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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]