\(\int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx\) [1868]

Optimal result

Integrand size = 47, antiderivative size = 128 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=-\frac {\text {RootSum}\left [b^2-2 b \text {$\#$1}^2-2 b \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{b+2 b \text {$\#$1}^2+3 \text {$\#$1}^4-2 \text {$\#$1}^6}\&\right ]}{a} \]

[Out]

Unintegrable

Rubi [F]

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

[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*} \text {integral}& = \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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \text {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}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=-\frac {\text {RootSum}\left [b^2-2 b \text {$\#$1}^2-2 b \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {b \log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {a x-\sqrt {b+a^2 x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{b+2 b \text {$\#$1}^2+3 \text {$\#$1}^4-2 \text {$\#$1}^6}\&\right ]}{a} \]

[In]

Integrate[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)

Maple [N/A] (verified)

Not integrable

Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.32

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

[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)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 2.34 (sec) , antiderivative size = 678002, normalized size of antiderivative = 5296.89 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=\text {Too large to display} \]

[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]

Too large to include

Sympy [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=\int \frac {\sqrt {a x - \sqrt {a^{2} x^{2} + b}}}{a^{2} x^{2} + \sqrt {a^{2} x^{2} + b}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=\int { \frac {\sqrt {a x - \sqrt {a^{2} x^{2} + b}}}{a^{2} x^{2} + \sqrt {a^{2} x^{2} + b}} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 2.49 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=\int { \frac {\sqrt {a x - \sqrt {a^{2} x^{2} + b}}}{a^{2} x^{2} + \sqrt {a^{2} x^{2} + b}} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 5.68 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {a x-\sqrt {b+a^2 x^2}}}{a^2 x^2+\sqrt {b+a^2 x^2}} \, dx=\int \frac {\sqrt {a\,x-\sqrt {a^2\,x^2+b}}}{\sqrt {a^2\,x^2+b}+a^2\,x^2} \,d x \]

[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)