\(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{(b^2+a x^2)^2} \, dx\) [1846]

Optimal result

Integrand size = 31, antiderivative size = 125 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\frac {x}{2 b \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {3 x}{4 b^2 \sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {3 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{4 \sqrt {a} b^{5/2}} \]

[Out]

1/2*x/b/(a*x^2+b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+3/4*x/b^2/(a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+3/4*ar
ctan(a^(1/2)*x/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2))/a^(1/2)/b^(5/2)

Rubi [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx \]

[In]

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

[Out]

Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(b - Sqrt[-a]*x), x]/(4*b^3) + Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(
b + Sqrt[-a]*x), x]/(4*b^3) - (a*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b - a*x)^2, x])/(4*b^2) - (a
*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b + a*x)^2, x])/(4*b^2)

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a \sqrt {b+\sqrt {b^2+a x^2}}}{4 b^2 \left (\sqrt {-a} b-a x\right )^2}-\frac {a \sqrt {b+\sqrt {b^2+a x^2}}}{4 b^2 \left (\sqrt {-a} b+a x\right )^2}-\frac {a \sqrt {b+\sqrt {b^2+a x^2}}}{2 b^2 \left (-a b^2-a^2 x^2\right )}\right ) \, dx \\ & = -\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{4 b^2}-\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{4 b^2}-\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{-a b^2-a^2 x^2} \, dx}{2 b^2} \\ & = -\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{4 b^2}-\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{4 b^2}-\frac {a \int \left (-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b-\sqrt {-a} x\right )}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{2 a b \left (b+\sqrt {-a} x\right )}\right ) \, dx}{2 b^2} \\ & = \frac {\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {-a} x} \, dx}{4 b^3}+\frac {\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {-a} x} \, dx}{4 b^3}-\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{4 b^2}-\frac {a \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{4 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\frac {x \left (3 b^2+3 a x^2+2 b \sqrt {b^2+a x^2}\right )}{4 b^2 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {3 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{4 \sqrt {a} b^{5/2}} \]

[In]

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

[Out]

(x*(3*b^2 + 3*a*x^2 + 2*b*Sqrt[b^2 + a*x^2]))/(4*b^2*(b^2 + a*x^2)^(3/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (3*Arc
Tan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(4*Sqrt[a]*b^(5/2))

Maple [F]

\[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\left (a x^{2} + b^{2}\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{{\left (b^2+a\,x^2\right )}^2} \,d x \]

[In]

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

[Out]

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