\(\int x^2 (4-x^2)^2 \, dx\) [461]

Optimal result

Integrand size = 13, antiderivative size = 22 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {16 x^3}{3}-\frac {8 x^5}{5}+\frac {x^7}{7} \]

[Out]

16/3*x^3-8/5*x^5+1/7*x^7

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276} \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {x^7}{7}-\frac {8 x^5}{5}+\frac {16 x^3}{3} \]

[In]

Int[x^2*(4 - x^2)^2,x]

[Out]

(16*x^3)/3 - (8*x^5)/5 + x^7/7

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (16 x^2-8 x^4+x^6\right ) \, dx \\ & = \frac {16 x^3}{3}-\frac {8 x^5}{5}+\frac {x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {16 x^3}{3}-\frac {8 x^5}{5}+\frac {x^7}{7} \]

[In]

Integrate[x^2*(4 - x^2)^2,x]

[Out]

(16*x^3)/3 - (8*x^5)/5 + x^7/7

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {16}{3} x^{3}-\frac {8}{5} x^{5}+\frac {1}{7} x^{7}\)	\(17\)
norman	\(\frac {16}{3} x^{3}-\frac {8}{5} x^{5}+\frac {1}{7} x^{7}\)	\(17\)
risch	\(\frac {16}{3} x^{3}-\frac {8}{5} x^{5}+\frac {1}{7} x^{7}\)	\(17\)
parallelrisch	\(\frac {16}{3} x^{3}-\frac {8}{5} x^{5}+\frac {1}{7} x^{7}\)	\(17\)
gosper	\(\frac {x^{3} \left (15 x^{4}-168 x^{2}+560\right )}{105}\)	\(18\)

[In]

int(x^2*(-x^2+4)^2,x,method=_RETURNVERBOSE)

[Out]

16/3*x^3-8/5*x^5+1/7*x^7

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {1}{7} \, x^{7} - \frac {8}{5} \, x^{5} + \frac {16}{3} \, x^{3} \]

[In]

integrate(x^2*(-x^2+4)^2,x, algorithm="fricas")

[Out]

1/7*x^7 - 8/5*x^5 + 16/3*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {x^{7}}{7} - \frac {8 x^{5}}{5} + \frac {16 x^{3}}{3} \]

[In]

integrate(x**2*(-x**2+4)**2,x)

[Out]

x**7/7 - 8*x**5/5 + 16*x**3/3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {1}{7} \, x^{7} - \frac {8}{5} \, x^{5} + \frac {16}{3} \, x^{3} \]

[In]

integrate(x^2*(-x^2+4)^2,x, algorithm="maxima")

[Out]

1/7*x^7 - 8/5*x^5 + 16/3*x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {1}{7} \, x^{7} - \frac {8}{5} \, x^{5} + \frac {16}{3} \, x^{3} \]

[In]

integrate(x^2*(-x^2+4)^2,x, algorithm="giac")

[Out]

1/7*x^7 - 8/5*x^5 + 16/3*x^3

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int x^2 \left (4-x^2\right )^2 \, dx=\frac {x^3\,\left (15\,x^4-168\,x^2+560\right )}{105} \]

[In]

int(x^2*(x^2 - 4)^2,x)

[Out]

(x^3*(15*x^4 - 168*x^2 + 560))/105