\(\int \frac {(b^2+a x^2)^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [2280]

Optimal result

Integrand size = 33, antiderivative size = 173 \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \sqrt {b^2+a x^2} \left (46 b^2+15 a x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 x \left (46 b^3+3 a b x^2\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {2} b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

[Out]

2/105*x*(a*x^2+b^2)^(1/2)*(15*a*x^2+46*b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-2/105*x*(3*a*b*x^2+46*b^3)/(b+(a*x^2+b
^2)^(1/2))^(1/2)+2*2^(1/2)*b^(7/2)*arctan(1/2*a^(1/2)*x*2^(1/2)/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-1/2*(b+(a*
x^2+b^2)^(1/2))^(1/2)*2^(1/2)/b^(1/2))/a^(1/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.74 \[ \int \frac {\left (b^2+a x^2\right )^{3/2}}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {2 x \left (-46 b^3-3 a b x^2+46 b^2 \sqrt {b^2+a x^2}+15 a x^2 \sqrt {b^2+a x^2}\right )}{105 \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} b^{7/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

(2*x*(-46*b^3 - 3*a*b*x^2 + 46*b^2*Sqrt[b^2 + a*x^2] + 15*a*x^2*Sqrt[b^2 + a*x^2]))/(105*Sqrt[b + Sqrt[b^2 + a
*x^2]]) + (Sqrt[2]*b^(7/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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