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

Optimal. Leaf size=20 \[ \frac{2 b x^n \sqrt{b x^n}}{3 n} \]

[Out]

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

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Rubi [A]  time = 0.0034291, antiderivative size = 20, 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} \[ \frac{2 b x^n \sqrt{b x^n}}{3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0020905, size = 16, normalized size = 0.8 \[ \frac{2 \left (b x^n\right )^{3/2}}{3 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 13, normalized size = 0.7 \begin{align*}{\frac{2}{3\,n} \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,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.958, size = 34, normalized size = 1.7 \begin{align*} \frac{2 \, \sqrt{b x^{n}} b x^{n}}{3 \, n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*sqrt(b*x^n)*b*x^n/n

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Sympy [A]  time = 4.87608, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} \frac{2 b^{\frac{3}{2}} \left (x^{n}\right )^{\frac{3}{2}}}{3 n} & \text{for}\: n \neq 0 \\b^{\frac{3}{2}} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*b**(3/2)*(x**n)**(3/2)/(3*n), Ne(n, 0)), (b**(3/2)*log(x), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n)^(3/2)/x, x)