\(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{(b^2+a x^2)^{5/2}} \, dx\) [1953]

Optimal result

Integrand size = 33, antiderivative size = 137 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\frac {5 x}{12 b^2 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {x \left (23 b^2+15 a x^2\right )}{24 b^3 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {5 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{8 \sqrt {a} b^{7/2}} \]

[Out]

5/12*x/b^2/(a*x^2+b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+1/24*x*(15*a*x^2+23*b^2)/b^3/(a*x^2+b^2)^(3/2)/(b+(a*x^2+b^
2)^(1/2))^(1/2)+5/8*arctan(a^(1/2)*x/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2))/a^(1/2)/b^(7/2)

Rubi [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx \]

[In]

Int[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]

[Out]

Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\frac {x \left (23 b^2+15 a x^2+10 b \sqrt {b^2+a x^2}\right )}{24 b^3 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {5 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{8 \sqrt {a} b^{7/2}} \]

[In]

Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^(5/2),x]

[Out]

(x*(23*b^2 + 15*a*x^2 + 10*b*Sqrt[b^2 + a*x^2]))/(24*b^3*(b^2 + a*x^2)^(3/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (5
*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(8*Sqrt[a]*b^(7/2))

Maple [F]

\[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{\frac {5}{2}}}d x\]

[In]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),x)

[Out]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\left (a x^{2} + b^{2}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((b+(a*x**2+b**2)**(1/2))**(1/2)/(a*x**2+b**2)**(5/2),x)

[Out]

Integral(sqrt(b + sqrt(a*x**2 + b**2))/(a*x**2 + b**2)**(5/2), x)

Maxima [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^(5/2), x)

Giac [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^(5/2),x, algorithm="giac")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{{\left (b^2+a\,x^2\right )}^{5/2}} \,d x \]

[In]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(a*x^2 + b^2)^(5/2),x)

[Out]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(a*x^2 + b^2)^(5/2), x)