3.140 \(\int \frac{\left (b x^n\right )^{3/2}}{x^2} \, dx\)

Optimal. Leaf size=24 \[ -\frac{2 b x^{n-1} \sqrt{b x^n}}{2-3 n} \]

[Out]

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

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Rubi [A]  time = 0.0185558, antiderivative size = 24, 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 \[ -\frac{2 b x^{n-1} \sqrt{b x^n}}{2-3 n} \]

Antiderivative was successfully verified.

[In]  Int[(b*x^n)^(3/2)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.25365, size = 31, normalized size = 1.29 \[ - \frac{2 b x^{- \frac{n}{2}} x^{\frac{3 n}{2} - 1} \sqrt{b x^{n}}}{- 3 n + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0117847, size = 22, normalized size = 0.92 \[ \frac{\left (b x^n\right )^{3/2}}{\left (\frac{3 n}{2}-1\right ) x} \]

Antiderivative was successfully verified.

[In]  Integrate[(b*x^n)^(3/2)/x^2,x]

[Out]

(b*x^n)^(3/2)/((-1 + (3*n)/2)*x)

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Maple [A]  time = 0.002, size = 20, normalized size = 0.8 \[ 2\,{\frac{ \left ( b{x}^{n} \right ) ^{3/2}}{x \left ( -2+3\,n \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^n)^(3/2)/x^2,x)

[Out]

2/x/(-2+3*n)*(b*x^n)^(3/2)

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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((b*x^n)^(3/2)/x^2,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((b*x^n)^(3/2)/x^2,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((b*x**n)**(3/2)/x**2,x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n)^(3/2)/x^2, x)