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

Optimal result

Integrand size = 7, antiderivative size = 14 \[ \int \left (x^2\right )^{3/2} \, dx=\frac {1}{4} x^3 \sqrt {x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30} \[ \int \left (x^2\right )^{3/2} \, dx=\frac {1}{4} x^3 \sqrt {x^2} \]

[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*} \text {integral}& = \frac {\sqrt {x^2} \int x^3 \, dx}{x} \\ & = \frac {1}{4} x^3 \sqrt {x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \left (x^2\right )^{3/2} \, dx=\frac {1}{4} x \left (x^2\right )^{3/2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 2.

Time = 0.05 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.36 \[ \int \left (x^2\right )^{3/2} \, dx=\frac {1}{4} \, x^{4} \]

[In]

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

[Out]

1/4*x^4

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \left (x^2\right )^{3/2} \, dx=\frac {x^{3} \sqrt {x^{2}}}{4} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.57 \[ \int \left (x^2\right )^{3/2} \, dx=\frac {1}{4} \, {\left (x^{2}\right )}^{\frac {3}{2}} x \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.50 \[ \int \left (x^2\right )^{3/2} \, dx=\frac {1}{4} \, x^{4} \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \left (x^2\right )^{3/2} \, dx=\int x^2\,\sqrt {x^2} \,d x \]

[In]

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

[Out]

int(x^2*(x^2)^(1/2), x)