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

Optimal. Leaf size=28 \[ \frac{2 b \sqrt{b x^n} x^{m+n+1}}{2 m+3 n+2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0077379, antiderivative size = 28, 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} \[ \frac{2 b \sqrt{b x^n} x^{m+n+1}}{2 m+3 n+2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0063787, size = 25, normalized size = 0.89 \[ \frac{x^{m+1} \left (b x^n\right )^{3/2}}{m+\frac{3 n}{2}+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 0.9 \begin{align*} 2\,{\frac{{x}^{1+m} \left ( b{x}^{n} \right ) ^{3/2}}{2+2\,m+3\,n}} \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 = 0.990538, size = 32, normalized size = 1.14 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{m}{\left (x^{n}\right )}^{\frac{3}{2}}}{2 \, m + 3 \, n + 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*b^(3/2)*x*x^m*(x^n)^(3/2)/(2*m + 3*n + 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 [A]  time = 1.21096, size = 32, normalized size = 1.14 \begin{align*} \frac{2 \, b^{\frac{3}{2}} x x^{m} x^{\frac{3}{2} \, n}}{2 \, m + 3 \, n + 2} \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]

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