3.1.11 \(\int (x^2)^{3/2} \, dx\) [11]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 14, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{4} x^3 \sqrt {x^2} \end {gather*}

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 = 0.00, size = 12, normalized size = 0.86 \begin {gather*} \frac {1}{4} x \left (x^2\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 2.
time = 0.02, size = 8, normalized size = 0.57

method	result	size
default	\(\frac {x^{4} \mathrm {csgn}\left (x \right )}{4}\)	\(8\)
gosper	\(\frac {x^{3} \sqrt {x^{2}}}{4}\)	\(11\)
risch	\(\frac {x^{3} \sqrt {x^{2}}}{4}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 8, normalized size = 0.57 \begin {gather*} \frac {1}{4} \, {\left (x^{2}\right )}^{\frac {3}{2}} x \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.33, size = 5, normalized size = 0.36 \begin {gather*} \frac {1}{4} \, x^{4} \end {gather*}

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

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 3, normalized size = 0.21 \begin {gather*} \frac {x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**4/4

________________________________________________________________________________________

Giac [A]
time = 1.35, size = 7, normalized size = 0.50 \begin {gather*} \frac {1}{4} \, x^{4} \mathrm {sgn}\left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int x^2\,\sqrt {x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(x^2)^(1/2), x)

________________________________________________________________________________________