\log \left (a x+\sqrt {b^2+a^2 x^2}\right )}{c}+\frac {2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {a d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+a \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 a d+3 a \text {$\#$1}+2 c \text {$\#$1}^2}\&\right ]}{c} \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx \]

[In]

Int[(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])^(-1),x]

[Out]

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) - (a*d^
2*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*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] - (a*c*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*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] + c*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] + d^2*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] - (a - 2*c*d)*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] + c^2*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] + Defe
r[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]

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*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.98 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=-\frac {\log \left (a x+\sqrt {b^2+a^2 x^2}\right )-2 \text {RootSum}\left [b^2 c-2 a d \text {$\#$1}^2-2 a \text {$\#$1}^3-c \text {$\#$1}^4\&,\frac {a d \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right )+a \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+c \log \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^2}{2 a d+3 a \text {$\#$1}+2 c \text {$\#$1}^2}\&\right ]}{c} \]

[In]

Integrate[(d + c*x + Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])^(-1),x]

[Out]

-((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 & , (a*d*Log[Sqrt[a*x + S
qrt[b^2 + a^2*x^2]] - #1] + a*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x^2]] - #1]*#1 + c*Log[Sqrt[a*x + Sqrt[b^2 + a^2*x
^2]] - #1]*#1^2)/(2*a*d + 3*a*#1 + 2*c*#1^2) & ])/c)

Maple [N/A] (verified)

Not integrable

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.14

\[\int \frac {1}{d +c x +\sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\text {Timed out} \]

[In]

integrate(1/(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 = 0.77 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \]

[In]

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

[Out]

Integral(1/(c*x + d + sqrt(a*x + sqrt(a**2*x**2 + b**2))), x)

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

[In]

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

[Out]

integrate(1/(c*x + d + sqrt(a*x + sqrt(a^2*x^2 + b^2))), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int { \frac {1}{c x + d + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}} \,d x } \]

[In]

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

[Out]

integrate(1/(c*x + d + sqrt(a*x + sqrt(a^2*x^2 + b^2))), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.15 \[ \int \frac {1}{d+c x+\sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx=\int \frac {1}{d+c\,x+\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \]

[In]

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

[Out]

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