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

Optimal. Leaf size=26 \[ \frac{2 \sqrt{\sqrt{a+b^2 x^2}+b x}}{b} \]

[Out]

(2*Sqrt[b*x + Sqrt[a + b^2*x^2]])/b

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Rubi [A]  time = 0.160506, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{2 \sqrt{\sqrt{a+b^2 x^2}+b x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b*x + Sqrt[a + b^2*x^2]])/b

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Rubi in Sympy [A]  time = 4.68085, size = 20, normalized size = 0.77 \[ \frac{2 \sqrt{b x + \sqrt{a + b^{2} x^{2}}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*x + sqrt(a + b**2*x**2))/b

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Mathematica [A]  time = 0.0409988, size = 26, normalized size = 1. \[ \frac{2 \sqrt{\sqrt{a+b^2 x^2}+b x}}{b} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[b*x + Sqrt[a + b^2*x^2]])/b

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{1\sqrt{bx+\sqrt{{b}^{2}{x}^{2}+a}}{\frac{1}{\sqrt{{b}^{2}{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222811, size = 30, normalized size = 1.15 \[ \frac{2 \, \sqrt{b x + \sqrt{b^{2} x^{2} + a}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*x + sqrt(b^2*x^2 + a))/b

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Sympy [A]  time = 1.34415, size = 27, normalized size = 1.04 \[ \begin{cases} \frac{2 \sqrt{b x + \sqrt{a + b^{2} x^{2}}}}{b} & \text{for}\: b \neq 0 \\\frac{x}{\sqrt [4]{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((2*sqrt(b*x + sqrt(a + b**2*x**2))/b, Ne(b, 0)), (x/a**(1/4), True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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