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

Optimal. Leaf size=66 \[ \frac{2 a x}{\sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}}+\frac{2 b^2 c x^3}{3 \left (b \sqrt{\frac{a^2}{b^2}+c x^2}+a\right )^{3/2}} \]

[Out]

(2*b^2*c*x^3)/(3*(a + b*Sqrt[a^2/b^2 + c*x^2])^(3/2)) + (2*a*x)/Sqrt[a + b*Sqrt[
a^2/b^2 + c*x^2]]

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Rubi [A]  time = 0.0755461, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{2 a x}{\sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}}+\frac{2 b^2 c x^3}{3 \left (b \sqrt{\frac{a^2}{b^2}+c x^2}+a\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*b^2*c*x^3)/(3*(a + b*Sqrt[a^2/b^2 + c*x^2])^(3/2)) + (2*a*x)/Sqrt[a + b*Sqrt[
a^2/b^2 + c*x^2]]

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Rubi in Sympy [A]  time = 2.37248, size = 60, normalized size = 0.91 \[ \frac{2 a x}{\sqrt{a + b \sqrt{\frac{a^{2}}{b^{2}} + c x^{2}}}} + \frac{2 b^{2} c x^{3}}{3 \left (a + b \sqrt{\frac{a^{2}}{b^{2}} + c x^{2}}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*x/sqrt(a + b*sqrt(a**2/b**2 + c*x**2)) + 2*b**2*c*x**3/(3*(a + b*sqrt(a**2/b
**2 + c*x**2))**(3/2))

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Mathematica [A]  time = 0.306316, size = 55, normalized size = 0.83 \[ \frac{2 b x \sqrt{\frac{a^2}{b^2}+c x^2}+4 a x}{3 \sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}} \]

Antiderivative was successfully verified.

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

[Out]

(4*a*x + 2*b*x*Sqrt[a^2/b^2 + c*x^2])/(3*Sqrt[a + b*Sqrt[a^2/b^2 + c*x^2]])

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int \sqrt{a+b\sqrt{{\frac{{a}^{2}}{{b}^{2}}}+c{x}^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.352361, size = 95, normalized size = 1.44 \[ \frac{2 \,{\left (b^{2} c x^{2} + a b \sqrt{\frac{b^{2} c x^{2} + a^{2}}{b^{2}}} - a^{2}\right )} \sqrt{b \sqrt{\frac{b^{2} c x^{2} + a^{2}}{b^{2}}} + a}}{3 \, b^{2} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(b^2*c*x^2 + a*b*sqrt((b^2*c*x^2 + a^2)/b^2) - a^2)*sqrt(b*sqrt((b^2*c*x^2 +
a^2)/b^2) + a)/(b^2*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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