\(\int \frac {(b^2+a x^2) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx\) [3090]

Optimal result

Integrand size = 42, antiderivative size = 530 \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\frac {4 b x}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \sqrt {b^2+a x^2}}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 i \left (\sqrt {-1+\sqrt {2}} b^{3/2}+\sqrt {2 \left (-1+\sqrt {2}\right )} b^{3/2}\right ) \arctan \left (\frac {\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}-i \sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {-1+\sqrt {2}} b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 i \left (-\sqrt {1+\sqrt {2}} b^{3/2}+\sqrt {2 \left (1+\sqrt {2}\right )} b^{3/2}\right ) \text {arctanh}\left (\frac {\frac {i a x}{\sqrt {b+\sqrt {b^2+a x^2}}}-i \sqrt {a} \sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}} \]

[Out]

4/3*b*x/(b+(a*x^2+b^2)^(1/2))^(1/2)+2/3*x*(a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+2*I*((2^(1/2)-1)^(1/2)
*b^(3/2)+(-2+2*2^(1/2))^(1/2)*b^(3/2))*arctan((I*a*x/(b+(a*x^2+b^2)^(1/2))^(1/2)-I*a^(1/2)*(b+(a*x^2+b^2)^(1/2
))^(1/2))/(2+2*2^(1/2))^(1/2)/a^(1/2)/b^(1/2))/a^(1/2)+2*(2^(1/2)-1)^(1/2)*b^(3/2)*arctan(a^(1/2)*x/(2+2*2^(1/
2))^(1/2)/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-(b+(a*x^2+b^2)^(1/2))^(1/2)/(2+2*2^(1/2))^(1/2)/b^(1/2))/a^(1/2)
-2*I*(-(1+2^(1/2))^(1/2)*b^(3/2)+(2+2*2^(1/2))^(1/2)*b^(3/2))*arctanh((I*a*x/(b+(a*x^2+b^2)^(1/2))^(1/2)-I*a^(
1/2)*(b+(a*x^2+b^2)^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2)/a^(1/2)/b^(1/2))/a^(1/2)+2*(1+2^(1/2))^(1/2)*b^(3/2)*ar
ctanh(a^(1/2)*x/(-2+2*2^(1/2))^(1/2)/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-(b+(a*x^2+b^2)^(1/2))^(1/2)/(-2+2*2^(
1/2))^(1/2)/b^(1/2))/a^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.33 \[ \int \frac {\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}{-b^2+a x^2} \, dx=\frac {2 x \left (2 b+\sqrt {b^2+a x^2}\right )}{3 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {1+\sqrt {2}} \sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} b^{3/2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

(2*x*(2*b + Sqrt[b^2 + a*x^2]))/(3*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*b^(3/2)*ArcTan[(Sqrt[-1 + Sqrt[2]]*Sqrt[a
]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/(Sqrt[1 + Sqrt[2]]*Sqrt[a]) - (2*Sqrt[1 + Sqrt[2]]*b^(3/2)*ArcTan
h[(Sqrt[1 + Sqrt[2]]*Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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