\(\int \frac {(-b+a^2 x^4) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx\) [2094]

Optimal result

Integrand size = 48, antiderivative size = 151 \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {3 \sqrt {a} b x^2+2 a^{5/2} x^6+2 a^{3/2} x^4 \sqrt {b+a^2 x^4}}{8 \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \]

[Out]

1/8*(3*a^(1/2)*b*x^2+2*a^(5/2)*x^6+2*a^(3/2)*x^4*(a^2*x^4+b)^(1/2))/a^(1/2)/x/(a*x^2+(a^2*x^4+b)^(1/2))^(1/2)-
11/16*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]

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

[In]

Int[((-b + a^2*x^4)*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^4*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^4 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}}\right ) \, dx \\ & = a^2 \int \frac {x^4 \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^4 \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^4 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.85 \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\frac {3 b x+2 a x^3 \left (a x^2+\sqrt {b+a^2 x^4}\right )}{8 \sqrt {a x^2+\sqrt {b+a^2 x^4}}}-\frac {11 b \log \left (a x^2+\sqrt {b+a^2 x^4}+\sqrt {2} \sqrt {a} x \sqrt {a x^2+\sqrt {b+a^2 x^4}}\right )}{8 \sqrt {2} \sqrt {a}} \]

[In]

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

[Out]

(3*b*x + 2*a*x^3*(a*x^2 + Sqrt[b + a^2*x^4]))/(8*Sqrt[a*x^2 + Sqrt[b + a^2*x^4]]) - (11*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]]])/(8*Sqrt[2]*Sqrt[a])

Maple [F]

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

[In]

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

Fricas [A] (verification not implemented)

none

Time = 1.49 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.56 \[ \int \frac {\left (-b+a^2 x^4\right ) \sqrt {a x^2+\sqrt {b+a^2 x^4}}}{\sqrt {b+a^2 x^4}} \, dx=\left [\frac {11 \, \sqrt {2} \sqrt {a} b \log \left (4 \, a^{2} x^{4} + 4 \, \sqrt {a^{2} x^{4} + b} a x^{2} - 2 \, {\left (\sqrt {2} a^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {a^{2} x^{4} + b} \sqrt {a} x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} + b\right ) - 4 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{32 \, a}, \frac {11 \, \sqrt {2} \sqrt {-a} b \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}} \sqrt {-a}}{2 \, a x}\right ) - 2 \, {\left (a^{2} x^{3} - 3 \, \sqrt {a^{2} x^{4} + b} a x\right )} \sqrt {a x^{2} + \sqrt {a^{2} x^{4} + b}}}{16 \, a}\right ] \]

[In]

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

[Out]

[1/32*(11*sqrt(2)*sqrt(a)*b*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)*sqrt(
a^2*x^4 + b)*sqrt(a)*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)) + b) - 4*(a^2*x^3 - 3*sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^
2 + sqrt(a^2*x^4 + b)))/a, 1/16*(11*sqrt(2)*sqrt(-a)*b*arctan(1/2*sqrt(2)*sqrt(a*x^2 + sqrt(a^2*x^4 + b))*sqrt
(-a)/(a*x)) - 2*(a^2*x^3 - 3*sqrt(a^2*x^4 + b)*a*x)*sqrt(a*x^2 + sqrt(a^2*x^4 + b)))/a]

Sympy [F]

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

[In]

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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

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

[In]

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