3.800 \(\int \frac{x^{-1+m} \left (2 a m+b (2 m-n) x^n\right )}{2 \left (a+b x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=15 \[ \frac{x^m}{\sqrt{a+b x^n}} \]

[Out]

x^m/Sqrt[a + b*x^n]

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Rubi [A]  time = 0.0580868, 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]

[Out]

x^m/Sqrt[a + b*x^n]

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Rubi in Sympy [A]  time = 3.8424, 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)

[Out]

x**m/sqrt(a + b*x**n)

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Mathematica [A]  time = 0.0861637, 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]

x^m/Sqrt[a + b*x^n]

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Maple [F]  time = 0.069, 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)

[Out]

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 = 0.834525, 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")

[Out]

x^m/sqrt(b*x^n + a)

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Fricas [A]  time = 0.29713, 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]

x*x^(m - 1)/sqrt(b*x^n + a)

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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]

Timed 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]

integrate(1/2*(b*(2*m - n)*x^n + 2*a*m)*x^(m - 1)/(b*x^n + a)^(3/2), x)