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

Optimal. Leaf size=32 \[ \frac{2 x^{m-n+1}}{b (2 m-3 n+2) \sqrt{b x^n}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0061174, antiderivative size = 32, 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} \[ \frac{2 x^{m-n+1}}{b (2 m-3 n+2) \sqrt{b x^n}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0057825, size = 25, normalized size = 0.78 \[ \frac{x^{m+1}}{\left (m-\frac{3 n}{2}+1\right ) \left (b x^n\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \begin{align*} 2\,{\frac{{x}^{1+m}}{ \left ( 2+2\,m-3\,n \right ) \left ( b{x}^{n} \right ) ^{3/2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.01226, size = 32, normalized size = 1. \begin{align*} \frac{2 \, x x^{m}}{b^{\frac{3}{2}}{\left (2 \, m - 3 \, n + 2\right )}{\left (x^{n}\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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^m/(b*x^n)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

integrate(x^m/(b*x^n)^(3/2), x)