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

Optimal. Leaf size=21 \[ \frac{2 x^2}{(4-n) \sqrt{b x^n}} \]

[Out]

(2*x^2)/((4 - n)*Sqrt[b*x^n])

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Rubi [A]  time = 0.0134409, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 x^2}{(4-n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[b*x^n],x]

[Out]

(2*x^2)/((4 - n)*Sqrt[b*x^n])

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Rubi in Sympy [A]  time = 3.41738, size = 26, normalized size = 1.24 \[ \frac{2 x^{- \frac{n}{2}} x^{- \frac{n}{2} + 2} \sqrt{b x^{n}}}{b \left (- n + 4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*x**(-n/2)*x**(-n/2 + 2)*sqrt(b*x**n)/(b*(-n + 4))

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Mathematica [A]  time = 0.00654173, size = 19, normalized size = 0.9 \[ -\frac{2 x^2}{(n-4) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[b*x^n],x]

[Out]

(-2*x^2)/((-4 + n)*Sqrt[b*x^n])

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Maple [A]  time = 0.002, size = 18, normalized size = 0.9 \[ -2\,{\frac{{x}^{2}}{ \left ( n-4 \right ) \sqrt{b{x}^{n}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^n)^(1/2),x)

[Out]

-2*x^2/(n-4)/(b*x^n)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(b*x^n), x)