\(\int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx\) [2829]

Optimal result

Integrand size = 40, antiderivative size = 287 \[ \int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {\sqrt {a} x \sqrt {b+a^2 x^4} \left (104 a b^2 x^2+264 a^3 b x^6+192 a^5 x^{10}\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}+\sqrt {a} x \left (39 b^3+212 a^2 b^2 x^4+360 a^4 b x^8+192 a^6 x^{12}\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{1152 a^{7/2} b x^2+1536 a^{11/2} x^6+384 a^{5/2} b \sqrt {b+a^2 x^4}+1536 a^{9/2} x^4 \sqrt {b+a^2 x^4}}-\frac {13 b^2 \log \left (i a x^2+i \sqrt {b+a^2 x^4}+i \sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{128 \sqrt {2} a^{5/2}} \]

[Out]

(a^(1/2)*x*(a^2*x^4+b)^(1/2)*(192*a^5*x^10+264*a^3*b*x^6+104*a*b^2*x^2)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)+a^(1/2
)*x*(192*a^6*x^12+360*a^4*b*x^8+212*a^2*b^2*x^4+39*b^3)*(a*x^2+(a^2*x^4+b)^(1/2))^(1/2))/(1152*a^(7/2)*b*x^2+1
536*a^(11/2)*x^6+384*a^(5/2)*b*(a^2*x^4+b)^(1/2)+1536*a^(9/2)*x^4*(a^2*x^4+b)^(1/2))-13/256*b^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)/a^(5/2)

Rubi [F]

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

[In]

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

[Out]

Defer[Int][x^4*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^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.90 \[ \int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\frac {\frac {2 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}} \left (39 b^3+192 a^5 x^{10} \left (a x^2+\sqrt {b+a^2 x^4}\right )+24 a^3 b x^6 \left (15 a x^2+11 \sqrt {b+a^2 x^4}\right )+4 a b^2 x^2 \left (53 a x^2+26 \sqrt {b+a^2 x^4}\right )\right )}{3 a b x^2+4 a^3 x^6+b \sqrt {b+a^2 x^4}+4 a^2 x^4 \sqrt {b+a^2 x^4}}-39 \sqrt {2} b^2 \log \left (i \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )\right )}{768 a^{5/2}} \]

[In]

Integrate[x^4*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]]*(39*b^3 + 192*a^5*x^10*(a*x^2 + Sqrt[b + a^2*x^4]) + 24*a^3*b*x^
6*(15*a*x^2 + 11*Sqrt[b + a^2*x^4]) + 4*a*b^2*x^2*(53*a*x^2 + 26*Sqrt[b + a^2*x^4])))/(3*a*b*x^2 + 4*a^3*x^6 +
b*Sqrt[b + a^2*x^4] + 4*a^2*x^4*Sqrt[b + a^2*x^4]) - 39*Sqrt[2]*b^2*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]])])/(768*a^(5/2))

Maple [F]

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

[In]

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

[Out]

int(x^4*(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 = 1.46 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.99 \[ \int x^4 \sqrt {b+a^2 x^4} \sqrt {a x^2+\sqrt {b+a^2 x^4}} \, dx=\left [\frac {39 \, \sqrt {2} \sqrt {a} b^{2} \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) - 4 \, {\left (8 \, a^{4} x^{7} + 13 \, a^{2} b x^{3} - {\left (56 \, a^{3} x^{5} + 39 \, a b x\right )} \sqrt {a^{2} x^{4} + b}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{1536 \, a^{3}}, \frac {39 \, \sqrt {2} \sqrt {-a} b^{2} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (8 \, a^{4} x^{7} + 13 \, a^{2} b x^{3} - {\left (56 \, a^{3} x^{5} + 39 \, a b x\right )} \sqrt {a^{2} x^{4} + b}\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{768 \, a^{3}}\right ] \]

[In]

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

[Out]

[1/1536*(39*sqrt(2)*sqrt(a)*b^2*log(4*a^2*x^4 + 4*sqrt(a^2*x^4 + b)*a*x^2 - 2*(sqrt(2)*a^(3/2)*x^3 + sqrt(2)*s
qrt(a^2*x^4 + b)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b) - 4*(8*a^4*x^7 + 13*a^2*b*x^3 - (56*a^3*x^5 +
39*a*b*x)*sqrt(a^2*x^4 + b))*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a^3, 1/768*(39*sqrt(2)*sqrt(-a)*b^2*arctan(1/2*
sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*sqrt(-a)/(a*x)) - 2*(8*a^4*x^7 + 13*a^2*b*x^3 - (56*a^3*x^5 + 39*a*b*x
)*sqrt(a^2*x^4 + b))*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a^3]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate(x^4*(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^4, x)

Giac [F]

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

[In]

integrate(x^4*(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^4, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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