3.282 \(\int \frac{1}{\sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}} \sqrt{a^2+b^2 x}} \, dx\)

Optimal. Leaf size=75 \[ -\frac{2 \sqrt{a^2-b^2 x} \tan ^{-1}\left (\frac{\sqrt{a^2-b^2 x}}{\sqrt{a^2+b^2 x}}\right )}{b^2 \sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.146345, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073 \[ -\frac{2 \sqrt{a^2-b^2 x} \tan ^{-1}\left (\frac{\sqrt{a^2-b^2 x}}{\sqrt{a^2+b^2 x}}\right )}{b^2 \sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]]*Sqrt[a^2 + b^2*x]),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.2102, size = 65, normalized size = 0.87 \[ \frac{2 \sqrt{a - b \sqrt{x}} \sqrt{a + b \sqrt{x}} \operatorname{atan}{\left (\frac{\sqrt{a^{2} + b^{2} x}}{\sqrt{a^{2} - b^{2} x}} \right )}}{b^{2} \sqrt{a^{2} - b^{2} x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b**2*x+a**2)**(1/2)/(a-b*x**(1/2))**(1/2)/(a+b*x**(1/2))**(1/2),x)

[Out]

2*sqrt(a - b*sqrt(x))*sqrt(a + b*sqrt(x))*atan(sqrt(a**2 + b**2*x)/sqrt(a**2 - b
**2*x))/(b**2*sqrt(a**2 - b**2*x))

_______________________________________________________________________________________

Mathematica [A]  time = 1.96338, size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a-b \sqrt{x}} \sqrt{a+b \sqrt{x}} \sqrt{a^2+b^2 x}} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[1/(Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]]*Sqrt[a^2 + b^2*x]),x]

[Out]

Integrate[1/(Sqrt[a - b*Sqrt[x]]*Sqrt[a + b*Sqrt[x]]*Sqrt[a^2 + b^2*x]), x]

_______________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt{{b}^{2}x+{a}^{2}}}}{\frac{1}{\sqrt{a-b\sqrt{x}}}}{\frac{1}{\sqrt{a+b\sqrt{x}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b^{2} x + a^{2}} \sqrt{b \sqrt{x} + a} \sqrt{-b \sqrt{x} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.242236, size = 68, normalized size = 0.91 \[ -\frac{2 \, \arctan \left (-\frac{a^{2} - \sqrt{b^{2} x + a^{2}} \sqrt{b \sqrt{x} + a} \sqrt{-b \sqrt{x} + a}}{b^{2} x}\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a - b \sqrt{x}} \sqrt{a + b \sqrt{x}} \sqrt{a^{2} + b^{2} x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{b^{2} x + a^{2}} \sqrt{b \sqrt{x} + a} \sqrt{-b \sqrt{x} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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