\(\int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\) [2571]

Optimal result

Integrand size = 40, antiderivative size = 219 \[ \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {\sqrt {a} x \sqrt {b+a^2 x^4} \left (28 a b x^2+16 a^3 x^6\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {a} x \left (9 b^2+36 a^2 b x^4+16 a^4 x^8\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{48 a^{3/2} b+96 a^{7/2} x^4+96 a^{5/2} x^2 \sqrt {b+a^2 x^4}}-\frac {3 b^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{16 \sqrt {2} a^{3/2}} \]

[Out]

(a^(1/2)*x*(a^2*x^4+b)^(1/2)*(16*a^3*x^6+28*a*b*x^2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)+a^(1/2)*x*(16*a^4*x^8+36*
a^2*b*x^4+9*b^2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2))/(48*a^(3/2)*b+96*a^(7/2)*x^4+96*a^(5/2)*x^2*(a^2*x^4+b)^(1/2
))-3/32*b^(3/2)*arctan(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)/a^(3/2)

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.84 \[ \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {\frac {2 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}} \left (9 b^2+16 a^3 x^6 \left (a x^2+\sqrt {b+a^2 x^4}\right )+4 a b x^2 \left (9 a x^2+7 \sqrt {b+a^2 x^4}\right )\right )}{b+2 a x^2 \left (a x^2+\sqrt {b+a^2 x^4}\right )}-9 \sqrt {2} b^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b}}\right )}{96 a^{3/2}} \]

[In]

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

[Out]

((2*Sqrt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]*(9*b^2 + 16*a^3*x^6*(a*x^2 + Sqrt[b + a^2*x^4]) + 4*a*b*x^2*(9*a
*x^2 + 7*Sqrt[b + a^2*x^4])))/(b + 2*a*x^2*(a*x^2 + Sqrt[b + a^2*x^4])) - 9*Sqrt[2]*b^(3/2)*ArcTan[(Sqrt[2]*Sq
rt[a]*x*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]])/Sqrt[b]])/(96*a^(3/2))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 1.99 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.46 \[ \int x^2 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\left [\frac {9 \, \sqrt {\frac {1}{2}} b \sqrt {-\frac {b}{a}} \log \left (4 \, a^{2} b x^{4} - 4 \, \sqrt {a^{2} x^{4} + b} a b x^{2} + b^{2} - 4 \, {\left (2 \, \sqrt {\frac {1}{2}} \sqrt {a^{2} x^{4} + b} a^{2} x^{3} \sqrt {-\frac {b}{a}} - \sqrt {\frac {1}{2}} {\left (2 \, a^{3} x^{5} + a b x\right )} \sqrt {-\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}\right ) - 2 \, {\left (2 \, a^{2} x^{5} - 10 \, \sqrt {a^{2} x^{4} + b} a x^{3} - 9 \, b x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{96 \, a}, \frac {9 \, \sqrt {\frac {1}{2}} b \sqrt {\frac {b}{a}} \arctan \left (-\frac {{\left (\sqrt {\frac {1}{2}} a x^{2} \sqrt {\frac {b}{a}} - \sqrt {\frac {1}{2}} \sqrt {a^{2} x^{4} + b} \sqrt {\frac {b}{a}}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{b x}\right ) - {\left (2 \, a^{2} x^{5} - 10 \, \sqrt {a^{2} x^{4} + b} a x^{3} - 9 \, b x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{48 \, a}\right ] \]

[In]

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

[Out]

[1/96*(9*sqrt(1/2)*b*sqrt(-b/a)*log(4*a^2*b*x^4 - 4*sqrt(a^2*x^4 + b)*a*b*x^2 + b^2 - 4*(2*sqrt(1/2)*sqrt(a^2*
x^4 + b)*a^2*x^3*sqrt(-b/a) - sqrt(1/2)*(2*a^3*x^5 + a*b*x)*sqrt(-b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b))) - 2*(
2*a^2*x^5 - 10*sqrt(a^2*x^4 + b)*a*x^3 - 9*b*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a, 1/48*(9*sqrt(1/2)*b*sqrt(b
/a)*arctan(-(sqrt(1/2)*a*x^2*sqrt(b/a) - sqrt(1/2)*sqrt(a^2*x^4 + b)*sqrt(b/a))*sqrt(a*x^2 + sqrt(a^2*x^4 + b)
)/(b*x)) - (2*a^2*x^5 - 10*sqrt(a^2*x^4 + b)*a*x^3 - 9*b*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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