Optimal. Leaf size=38 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2} \]
[Out]
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Rubi [A] time = 0.0578148, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2 a \left (a+\frac{b}{x}\right )^{3/2}}{3 b^2}-\frac{2 \left (a+\frac{b}{x}\right )^{5/2}}{5 b^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 6.78929, size = 31, normalized size = 0.82 \[ \frac{2 a \left (a + \frac{b}{x}\right )^{\frac{3}{2}}}{3 b^{2}} - \frac{2 \left (a + \frac{b}{x}\right )^{\frac{5}{2}}}{5 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0249001, size = 40, normalized size = 1.05 \[ \frac{2 \sqrt{a+\frac{b}{x}} \left (2 a^2 x^2-a b x-3 b^2\right )}{15 b^2 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x]/x^3,x]
[Out]
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Maple [A] time = 0.008, size = 33, normalized size = 0.9 \[{\frac{ \left ( 2\,ax+2\,b \right ) \left ( 2\,ax-3\,b \right ) }{15\,{b}^{2}{x}^{2}}\sqrt{{\frac{ax+b}{x}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(1/2)/x^3,x)
[Out]
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Maxima [A] time = 1.43872, size = 41, normalized size = 1.08 \[ -\frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}}{5 \, b^{2}} + \frac{2 \,{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}} a}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227165, size = 51, normalized size = 1.34 \[ \frac{2 \,{\left (2 \, a^{2} x^{2} - a b x - 3 \, b^{2}\right )} \sqrt{\frac{a x + b}{x}}}{15 \, b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.30176, size = 304, normalized size = 8. \[ \frac{4 a^{\frac{11}{2}} b^{\frac{3}{2}} x^{3} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} + \frac{2 a^{\frac{9}{2}} b^{\frac{5}{2}} x^{2} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{8 a^{\frac{7}{2}} b^{\frac{7}{2}} x \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{6 a^{\frac{5}{2}} b^{\frac{9}{2}} \sqrt{\frac{a x}{b} + 1}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{6} b x^{\frac{7}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} - \frac{4 a^{5} b^{2} x^{\frac{5}{2}}}{15 a^{\frac{7}{2}} b^{3} x^{\frac{7}{2}} + 15 a^{\frac{5}{2}} b^{4} x^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.262161, size = 155, normalized size = 4.08 \[ \frac{2 \,{\left (15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{3} a^{\frac{3}{2}}{\rm sign}\left (x\right ) + 25 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{2} a b{\rm sign}\left (x\right ) + 15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} b^{2}{\rm sign}\left (x\right ) + 3 \, b^{3}{\rm sign}\left (x\right )\right )}}{15 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/x^3,x, algorithm="giac")
[Out]