\(\int \frac {(-b+a^2 x^2) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\) [2352]

Optimal result

Integrand size = 48, antiderivative size = 186 \[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {1}{2} a x \sqrt {a x^2+\sqrt {b+a^2 x^4}}-\frac {\sqrt {a} \sqrt {b} \arctan \left (\frac {a x^2}{\sqrt {b}}+\frac {\sqrt {b+a^2 x^4}}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{\sqrt {2}}-\frac {b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{\sqrt {2} \sqrt {a}} \]

[Out]

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

Rubi [F]

\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \]

[In]

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

[Out]

-((b*ArcTanh[(Sqrt[2]*Sqrt[a]*x)/Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(Sqrt[2]*Sqrt[a])) + a^2*Defer[Int][(x^2*Sq
rt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b + a^2*x^4], x]

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {a^{3/2} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {2} a \sqrt {b} \arctan \left (\frac {a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )+\sqrt {2} b \log \left (a x^2+\sqrt {b+a^2 x^4}-\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{2 \sqrt {a}} \]

[In]

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

[Out]

(a^(3/2)*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]] + Sqrt[2]*a*Sqrt[b]*ArcTan[(a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqr
t[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]] + Sqrt[2]*b*Log[a*x^2 + Sqrt[b + a^2*x^4] - Sqrt[2]*Sqrt[a]*x
*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]])/(2*Sqrt[a])

Maple [F]

\[\int \frac {\left (a^{2} x^{2}-b \right ) \sqrt {a \,x^{2}+\sqrt {a^{2} x^{4}+b}}}{\sqrt {a^{2} x^{4}+b}}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int \frac {\sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \left (a^{2} x^{2} - b\right )}{\sqrt {a^{2} x^{4} + b}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {{\left (a^{2} x^{2} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (-b+a^2 x^2\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\int { \frac {{\left (a^{2} x^{2} - b\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{\sqrt {a^{2} x^{4} + b}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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