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

Optimal. Leaf size=28 \[ \frac{2 x^{2-n}}{b (4-3 n) \sqrt{b x^n}} \]

[Out]

(2*x^(2 - n))/(b*(4 - 3*n)*Sqrt[b*x^n])

________________________________________________________________________________________

Rubi [A]  time = 0.007607, antiderivative size = 28, 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, Rules used = {15, 30} \[ \frac{2 x^{2-n}}{b (4-3 n) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x^(2 - n))/(b*(4 - 3*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}{\left (b x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int x^{1-\frac{3 n}{2}} \, dx}{b \sqrt{b x^n}}\\ &=\frac{2 x^{2-n}}{b (4-3 n) \sqrt{b x^n}}\\ \end{align*}

Mathematica [A]  time = 0.0059329, size = 22, normalized size = 0.79 \[ \frac{x^2}{\left (2-\frac{3 n}{2}\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^2/((2 - (3*n)/2)*(b*x^n)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 20, normalized size = 0.7 \begin{align*} -2\,{\frac{{x}^{2}}{ \left ( 3\,n-4 \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^2/(3*n-4)/(b*x^n)^(3/2)

________________________________________________________________________________________

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/(b*x^n)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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/(b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\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/(b*x^n)^(3/2),x, algorithm="giac")

[Out]

integrate(x/(b*x^n)^(3/2), x)