\(\int \frac {(-b^2+a x^2)^2}{(b^2+a x^2)^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\) [2527]

Optimal result

Integrand size = 44, antiderivative size = 211 \[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=-\frac {b x}{\sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (2 b^2+a x^2\right )}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]

[Out]

-b*x/(a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+2*x*(a*x^2+2*b^2)/(a*x^2+b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-b
^(1/2)*arctan(a^(1/2)*x/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2))/a^(1/2)-2*2^(1/2)*b^(1/2)*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]

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

[In]

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

[Out]

Defer[Int][1/Sqrt[b + Sqrt[b^2 + a*x^2]], x] - b*Defer[Int][1/((b - Sqrt[-a]*x)*Sqrt[b + Sqrt[b^2 + a*x^2]]),
x] - b*Defer[Int][1/((b + Sqrt[-a]*x)*Sqrt[b + Sqrt[b^2 + a*x^2]]), x] - a*b^2*Defer[Int][1/((Sqrt[-a]*b - a*x
)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]), x] - a*b^2*Defer[Int][1/((Sqrt[-a]*b + a*x)^2*Sqrt[b + Sqrt[b^2 + a*x^2]]),
x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}}+\frac {4 b^4}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 b^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx \\ & = -\left (\left (4 b^2\right ) \int \frac {1}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )+\left (4 b^4\right ) \int \frac {1}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = -\left (\left (4 b^2\right ) \int \left (\frac {1}{2 b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {1}{2 b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx\right )+\left (4 b^4\right ) \int \left (-\frac {a}{4 b^2 \left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {a}{4 b^2 \left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {a}{2 b^2 \left (-a b^2-a^2 x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (2 a b^2\right ) \int \frac {1}{\left (-a b^2-a^2 x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (2 a b^2\right ) \int \left (-\frac {1}{2 a b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {1}{2 a b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ & = b \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+b \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-(2 b) \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-(2 b) \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b-a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx-\left (a b^2\right ) \int \frac {1}{\left (\sqrt {-a} b+a x\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.45 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-b^2+a x^2\right )^2}{\left (b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}} \, dx=\frac {x \left (4 b^2+2 a x^2-b \sqrt {b^2+a x^2}\right )}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

(x*(4*b^2 + 2*a*x^2 - b*Sqrt[b^2 + a*x^2]))/((b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (Sqrt[b]*ArcTan[(Sqr
t[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a] - (Sqrt[2]*Sqrt[b]*ArcTan[(Sqrt[a]*x)/(Sqrt[2]*Sqrt[b]
*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqrt[a]

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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