\(\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{(b^2+a x^2)^3} \, dx\) [2069]

Optimal result

Integrand size = 31, antiderivative size = 149 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\frac {7 x \left (23 b^2+15 a x^2\right )}{192 b^4 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {x \left (59 b^2+35 a x^2\right )}{96 b^3 \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {35 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{64 \sqrt {a} b^{9/2}} \]

[Out]

7/192*x*(15*a*x^2+23*b^2)/b^4/(a*x^2+b^2)^(3/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+1/96*x*(35*a*x^2+59*b^2)/b^3/(a*x^
2+b^2)^2/(b+(a*x^2+b^2)^(1/2))^(1/2)+35/64*arctan(a^(1/2)*x/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2))/a^(1/2)/b^(9/
2)

Rubi [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx \]

[In]

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

[Out]

(3*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(b - Sqrt[-a]*x), x])/(16*b^5) + (3*Defer[Int][Sqrt[b + Sqrt[b^2 + a
*x^2]]/(b + Sqrt[-a]*x), x])/(16*b^5) + ((-a)^(3/2)*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b - a*x)^
3, x])/(8*b^3) - (3*a*Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b - a*x)^2, x])/(16*b^4) + ((-a)^(3/2)*
Defer[Int][Sqrt[b + Sqrt[b^2 + a*x^2]]/(Sqrt[-a]*b + a*x)^3, x])/(8*b^3) - (3*a*Defer[Int][Sqrt[b + Sqrt[b^2 +
a*x^2]]/(Sqrt[-a]*b + a*x)^2, x])/(16*b^4)

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {a^3 \sqrt {b+\sqrt {b^2+a x^2}}}{8 (-a)^{3/2} b^3 \left (\sqrt {-a} b-a x\right )^3}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{16 b^4 \left (\sqrt {-a} b-a x\right )^2}-\frac {a^3 \sqrt {b+\sqrt {b^2+a x^2}}}{8 (-a)^{3/2} b^3 \left (\sqrt {-a} b+a x\right )^3}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{16 b^4 \left (\sqrt {-a} b+a x\right )^2}-\frac {3 a \sqrt {b+\sqrt {b^2+a x^2}}}{8 b^4 \left (-a b^2-a^2 x^2\right )}\right ) \, dx \\ & = -\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{-a b^2-a^2 x^2} \, dx}{8 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3} \\ & = -\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}-\frac {(3 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}{8 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3} \\ & = \frac {3 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {-a} x} \, dx}{16 b^5}+\frac {3 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {-a} x} \, dx}{16 b^5}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2} \, dx}{16 b^4}-\frac {(3 a) \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2} \, dx}{16 b^4}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^3} \, dx}{8 b^3}+\frac {(-a)^{3/2} \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^3} \, dx}{8 b^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\frac {7 x \left (23 b^2+15 a x^2\right )}{192 b^4 \left (b^2+a x^2\right )^{3/2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {x \left (59 b^2+35 a x^2\right )}{96 b^3 \left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {35 \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{64 \sqrt {a} b^{9/2}} \]

[In]

Integrate[Sqrt[b + Sqrt[b^2 + a*x^2]]/(b^2 + a*x^2)^3,x]

[Out]

(7*x*(23*b^2 + 15*a*x^2))/(192*b^4*(b^2 + a*x^2)^(3/2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (x*(59*b^2 + 35*a*x^2))/
(96*b^3*(b^2 + a*x^2)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (35*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x
^2]])])/(64*Sqrt[a]*b^(9/2))

Maple [F]

\[\int \frac {\sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{3}}d x\]

[In]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^3,x)

[Out]

int((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^3,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\text {Timed out} \]

[In]

integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^3,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{\left (a x^{2} + b^{2}\right )^{3}}\, dx \]

[In]

integrate((b+(a*x**2+b**2)**(1/2))**(1/2)/(a*x**2+b**2)**3,x)

[Out]

Integral(sqrt(b + sqrt(a*x**2 + b**2))/(a*x**2 + b**2)**3, x)

Maxima [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{3}} \,d x } \]

[In]

integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^3,x, algorithm="maxima")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^3, x)

Giac [F]

\[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\int { \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{3}} \,d x } \]

[In]

integrate((b+(a*x^2+b^2)^(1/2))^(1/2)/(a*x^2+b^2)^3,x, algorithm="giac")

[Out]

integrate(sqrt(b + sqrt(a*x^2 + b^2))/(a*x^2 + b^2)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^3} \, dx=\int \frac {\sqrt {b+\sqrt {b^2+a\,x^2}}}{{\left (b^2+a\,x^2\right )}^3} \,d x \]

[In]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(a*x^2 + b^2)^3,x)

[Out]

int((b + (a*x^2 + b^2)^(1/2))^(1/2)/(a*x^2 + b^2)^3, x)