3.2978 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c} \]

[Out]

(4*a*(a + b*Sqrt[c/x])^(3/2))/(3*b^2*c) - (4*(a + b*Sqrt[c/x])^(5/2))/(5*b^2*c)

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Rubi [A]  time = 0.095529, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*a*(a + b*Sqrt[c/x])^(3/2))/(3*b^2*c) - (4*(a + b*Sqrt[c/x])^(5/2))/(5*b^2*c)

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Rubi in Sympy [A]  time = 9.95141, size = 44, normalized size = 0.79 \[ \frac{4 a \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{3 b^{2} c} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{5}{2}}}{5 b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a*(a + b*sqrt(c/x))**(3/2)/(3*b**2*c) - 4*(a + b*sqrt(c/x))**(5/2)/(5*b**2*c)

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Mathematica [A]  time = 0.0434579, size = 43, normalized size = 0.77 \[ \frac{4 \left (2 a-3 b \sqrt{\frac{c}{x}}\right ) \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{15 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

(4*(2*a - 3*b*Sqrt[c/x])*(a + b*Sqrt[c/x])^(3/2))/(15*b^2*c)

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Maple [A]  time = 0.03, size = 70, normalized size = 1.3 \[ -{\frac{4}{15\,cx{b}^{2}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 3\,b\sqrt{{\frac{c}{x}}}-2\,a \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*(a+b*(c/x)^(1/2))^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)/x/c*(3*b*(c/x)^(1/2)-2
*a)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^2

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Maxima [A]  time = 1.33199, size = 58, normalized size = 1.04 \[ -\frac{4 \,{\left (\frac{3 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*(3*(b*sqrt(c/x) + a)^(5/2)/b^2 - 5*(b*sqrt(c/x) + a)^(3/2)*a/b^2)/c

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Fricas [A]  time = 0.249477, size = 65, normalized size = 1.16 \[ -\frac{4 \,{\left (a b x \sqrt{\frac{c}{x}} + 3 \, b^{2} c - 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{15 \, b^{2} c x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/15*(a*b*x*sqrt(c/x) + 3*b^2*c - 2*a^2*x)*sqrt(b*sqrt(c/x) + a)/(b^2*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out