Optimal. Leaf size=44 \[ \frac{2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]
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Rubi [A] time = 0.0392088, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{2 x^{3 (1-n)} \left (a x^{-2 (1-n)}+b x^{3 n-2}\right )^{3/2}}{3 b n} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x^(2*(-1 + n))*(a + b*x^n)],x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{2 n - 2} \left (a + b x^{n}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**(-2+2*n)*(a+b*x**n))**(1/2),x)
[Out]
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Mathematica [A] time = 0.0608137, size = 36, normalized size = 0.82 \[ \frac{2 x^{3-3 n} \left (x^{2 n-2} \left (a+b x^n\right )\right )^{3/2}}{3 b n} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x^(2*(-1 + n))*(a + b*x^n)],x]
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Maple [A] time = 0.051, size = 40, normalized size = 0.9 \[{\frac{ \left ( 2\,a+2\,b{x}^{n} \right ) x}{3\,b{x}^{n}n}\sqrt{{\frac{ \left ({x}^{n} \right ) ^{2} \left ( a+b{x}^{n} \right ) }{{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^(-2+2*n)*(a+b*x^n))^(1/2),x)
[Out]
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Maxima [A] time = 1.40278, size = 23, normalized size = 0.52 \[ \frac{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}{3 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^n + a)*x^(2*n - 2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.24123, size = 59, normalized size = 1.34 \[ \frac{2 \,{\left (b x x^{n} + a x\right )} \sqrt{\frac{b x^{3 \, n} + a x^{2 \, n}}{x^{2}}}}{3 \, b n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^n + a)*x^(2*n - 2)),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**(-2+2*n)*(a+b*x**n))**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{{\left (b x^{n} + a\right )} x^{2 \, n - 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^n + a)*x^(2*n - 2)),x, algorithm="giac")
[Out]