\(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx\) [206]

Optimal result

Integrand size = 33, antiderivative size = 22 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx=\frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx=\frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}} \]

[In]

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

[Out]

(2*x)/Sqrt[b + Sqrt[b^2 + a*x^2]]

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 38 vs. \(2 (18) = 36\).

Time = 0.28 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx=-\frac {2 \, \sqrt {b + \sqrt {a x^{2} + b^{2}}} {\left (b - \sqrt {a x^{2} + b^{2}}\right )}}{a x} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (19) = 38\).

Time = 0.44 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {b^2+a x^2}} \, dx=\frac {\sqrt {2} x \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right )}{\pi \sqrt {b} \sqrt {\sqrt {\frac {a x^{2}}{b^{2}} + 1} + 1}} \]

[In]

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

[Out]

sqrt(2)*x*gamma(1/4)*gamma(3/4)/(pi*sqrt(b)*sqrt(sqrt(a*x**2/b**2 + 1) + 1))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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