Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^n}} \]
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Rubi [A] time = 0.0638875, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054 \[ \frac{x^m}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Int[(x^(-1 + m)*(2*a*m + b*(2*m - n)*x^n))/(2*(a + b*x^n)^(3/2)),x]
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Rubi in Sympy [A] time = 7.56898, size = 12, normalized size = 0.8 \[ \frac{x^{m}}{\sqrt{a + b x^{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/2*x**(-1+m)*(2*a*m+b*(2*m-n)*x**n)/(a+b*x**n)**(3/2),x)
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Mathematica [A] time = 0.0843373, size = 15, normalized size = 1. \[ \frac{x^m}{\sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(-1 + m)*(2*a*m + b*(2*m - n)*x^n))/(2*(a + b*x^n)^(3/2)),x]
[Out]
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Maple [F] time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{{x}^{-1+m} \left ( 2\,am+b \left ( 2\,m-n \right ){x}^{n} \right ) }{2} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/2*x^(-1+m)*(2*a*m+b*(2*m-n)*x^n)/(a+b*x^n)^(3/2),x)
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Maxima [A] time = 1.63231, size = 18, normalized size = 1.2 \[ \frac{x^{m}}{\sqrt{b x^{n} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/2*(b*(2*m - n)*x^n + 2*a*m)*x^(m - 1)/(b*x^n + a)^(3/2),x, algorithm="maxima")
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Fricas [A] time = 0.238836, size = 22, normalized size = 1.47 \[ \frac{x x^{m - 1}}{\sqrt{b x^{n} + a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/2*(b*(2*m - n)*x^n + 2*a*m)*x^(m - 1)/(b*x^n + a)^(3/2),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(1/2*x**(-1+m)*(2*a*m+b*(2*m-n)*x**n)/(a+b*x**n)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b{\left (2 \, m - n\right )} x^{n} + 2 \, a m\right )} x^{m - 1}}{2 \,{\left (b x^{n} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/2*(b*(2*m - n)*x^n + 2*a*m)*x^(m - 1)/(b*x^n + a)^(3/2),x, algorithm="giac")
[Out]