Optimal. Leaf size=44 \[ \frac{2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}-\frac{2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]
[Out]
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Rubi [A] time = 0.0618802, antiderivative size = 44, 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 \[ \frac{2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}-\frac{2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]
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 = 8.30575, size = 37, normalized size = 0.84 \[ - \frac{2 a \left (a + b x^{n}\right )^{\frac{3}{2}}}{3 b^{2} n} + \frac{2 \left (a + b x^{n}\right )^{\frac{5}{2}}}{5 b^{2} n} \]
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 = 0.0364608, size = 43, normalized size = 0.98 \[ \frac{2 \sqrt{a+b x^n} \left (-2 a^2+a b x^n+3 b^2 x^{2 n}\right )}{15 b^2 n} \]
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 = 41, normalized size = 0.9 \[ -{\frac{-6\,{b}^{2} \left ({x}^{n} \right ) ^{2}-2\,a{x}^{n}b+4\,{a}^{2}}{15\,{b}^{2}n}\sqrt{a+b{x}^{n}}} \]
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 = 1.47734, size = 53, normalized size = 1.2 \[ \frac{2 \,{\left (3 \, b^{2} x^{2 \, n} + a b x^{n} - 2 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(2*n - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218373, size = 53, normalized size = 1.2 \[ \frac{2 \,{\left (3 \, b^{2} x^{2 \, n} + a b x^{n} - 2 \, a^{2}\right )} \sqrt{b x^{n} + a}}{15 \, b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(2*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+2*n)*(a+b*x**n)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{n} + a} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(b*x^n + a)*x^(2*n - 1),x, algorithm="giac")
[Out]