3.19.68 \(\int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx\)

Optimal. Leaf size=128 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^6-2 \text {$\#$1}^4 b-2 \text {$\#$1}^2 b+b^2\& ,\frac {\text {$\#$1}^5 \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )+\text {$\#$1} b \log \left (\sqrt {a x-\sqrt {a^2 x^2+b}}-\text {$\#$1}\right )}{-2 \text {$\#$1}^6+3 \text {$\#$1}^4+2 \text {$\#$1}^2 b+b}\& \right ]}{a} \]

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Rubi [F]  time = 49.68, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sqrt[a*x - Sqrt[b + a^2*x^2]]/(a^2*x^2 + Sqrt[b + a^2*x^2]),x]

[Out]

-(((1 - 1/Sqrt[1 + 4*b])*(1 + 2*b - Sqrt[1 + 4*b] - Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sqrt[1 + 4*b]])*Arc
Tan[(2^(1/4)*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] - Sqrt[1 + 2*b - Sqrt[1 + 4*b]]]])/(2
^(3/4)*a*Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] - Sqrt[1 + 2*b - S
qrt[1 + 4*b]]])) + ((1 - 1/Sqrt[1 + 4*b])*(1 + 2*b - Sqrt[1 + 4*b] + Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sq
rt[1 + 4*b]])*ArcTan[(2^(1/4)*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] + Sqrt[1 + 2*b - Sqr
t[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] +
Sqrt[1 + 2*b - Sqrt[1 + 4*b]]]) - (Sqrt[1 + Sqrt[1 + 4*b]]*(1 + 2*b + Sqrt[1 + 4*b] - Sqrt[1 + Sqrt[1 + 4*b]]
*Sqrt[1 + 2*b + Sqrt[1 + 4*b]])*ArcTan[(2^(1/4)*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 + Sqrt[1 + 4*b]] -
Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1 + 4*b]*Sqrt[1 + 2*b + Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 + Sqrt[1 +
4*b]] - Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]) + (Sqrt[1 + Sqrt[1 + 4*b]]*(1 + 2*b + Sqrt[1 + 4*b] + Sqrt[1 + Sqrt[1
+ 4*b]]*Sqrt[1 + 2*b + Sqrt[1 + 4*b]])*ArcTan[(2^(1/4)*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 + Sqrt[1 +
4*b]] + Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1 + 4*b]*Sqrt[1 + 2*b + Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 +
Sqrt[1 + 4*b]] + Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]) + ((1 - 1/Sqrt[1 + 4*b])*(1 + 2*b - Sqrt[1 + 4*b] - Sqrt[1 -
Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sqrt[1 + 4*b]])*ArcTanh[(2^(1/4)*Sqrt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 - Sq
rt[1 + 4*b]] - Sqrt[1 + 2*b - Sqrt[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sqrt[1 + 4*b
]]*Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] - Sqrt[1 + 2*b - Sqrt[1 + 4*b]]]) - ((1 - 1/Sqrt[1 + 4*b])*(1 + 2*b - Sqrt[1 +
4*b] + Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2*b - Sqrt[1 + 4*b]])*ArcTanh[(2^(1/4)*Sqrt[a*x - Sqrt[b + a^2*x^2]])
/Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] + Sqrt[1 + 2*b - Sqrt[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1 - Sqrt[1 + 4*b]]*Sqrt[1 + 2
*b - Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 - Sqrt[1 + 4*b]] + Sqrt[1 + 2*b - Sqrt[1 + 4*b]]]) + (Sqrt[1 + Sqrt[1 + 4*b]]*
(1 + 2*b + Sqrt[1 + 4*b] - Sqrt[1 + Sqrt[1 + 4*b]]*Sqrt[1 + 2*b + Sqrt[1 + 4*b]])*ArcTanh[(2^(1/4)*Sqrt[a*x -
Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 + Sqrt[1 + 4*b]] - Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1 + 4*b]*S
qrt[1 + 2*b + Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 + Sqrt[1 + 4*b]] - Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]) - (Sqrt[1 + Sqrt[1
+ 4*b]]*(1 + 2*b + Sqrt[1 + 4*b] + Sqrt[1 + Sqrt[1 + 4*b]]*Sqrt[1 + 2*b + Sqrt[1 + 4*b]])*ArcTanh[(2^(1/4)*Sq
rt[a*x - Sqrt[b + a^2*x^2]])/Sqrt[Sqrt[1 + Sqrt[1 + 4*b]] + Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]])/(2^(3/4)*a*Sqrt[1
+ 4*b]*Sqrt[1 + 2*b + Sqrt[1 + 4*b]]*Sqrt[Sqrt[1 + Sqrt[1 + 4*b]] + Sqrt[1 + 2*b + Sqrt[1 + 4*b]]]) - Defer[I
nt][(Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(Sqrt[1 - Sqrt[1 + 4*b]] - Sqrt[2]*a*x), x]/(Sqrt[1 + 4*
b]*Sqrt[1 - Sqrt[1 + 4*b]]) + Defer[Int][(Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(Sqrt[1 + Sqrt[1 +
4*b]] - Sqrt[2]*a*x), x]/(Sqrt[1 + 4*b]*Sqrt[1 + Sqrt[1 + 4*b]]) - Defer[Int][(Sqrt[b + a^2*x^2]*Sqrt[a*x - Sq
rt[b + a^2*x^2]])/(Sqrt[1 - Sqrt[1 + 4*b]] + Sqrt[2]*a*x), x]/(Sqrt[1 + 4*b]*Sqrt[1 - Sqrt[1 + 4*b]]) + Defer[
Int][(Sqrt[b + a^2*x^2]*Sqrt[a*x - Sqrt[b + a^2*x^2]])/(Sqrt[1 + Sqrt[1 + 4*b]] + Sqrt[2]*a*x), x]/(Sqrt[1 + 4
*b]*Sqrt[1 + Sqrt[1 + 4*b]])

Rubi steps

\begin {align*} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx &=\int \left (\frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+a^2 x^2-a^4 x^4}+\frac {a^2 x^2 \sqrt {a x-\sqrt {b+a^2 x^2}}}{-b-a^2 x^2+a^4 x^4}\right ) \, dx\\ &=a^2 \int \frac {x^2 \sqrt {a x-\sqrt {b+a^2 x^2}}}{-b-a^2 x^2+a^4 x^4} \, dx+\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{b+a^2 x^2-a^4 x^4} \, dx\\ &=a^2 \int \left (\frac {\left (1+\frac {1}{\sqrt {1+4 b}}\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{-a^2-a^2 \sqrt {1+4 b}+2 a^4 x^2}+\frac {\left (1-\frac {1}{\sqrt {1+4 b}}\right ) \sqrt {a x-\sqrt {b+a^2 x^2}}}{-a^2+a^2 \sqrt {1+4 b}+2 a^4 x^2}\right ) \, dx+\int \left (\frac {2 a^2 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+4 b} \left (a^2+a^2 \sqrt {1+4 b}-2 a^4 x^2\right )}+\frac {2 a^2 \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+4 b} \left (-a^2+a^2 \sqrt {1+4 b}+2 a^4 x^2\right )}\right ) \, dx\\ &=\frac {\left (2 a^2\right ) \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2+a^2 \sqrt {1+4 b}-2 a^4 x^2} \, dx}{\sqrt {1+4 b}}+\frac {\left (2 a^2\right ) \int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{-a^2+a^2 \sqrt {1+4 b}+2 a^4 x^2} \, dx}{\sqrt {1+4 b}}+\left (a^2 \left (1-\frac {1}{\sqrt {1+4 b}}\right )\right ) \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{-a^2+a^2 \sqrt {1+4 b}+2 a^4 x^2} \, dx+\left (a^2 \left (1+\frac {1}{\sqrt {1+4 b}}\right )\right ) \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{-a^2-a^2 \sqrt {1+4 b}+2 a^4 x^2} \, dx\\ &=\frac {\left (2 a^2\right ) \int \left (\frac {\sqrt {1-\sqrt {1+4 b}} \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (-a^2+a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x\right )}+\frac {\sqrt {1-\sqrt {1+4 b}} \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (-a^2+a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x\right )}\right ) \, dx}{\sqrt {1+4 b}}+\frac {\left (2 a^2\right ) \int \left (\frac {\sqrt {1+\sqrt {1+4 b}} \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (a^2+a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x\right )}+\frac {\sqrt {1+\sqrt {1+4 b}} \sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (a^2+a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x\right )}\right ) \, dx}{\sqrt {1+4 b}}+\left (a^2 \left (1-\frac {1}{\sqrt {1+4 b}}\right )\right ) \int \left (\frac {\sqrt {1-\sqrt {1+4 b}} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (-a^2+a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x\right )}+\frac {\sqrt {1-\sqrt {1+4 b}} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (-a^2+a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x\right )}\right ) \, dx+\left (a^2 \left (1+\frac {1}{\sqrt {1+4 b}}\right )\right ) \int \left (\frac {\sqrt {1+\sqrt {1+4 b}} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (-a^2-a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x\right )}+\frac {\sqrt {1+\sqrt {1+4 b}} \sqrt {a x-\sqrt {b+a^2 x^2}}}{2 \left (-a^2-a^2 \sqrt {1+4 b}\right ) \left (\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\sqrt {1-\sqrt {1+4 b}} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{2 \sqrt {1+4 b}}+\frac {\sqrt {1-\sqrt {1+4 b}} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{2 \sqrt {1+4 b}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{2 \sqrt {1+4 b}}-\frac {\sqrt {1+\sqrt {1+4 b}} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{2 \sqrt {1+4 b}}\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {b+x^2}{\sqrt {x} \left (\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x-\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {b+x^2}{\sqrt {x} \left (-\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x+\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {b+x^2}{\sqrt {x} \left (\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x-\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {b+x^2}{\sqrt {x} \left (-\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x+\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {2} a \sqrt {x}}+\frac {2 b+\sqrt {2} \sqrt {1-\sqrt {1+4 b}} x}{\sqrt {x} \left (\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x-\sqrt {2} a x^2\right )}\right ) \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {2} a \sqrt {x}}+\frac {2 b-\sqrt {2} \sqrt {1-\sqrt {1+4 b}} x}{\sqrt {x} \left (-\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x+\sqrt {2} a x^2\right )}\right ) \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt {2} a \sqrt {x}}+\frac {2 b+\sqrt {2} \sqrt {1+\sqrt {1+4 b}} x}{\sqrt {x} \left (\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x-\sqrt {2} a x^2\right )}\right ) \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {2} a \sqrt {x}}+\frac {2 b-\sqrt {2} \sqrt {1+\sqrt {1+4 b}} x}{\sqrt {x} \left (-\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x+\sqrt {2} a x^2\right )}\right ) \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b+\sqrt {2} \sqrt {1-\sqrt {1+4 b}} x}{\sqrt {x} \left (\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x-\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b-\sqrt {2} \sqrt {1-\sqrt {1+4 b}} x}{\sqrt {x} \left (-\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x+\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b+\sqrt {2} \sqrt {1+\sqrt {1+4 b}} x}{\sqrt {x} \left (\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x-\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b-\sqrt {2} \sqrt {1+\sqrt {1+4 b}} x}{\sqrt {x} \left (-\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x+\sqrt {2} a x^2\right )} \, dx,x,a x-\sqrt {b+a^2 x^2}\right )}{2 \sqrt {1+4 b}}\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b+\sqrt {2} \sqrt {1-\sqrt {1+4 b}} x^2}{\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x^2-\sqrt {2} a x^4} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {1+4 b}}+\frac {\sqrt {1-\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b-\sqrt {2} \sqrt {1-\sqrt {1+4 b}} x^2}{-\sqrt {2} a b+2 a \sqrt {1-\sqrt {1+4 b}} x^2+\sqrt {2} a x^4} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {1+4 b}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b+\sqrt {2} \sqrt {1+\sqrt {1+4 b}} x^2}{\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x^2-\sqrt {2} a x^4} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {1+4 b}}-\frac {\sqrt {1+\sqrt {1+4 b}} \operatorname {Subst}\left (\int \frac {2 b-\sqrt {2} \sqrt {1+\sqrt {1+4 b}} x^2}{-\sqrt {2} a b+2 a \sqrt {1+\sqrt {1+4 b}} x^2+\sqrt {2} a x^4} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {1+4 b}}\\ &=-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}-\frac {\left (\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}-\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1-\sqrt {1+4 b}}-a \sqrt {1+2 b-\sqrt {1+4 b}}-\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}}}+\frac {\left (\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}-\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1-\sqrt {1+4 b}}-a \sqrt {1+2 b-\sqrt {1+4 b}}+\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}}}+\frac {\left (\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}+\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1-\sqrt {1+4 b}}+a \sqrt {1+2 b-\sqrt {1+4 b}}-\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}}}-\frac {\left (\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}+\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1-\sqrt {1+4 b}}+a \sqrt {1+2 b-\sqrt {1+4 b}}+\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}}}+\frac {\left (\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}-\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1+\sqrt {1+4 b}}-a \sqrt {1+2 b+\sqrt {1+4 b}}-\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}}}-\frac {\left (\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}-\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1+\sqrt {1+4 b}}-a \sqrt {1+2 b+\sqrt {1+4 b}}+\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}}}-\frac {\left (\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}+\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1+\sqrt {1+4 b}}+a \sqrt {1+2 b+\sqrt {1+4 b}}-\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}}}+\frac {\left (\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}+\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a \sqrt {1+\sqrt {1+4 b}}+a \sqrt {1+2 b+\sqrt {1+4 b}}+\sqrt {2} a x^2} \, dx,x,\sqrt {a x-\sqrt {b+a^2 x^2}}\right )}{\sqrt {2} \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}}}\\ &=\frac {\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}-\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1-\sqrt {1+4 b}}-\sqrt {1+2 b-\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}} \sqrt {\sqrt {1-\sqrt {1+4 b}}-\sqrt {1+2 b-\sqrt {1+4 b}}}}-\frac {\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}+\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1-\sqrt {1+4 b}}+\sqrt {1+2 b-\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}} \sqrt {\sqrt {1-\sqrt {1+4 b}}+\sqrt {1+2 b-\sqrt {1+4 b}}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}-\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1+\sqrt {1+4 b}}-\sqrt {1+2 b+\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}} \sqrt {\sqrt {1+\sqrt {1+4 b}}-\sqrt {1+2 b+\sqrt {1+4 b}}}}+\frac {\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}+\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1+\sqrt {1+4 b}}+\sqrt {1+2 b+\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}} \sqrt {\sqrt {1+\sqrt {1+4 b}}+\sqrt {1+2 b+\sqrt {1+4 b}}}}-\frac {\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}-\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1-\sqrt {1+4 b}}-\sqrt {1+2 b-\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}} \sqrt {\sqrt {1-\sqrt {1+4 b}}-\sqrt {1+2 b-\sqrt {1+4 b}}}}+\frac {\sqrt {1-\sqrt {1+4 b}} \left (1+2 b-\sqrt {1+4 b}+\sqrt {1-\sqrt {1+4 b}} \sqrt {1+2 b-\sqrt {1+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1-\sqrt {1+4 b}}+\sqrt {1+2 b-\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b-\sqrt {1+4 b}} \sqrt {\sqrt {1-\sqrt {1+4 b}}+\sqrt {1+2 b-\sqrt {1+4 b}}}}+\frac {\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}-\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1+\sqrt {1+4 b}}-\sqrt {1+2 b+\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}} \sqrt {\sqrt {1+\sqrt {1+4 b}}-\sqrt {1+2 b+\sqrt {1+4 b}}}}-\frac {\sqrt {1+\sqrt {1+4 b}} \left (1+2 b+\sqrt {1+4 b}+\sqrt {1+\sqrt {1+4 b}} \sqrt {1+2 b+\sqrt {1+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {\sqrt {1+\sqrt {1+4 b}}+\sqrt {1+2 b+\sqrt {1+4 b}}}}\right )}{2^{3/4} a \sqrt {1+4 b} \sqrt {1+2 b+\sqrt {1+4 b}} \sqrt {\sqrt {1+\sqrt {1+4 b}}+\sqrt {1+2 b+\sqrt {1+4 b}}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}-\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1-\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1-\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}-\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}+\frac {\int \frac {\sqrt {b+a^2 x^2} \sqrt {a x-\sqrt {b+a^2 x^2}}}{\sqrt {1+\sqrt {1+4 b}}+\sqrt {2} a x} \, dx}{\sqrt {1+4 b} \sqrt {1+\sqrt {1+4 b}}}\\ \end {align*}

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Mathematica [F]  time = 3.82, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[a*x - Sqrt[b + a^2*x^2]]/(a^2*x^2 + Sqrt[b + a^2*x^2]),x]

[Out]

Integrate[Sqrt[a*x - Sqrt[b + a^2*x^2]]/(a^2*x^2 + Sqrt[b + a^2*x^2]), x]

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IntegrateAlgebraic [A]  time = 0.53, size = 128, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [b^2-2 b \text {$\#$1}^2-2 b \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{b+2 b \text {$\#$1}^2+3 \text {$\#$1}^4-2 \text {$\#$1}^6}\&\right ]}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[a*x - Sqrt[b + a^2*x^2]]/(a^2*x^2 + Sqrt[b + a^2*x^2]),x]

[Out]

-(RootSum[b^2 - 2*b*#1^2 - 2*b*#1^4 - 2*#1^6 + #1^8 & , (b*Log[Sqrt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1 + Log[Sq
rt[a*x - Sqrt[b + a^2*x^2]] - #1]*#1^5)/(b + 2*b*#1^2 + 3*#1^4 - 2*#1^6) & ]/a)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x-(a^2*x^2+b)^(1/2))^(1/2)/(a^2*x^2+(a^2*x^2+b)^(1/2)),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x - \sqrt {a^{2} x^{2} + b}}}{a^{2} x^{2} + \sqrt {a^{2} x^{2} + b}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x - sqrt(a^2*x^2 + b))/(a^2*x^2 + sqrt(a^2*x^2 + b)), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a x -\sqrt {a^{2} x^{2}+b}}}{a^{2} x^{2}+\sqrt {a^{2} x^{2}+b}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x - \sqrt {a^{2} x^{2} + b}}}{a^{2} x^{2} + \sqrt {a^{2} x^{2} + b}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x - sqrt(a^2*x^2 + b))/(a^2*x^2 + sqrt(a^2*x^2 + b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a\,x-\sqrt {a^2\,x^2+b}}}{\sqrt {a^2\,x^2+b}+a^2\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x - \sqrt {a^{2} x^{2} + b}}}{a^{2} x^{2} + \sqrt {a^{2} x^{2} + b}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*x - sqrt(a**2*x**2 + b))/(a**2*x**2 + sqrt(a**2*x**2 + b)), x)

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