3.160 \(\int \frac{x^m}{\sqrt{b x^n}} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 x^{m+1}}{(2 m-n+2) \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0186377, antiderivative size = 26, 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 \[ \frac{2 x^{m+1}}{(2 m-n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x^m/Sqrt[b*x^n],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.92869, size = 31, normalized size = 1.19 \[ \frac{2 x^{- \frac{n}{2}} x^{m - \frac{n}{2} + 1} \sqrt{b x^{n}}}{b \left (2 m - n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**m/(b*x**n)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0123315, size = 25, normalized size = 0.96 \[ \frac{x^{m+1}}{\left (m-\frac{n}{2}+1\right ) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^m/Sqrt[b*x^n],x]

[Out]

x^(1 + m)/((1 + m - n/2)*Sqrt[b*x^n])

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Maple [A]  time = 0.003, size = 25, normalized size = 1. \[ 2\,{\frac{{x}^{1+m}}{ \left ( 2+2\,m-n \right ) \sqrt{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m/(b*x^n)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.46164, size = 32, normalized size = 1.23 \[ \frac{2 \, x x^{m}}{\sqrt{b}{\left (2 \, m - n + 2\right )} \sqrt{x^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^n),x, algorithm="maxima")

[Out]

2*x*x^m/(sqrt(b)*(2*m - n + 2)*sqrt(x^n))

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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/sqrt(b*x^n),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [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/(b*x**n)**(1/2),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{m}}{\sqrt{b x^{n}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^m/sqrt(b*x^n),x, algorithm="giac")

[Out]

integrate(x^m/sqrt(b*x^n), x)