Optimal. Leaf size=42 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac{2 a \sqrt{a+b x^n}}{b^2 n} \]
[Out]
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Rubi [A] time = 0.0630181, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 \left (a+b x^n\right )^{3/2}}{3 b^2 n}-\frac{2 a \sqrt{a+b x^n}}{b^2 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 2*n)/Sqrt[a + b*x^n],x]
[Out]
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Rubi in Sympy [A] time = 8.21604, size = 36, normalized size = 0.86 \[ - \frac{2 a \sqrt{a + b x^{n}}}{b^{2} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+2*n)/(a+b*x**n)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0366854, size = 30, normalized size = 0.71 \[ \frac{2 \left (b x^n-2 a\right ) \sqrt{a+b x^n}}{3 b^2 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 2*n)/Sqrt[a + b*x^n],x]
[Out]
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Maple [A] time = 0.031, size = 28, normalized size = 0.7 \[ -{\frac{-2\,b{x}^{n}+4\,a}{3\,{b}^{2}n}\sqrt{a+b{x}^{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+2*n)/(a+b*x^n)^(1/2),x)
[Out]
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Maxima [A] time = 1.463, size = 53, normalized size = 1.26 \[ \frac{2 \,{\left (b^{2} x^{2 \, n} - a b x^{n} - 2 \, a^{2}\right )}}{3 \, \sqrt{b x^{n} + a} b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/sqrt(b*x^n + a),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232123, size = 35, normalized size = 0.83 \[ \frac{2 \, \sqrt{b x^{n} + a}{\left (b x^{n} - 2 \, a\right )}}{3 \, b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/sqrt(b*x^n + a),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(x**(-1+2*n)/(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2 \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(2*n - 1)/sqrt(b*x^n + a),x, algorithm="giac")
[Out]