Optimal. Leaf size=113 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{8 a^{5/2} n}+\frac{b^2 x^{-n} \sqrt{a+b x^n}}{8 a^2 n}-\frac{x^{-3 n} \sqrt{a+b x^n}}{3 n}-\frac{b x^{-2 n} \sqrt{a+b x^n}}{12 a n} \]
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Rubi [A] time = 0.14719, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ -\frac{b^3 \tanh ^{-1}\left (\frac{\sqrt{a+b x^n}}{\sqrt{a}}\right )}{8 a^{5/2} n}+\frac{b^2 x^{-n} \sqrt{a+b x^n}}{8 a^2 n}-\frac{x^{-3 n} \sqrt{a+b x^n}}{3 n}-\frac{b x^{-2 n} \sqrt{a+b x^n}}{12 a n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 3*n)*Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 16.599, size = 92, normalized size = 0.81 \[ - \frac{x^{- 3 n} \sqrt{a + b x^{n}}}{3 n} - \frac{b x^{- 2 n} \sqrt{a + b x^{n}}}{12 a n} + \frac{b^{2} x^{- n} \sqrt{a + b x^{n}}}{8 a^{2} n} - \frac{b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a + b x^{n}}}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-3*n)*(a+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.137018, size = 98, normalized size = 0.87 \[ \frac{-2 \sqrt{a} x^{-3 n} \sqrt{a+b x^n} \left (8 a^2+2 a b x^n-3 b^2 x^{2 n}\right )-3 b^3 \log \left (x^{-n} \left (2 \sqrt{a} \sqrt{a+b x^n}+2 a+b x^n\right )\right )}{48 a^{5/2} n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 3*n)*Sqrt[a + b*x^n],x]
[Out]
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Maple [F] time = 0.068, size = 0, normalized size = 0. \[ \int{x}^{-1-3\,n}\sqrt{a+b{x}^{n}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-3*n)*(a+b*x^n)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(-3*n - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237258, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, b^{3} x^{3 \, n} \log \left (\frac{\sqrt{a} b x^{n} - 2 \, \sqrt{b x^{n} + a} a + 2 \, a^{\frac{3}{2}}}{x^{n}}\right ) + 2 \,{\left (3 \, \sqrt{a} b^{2} x^{2 \, n} - 2 \, a^{\frac{3}{2}} b x^{n} - 8 \, a^{\frac{5}{2}}\right )} \sqrt{b x^{n} + a}}{48 \, a^{\frac{5}{2}} n x^{3 \, n}}, \frac{3 \, b^{3} x^{3 \, n} \arctan \left (\frac{a}{\sqrt{b x^{n} + a} \sqrt{-a}}\right ) +{\left (3 \, \sqrt{-a} b^{2} x^{2 \, n} - 2 \, \sqrt{-a} a b x^{n} - 8 \, \sqrt{-a} a^{2}\right )} \sqrt{b x^{n} + a}}{24 \, \sqrt{-a} a^{2} n x^{3 \, n}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(-3*n - 1),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1-3*n)*(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + a} x^{-3 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(-3*n - 1),x, algorithm="giac")
[Out]