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

Optimal. Leaf size=17 \[ \frac{2}{11} b x^4 \sqrt{b x^3} \]

[Out]

(2*b*x^4*Sqrt[b*x^3])/11

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Rubi [A]  time = 0.0016906, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac{2}{11} b x^4 \sqrt{b x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x^4*Sqrt[b*x^3])/11

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 \left (b x^3\right )^{3/2} \, dx &=\frac{\left (b \sqrt{b x^3}\right ) \int x^{9/2} \, dx}{x^{3/2}}\\ &=\frac{2}{11} b x^4 \sqrt{b x^3}\\ \end{align*}

Mathematica [A]  time = 0.0012278, size = 14, normalized size = 0.82 \[ \frac{2}{11} x \left (b x^3\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(b*x^3)^(3/2))/11

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Maple [A]  time = 0.002, size = 11, normalized size = 0.7 \begin{align*}{\frac{2\,x}{11} \left ( b{x}^{3} \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*x*(b*x^3)^(3/2)

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Maxima [A]  time = 0.96569, size = 14, normalized size = 0.82 \begin{align*} \frac{2}{11} \, \left (b x^{3}\right )^{\frac{3}{2}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(b*x^3)^(3/2)*x

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Fricas [A]  time = 1.67623, size = 32, normalized size = 1.88 \begin{align*} \frac{2}{11} \, \sqrt{b x^{3}} b x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*sqrt(b*x^3)*b*x^4

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Sympy [A]  time = 0.493833, size = 15, normalized size = 0.88 \begin{align*} \frac{2 b^{\frac{3}{2}} x \left (x^{3}\right )^{\frac{3}{2}}}{11} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b**(3/2)*x*(x**3)**(3/2)/11

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Giac [A]  time = 1.14539, size = 18, normalized size = 1.06 \begin{align*} \frac{2}{11} \, \sqrt{b x} b x^{5} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*sqrt(b*x)*b*x^5*sgn(x)