3.2984 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^2} \, dx\)

Optimal. Leaf size=54 \[ \frac{4 a \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c} \]

[Out]

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

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Rubi [A]  time = 0.0964886, antiderivative size = 54, 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 \[ \frac{4 a \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.8593, size = 42, normalized size = 0.78 \[ \frac{4 a \sqrt{a + b \sqrt{\frac{c}{x}}}}{b^{2} c} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{3 b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0764967, size = 42, normalized size = 0.78 \[ -\frac{4 \left (b \sqrt{\frac{c}{x}}-2 a\right ) \sqrt{a+b \sqrt{\frac{c}{x}}}}{3 b^2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.069, size = 266, normalized size = 4.9 \[ -{\frac{1}{3\,{b}^{3}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 3\,{a}^{3/2}\sqrt{{\frac{c}{x}}}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){x}^{2}b-3\,{a}^{3/2}\sqrt{{\frac{c}{x}}}\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){x}^{2}b+6\,{x}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{2}+6\,{x}^{3/2}\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{2}+4\,\sqrt{x}\sqrt{{\frac{c}{x}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}b-12\,\sqrt{x} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}a \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(a+b*(c/x)^(1/2))^(1/2)*(3*a^(3/2)*(c/x)^(1/2)*ln(1/2*(b*(c/x)^(1/2)*x^(1/2
)+2*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*x^2*b-3*a^(3/2)*(c
/x)^(1/2)*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*
a*x^(1/2))/a^(1/2))*x^2*b+6*x^(3/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^2+6*x^(3/2)*(x
*(a+b*(c/x)^(1/2)))^(1/2)*a^2+4*x^(1/2)*(c/x)^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*
b-12*x^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a)/x^(5/2)/(x*(a+b*(c/x)^(1/2)))^(1/2)/
b^3/(c/x)^(3/2)

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Maxima [A]  time = 1.34064, size = 57, normalized size = 1.06 \[ -\frac{4 \,{\left (\frac{{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}}}{b^{2}} - \frac{3 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a}{b^{2}}\right )}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.255511, size = 46, normalized size = 0.85 \[ -\frac{4 \, \sqrt{b \sqrt{\frac{c}{x}} + a}{\left (b \sqrt{\frac{c}{x}} - 2 \, a\right )}}{3 \, b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224926, size = 51, normalized size = 0.94 \[ -\frac{4 \,{\left ({\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a\right )}}{3 \, b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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