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

Optimal. Leaf size=14 \[ \frac{1}{4} x^3 \sqrt{x^2} \]

[Out]

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

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Rubi [A]  time = 0.001058, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {15, 30} \[ \frac{1}{4} x^3 \sqrt{x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (x^2\right )^{3/2} \, dx &=\frac{\sqrt{x^2} \int x^3 \, dx}{x}\\ &=\frac{1}{4} x^3 \sqrt{x^2}\\ \end{align*}

Mathematica [A]  time = 0.0031073, size = 12, normalized size = 0.86 \[ \frac{1}{4} x \left (x^2\right )^{3/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(x^2)^(3/2))/4

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Maple [A]  time = 0.002, size = 11, normalized size = 0.8 \begin{align*}{\frac{{x}^{3}}{4}\sqrt{{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^2)^(1/2),x)

[Out]

1/4*x^3*(x^2)^(1/2)

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Maxima [A]  time = 1.03718, size = 11, normalized size = 0.79 \begin{align*} \frac{1}{4} \,{\left (x^{2}\right )}^{\frac{3}{2}} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2)^(1/2),x, algorithm="maxima")

[Out]

1/4*(x^2)^(3/2)*x

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Fricas [A]  time = 1.16087, size = 12, normalized size = 0.86 \begin{align*} \frac{1}{4} \, x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2)^(1/2),x, algorithm="fricas")

[Out]

1/4*x^4

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Sympy [A]  time = 0.062199, size = 3, normalized size = 0.21 \begin{align*} \frac{x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**2)**(1/2),x)

[Out]

x**4/4

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Giac [A]  time = 1.15197, size = 9, normalized size = 0.64 \begin{align*} \frac{1}{4} \, x^{4} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2)^(1/2),x, algorithm="giac")

[Out]

1/4*x^4*sgn(x)