3.156 \(\int \frac{x^m}{\left (a x^n\right )^{3/2}} \, dx\)

Optimal. Leaf size=32 \[ \frac{2 x^{m-n+1}}{a (2 m-3 n+2) \sqrt{a x^n}} \]

[Out]

(2*x^(1 + m - n))/(a*(2 + 2*m - 3*n)*Sqrt[a*x^n])

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Rubi [A]  time = 0.0241837, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 x^{m-n+1}}{a (2 m-3 n+2) \sqrt{a x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/(a*x^n)^(3/2),x]

[Out]

(2*x^(1 + m - n))/(a*(2 + 2*m - 3*n)*Sqrt[a*x^n])

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Rubi in Sympy [A]  time = 4.08501, size = 36, normalized size = 1.12 \[ \frac{2 x^{- \frac{n}{2}} x^{m - \frac{3 n}{2} + 1} \sqrt{a x^{n}}}{a^{2} \left (2 m - 3 n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(a*x**n)**(3/2),x)

[Out]

2*x**(-n/2)*x**(m - 3*n/2 + 1)*sqrt(a*x**n)/(a**2*(2*m - 3*n + 2))

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Mathematica [A]  time = 0.0175706, size = 25, normalized size = 0.78 \[ \frac{x^{m+1}}{\left (m-\frac{3 n}{2}+1\right ) \left (a x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/(a*x^n)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \[ 2\,{\frac{{x}^{1+m}}{ \left ( 2+2\,m-3\,n \right ) \left ( a{x}^{n} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(a*x^n)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.4603, size = 32, normalized size = 1. \[ \frac{2 \, x x^{m}}{a^{\frac{3}{2}}{\left (2 \, m - 3 \, n + 2\right )}{\left (x^{n}\right )}^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(a*x^n)^(3/2),x, algorithm="maxima")

[Out]

2*x*x^m/(a^(3/2)*(2*m - 3*n + 2)*(x^n)^(3/2))

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Fricas [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^m/(a*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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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**m/(a*x**n)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\left (a x^{n}\right )^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/(a*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(x^m/(a*x^n)^(3/2), x)