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

Optimal. Leaf size=30 \[ \frac{2 x^{3-n}}{3 b (2-n) \sqrt{b x^n}} \]

[Out]

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

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Rubi [A]  time = 0.0078502, antiderivative size = 30, 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} \[ \frac{2 x^{3-n}}{3 b (2-n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(b*x^n)^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0076018, size = 22, normalized size = 0.73 \[ \frac{x^3}{\left (3-\frac{3 n}{2}\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(b*x^n)^(3/2),x]

[Out]

x^3/((3 - (3*n)/2)*(b*x^n)^(3/2))

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Maple [A]  time = 0.002, size = 18, normalized size = 0.6 \begin{align*} -{\frac{2\,{x}^{3}}{-6+3\,n} \left ( b{x}^{n} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/(b*x^n)^(3/2), x)