\(\int x^2 (3-4 x+x^2)^2 \, dx\) [2165]

Optimal result

Integrand size = 14, antiderivative size = 32 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=3 x^3-6 x^4+\frac {22 x^5}{5}-\frac {4 x^6}{3}+\frac {x^7}{7} \]

[Out]

3*x^3-6*x^4+22/5*x^5-4/3*x^6+1/7*x^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {712} \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=\frac {x^7}{7}-\frac {4 x^6}{3}+\frac {22 x^5}{5}-6 x^4+3 x^3 \]

[In]

Int[x^2*(3 - 4*x + x^2)^2,x]

[Out]

3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (9 x^2-24 x^3+22 x^4-8 x^5+x^6\right ) \, dx \\ & = 3 x^3-6 x^4+\frac {22 x^5}{5}-\frac {4 x^6}{3}+\frac {x^7}{7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=3 x^3-6 x^4+\frac {22 x^5}{5}-\frac {4 x^6}{3}+\frac {x^7}{7} \]

[In]

Integrate[x^2*(3 - 4*x + x^2)^2,x]

[Out]

3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7

Maple [A] (verified)

Time = 10.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {x^{3} \left (15 x^{4}-140 x^{3}+462 x^{2}-630 x +315\right )}{105}\)	\(26\)
default	\(3 x^{3}-6 x^{4}+\frac {22}{5} x^{5}-\frac {4}{3} x^{6}+\frac {1}{7} x^{7}\)	\(27\)
norman	\(3 x^{3}-6 x^{4}+\frac {22}{5} x^{5}-\frac {4}{3} x^{6}+\frac {1}{7} x^{7}\)	\(27\)
risch	\(3 x^{3}-6 x^{4}+\frac {22}{5} x^{5}-\frac {4}{3} x^{6}+\frac {1}{7} x^{7}\)	\(27\)
parallelrisch	\(3 x^{3}-6 x^{4}+\frac {22}{5} x^{5}-\frac {4}{3} x^{6}+\frac {1}{7} x^{7}\)	\(27\)

[In]

int(x^2*(x^2-4*x+3)^2,x,method=_RETURNVERBOSE)

[Out]

1/105*x^3*(15*x^4-140*x^3+462*x^2-630*x+315)

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=\frac {1}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \]

[In]

integrate(x^2*(x^2-4*x+3)^2,x, algorithm="fricas")

[Out]

1/7*x^7 - 4/3*x^6 + 22/5*x^5 - 6*x^4 + 3*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=\frac {x^{7}}{7} - \frac {4 x^{6}}{3} + \frac {22 x^{5}}{5} - 6 x^{4} + 3 x^{3} \]

[In]

integrate(x**2*(x**2-4*x+3)**2,x)

[Out]

x**7/7 - 4*x**6/3 + 22*x**5/5 - 6*x**4 + 3*x**3

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=\frac {1}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \]

[In]

integrate(x^2*(x^2-4*x+3)^2,x, algorithm="maxima")

[Out]

1/7*x^7 - 4/3*x^6 + 22/5*x^5 - 6*x^4 + 3*x^3

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=\frac {1}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \]

[In]

integrate(x^2*(x^2-4*x+3)^2,x, algorithm="giac")

[Out]

1/7*x^7 - 4/3*x^6 + 22/5*x^5 - 6*x^4 + 3*x^3

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int x^2 \left (3-4 x+x^2\right )^2 \, dx=\frac {x^7}{7}-\frac {4\,x^6}{3}+\frac {22\,x^5}{5}-6\,x^4+3\,x^3 \]

[In]

int(x^2*(x^2 - 4*x + 3)^2,x)

[Out]

3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7