3.85 \(\int (b x)^{3/2} \, dx\)

Optimal. Leaf size=14 \[ \frac{2 (b x)^{5/2}}{5 b} \]

[Out]

(2*(b*x)^(5/2))/(5*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0013268, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {32} \[ \frac{2 (b x)^{5/2}}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*x)^(5/2))/(5*b)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int (b x)^{3/2} \, dx &=\frac{2 (b x)^{5/2}}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.0010862, size = 12, normalized size = 0.86 \[ \frac{2}{5} x (b x)^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(b*x)^(3/2))/5

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 9, normalized size = 0.6 \begin{align*}{\frac{2\,x}{5} \left ( bx \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x*(b*x)^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 0.999249, size = 14, normalized size = 1. \begin{align*} \frac{2 \, \left (b x\right )^{\frac{5}{2}}}{5 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(b*x)^(5/2)/b

________________________________________________________________________________________

Fricas [A]  time = 1.72605, size = 28, normalized size = 2. \begin{align*} \frac{2}{5} \, \sqrt{b x} b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(b*x)*b*x^2

________________________________________________________________________________________

Sympy [A]  time = 0.068235, size = 10, normalized size = 0.71 \begin{align*} \frac{2 \left (b x\right )^{\frac{5}{2}}}{5 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x)**(5/2)/(5*b)

________________________________________________________________________________________

Giac [A]  time = 1.17478, size = 15, normalized size = 1.07 \begin{align*} \frac{2}{5} \, \sqrt{b x} b x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*sqrt(b*x)*b*x^2