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

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 = 0.0068027, 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, Rules used = {15, 30} \[ -\frac{2 b x^{n-1} \sqrt{b x^n}}{2-3 n} \]

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)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (b x^n\right )^{3/2}}{x^2} \, dx &=\left (b x^{-n/2} \sqrt{b x^n}\right ) \int x^{-2+\frac{3 n}{2}} \, dx\\ &=-\frac{2 b x^{-1+n} \sqrt{b x^n}}{2-3 n}\\ \end{align*}

Mathematica [A]  time = 0.0064998, 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 = 0.001, size = 20, normalized size = 0.8 \begin{align*} 2\,{\frac{ \left ( b{x}^{n} \right ) ^{3/2}}{x \left ( -2+3\,n \right ) }} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}

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)