3.142 \(\int \frac{(b x^n)^{3/2}}{x^4} \, dx\)

Optimal. Leaf size=26 \[ -\frac{2 b x^{n-3} \sqrt{b x^n}}{3 (2-n)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0054398, antiderivative size = 26, 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} \[ -\frac{2 b x^{n-3} \sqrt{b x^n}}{3 (2-n)} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^n)^(3/2)/x^4,x]

[Out]

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

Mathematica [A]  time = 0.0055524, size = 22, normalized size = 0.85 \[ \frac{\left (b x^n\right )^{3/2}}{\left (\frac{3 n}{2}-3\right ) x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^n)^(3/2)/x^4,x]

[Out]

(b*x^n)^(3/2)/((-3 + (3*n)/2)*x^3)

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 18, normalized size = 0.7 \begin{align*}{\frac{2}{3\,{x}^{3} \left ( -2+n \right ) } \left ( b{x}^{n} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^n)^(3/2)/x^4,x)

[Out]

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

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (b x^{n}\right )^{\frac{3}{2}}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^n)^(3/2)/x^4,x, algorithm="giac")

[Out]

integrate((b*x^n)^(3/2)/x^4, x)