\(\int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [2210]

Optimal result

Integrand size = 56, antiderivative size = 163 \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\left (-9-2 a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{24 b}+\frac {5}{12} x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}+\frac {3 \text {arctanh}\left (\sqrt {2} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{8 \sqrt {2} b} \]

[Out]

1/24*(-2*a*x^2-9)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/b+5/12*x*(-a/b^2+a^2*x^2/b^2)^(1/2)*(a*x^2+b*x*
(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)+3/16*arctanh(2^(1/2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2))*2^(1/2)/b

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 7.53 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.98 \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=-\frac {\sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \left (2 \sqrt {a} x \left (9+2 a x^2-10 b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )+9 \sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\frac {\sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}}{\sqrt {2} \sqrt {a} x}\right )\right )}{48 \sqrt {a} b x} \]

[In]

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

[Out]

-1/48*(Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*(2*Sqrt[a]*x*(9 + 2*a*x^2 - 10*b*x*Sqrt[(a*(-1 + a*x^2))/b
^2]) + 9*Sqrt[2]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*ArcTan[Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2
))/b^2])]/(Sqrt[2]*Sqrt[a]*x)]))/(Sqrt[a]*b*x)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 11.70 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.03 \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=-\frac {4 \, {\left (2 \, a x^{2} - 10 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} + 9\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} - 9 \, \sqrt {2} \log \left (4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (2 \, \sqrt {2} b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - \sqrt {2} {\left (2 \, a x^{2} - 1\right )}\right )} - 1\right )}{96 \, b} \]

[In]

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

[Out]

-1/96*(4*(2*a*x^2 - 10*b*x*sqrt((a^2*x^2 - a)/b^2) + 9)*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2)) - 9*sqrt(2)*
log(4*a*x^2 - 4*b*x*sqrt((a^2*x^2 - a)/b^2) - 2*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))*(2*sqrt(2)*b*x*sqrt(
(a^2*x^2 - a)/b^2) - sqrt(2)*(2*a*x^2 - 1)) - 1))/b

Sympy [F]

\[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\int \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )} \sqrt {\frac {a \left (a x^{2} - 1\right )}{b^{2}}}\, dx \]

[In]

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

[Out]

Integral(sqrt(x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2)))*sqrt(a*(a*x**2 - 1)/b**2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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