Optimal. Leaf size=49 \[ -\frac{\, _2F_1\left (1,-\frac{1}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}} \]
[Out]
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Rubi [A] time = 0.0671232, antiderivative size = 61, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*(a + b*x^n)^(3/2)),x]
[Out]
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Rubi in Sympy [A] time = 7.11889, size = 46, normalized size = 0.94 \[ - \frac{\sqrt{a + b x^{n}}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - \frac{1}{n} \\ \frac{n - 1}{n} \end{matrix}\middle |{- \frac{b x^{n}}{a}} \right )}}{a^{2} x \sqrt{1 + \frac{b x^{n}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(a+b*x**n)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0935102, size = 67, normalized size = 1.37 \[ \frac{2-(n+2) \sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};-\frac{b x^n}{a}\right )}{a n x \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*(a + b*x^n)^(3/2)),x]
[Out]
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Maple [F] time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(a+b*x^n)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^(3/2)*x^2),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(1/((b*x^n + a)^(3/2)*x^2),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(1/x**2/(a+b*x**n)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^n + a)^(3/2)*x^2),x, algorithm="giac")
[Out]