Optimal. Leaf size=51 \[ -\frac{\left (a+b x^n\right )^{3/2} \, _2F_1\left (1,\frac{3}{2}-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.0643137, antiderivative size = 60, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{a+b x^n} \, _2F_1\left (-\frac{1}{2},-\frac{2}{n};-\frac{2-n}{n};-\frac{b x^n}{a}\right )}{2 x^2 \sqrt{\frac{b x^n}{a}+1}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b*x^n]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 6.86546, size = 48, normalized size = 0.94 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{2}{n} \\ \frac{n - 2}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{2 x^{2} \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0689557, size = 76, normalized size = 1.49 \[ \frac{4 \left (a+b x^n\right )-a n \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{2}{n};\frac{n-2}{n};-\frac{b x^n}{a}\right )}{2 (n-4) x^2 \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b*x^n]/x^3,x]
[Out]
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Maple [F] time = 0.069, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt{a+b{x}^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^(1/2)/x^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x^3,x, algorithm="maxima")
[Out]
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b*x**n)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{b x^{n} + a}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)/x^3,x, algorithm="giac")
[Out]