\(\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx\) [1525]

Optimal result

Integrand size = 27, antiderivative size = 105 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=-\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x}+\frac {\sqrt {a} \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2}} \]

[Out]

-(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x+1/2*a^(1/2)*ln(I*a*x^2+I*(a^2*x^4+b)^(1/2)+I*2^(1/2)*a^(1/2)*x*(a*x^2+(a^2*
x^4+b)^(1/2))^(1/2))*2^(1/2)

Rubi [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=\int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx \]

[In]

Int[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x^2,x]

[Out]

Defer[Int][Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x^2, x]

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=-\frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x}+\frac {\sqrt {a} \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )\right )}{\sqrt {2}} \]

[In]

Integrate[Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x^2,x]

[Out]

-(Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]/x) + (Sqrt[a]*Log[I*(a*x^2 + Sqrt[b + a^2*x^4] + Sqrt[2]*Sqrt[a]*x*Sqrt[a*x^
2 + Sqrt[b + a^2*x^4]])])/Sqrt[2]

Maple [F]

\[\int \frac {\sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{x^{2}}d x\]

[In]

int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^2,x)

[Out]

int((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^2,x)

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=\text {Timed out} \]

[In]

integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{x^{2}}\, dx \]

[In]

integrate((a*x**2+(a**2*x**4+b)**(1/2))**(1/2)/x**2,x)

[Out]

Integral(sqrt(a*x**2 + sqrt(a**2*x**4 + b))/x**2, x)

Maxima [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{x^{2}} \,d x } \]

[In]

integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^2,x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/x^2, x)

Giac [F]

\[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=\int { \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{x^{2}} \,d x } \]

[In]

integrate((a*x^2+(a^2*x^4+b)^(1/2))^(1/2)/x^2,x, algorithm="giac")

[Out]

integrate(sqrt(a*x^2 + sqrt(a^2*x^4 + b))/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x^2+\sqrt {b+a^2 x^4}}}{x^2} \, dx=\int \frac {\sqrt {\sqrt {a^2\,x^4+b}+a\,x^2}}{x^2} \,d x \]

[In]

int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)/x^2,x)

[Out]

int(((b + a^2*x^4)^(1/2) + a*x^2)^(1/2)/x^2, x)