\(\int x (a+8 x-8 x^2+4 x^3-x^4)^2 \, dx\) [125]

Optimal result

Integrand size = 24, antiderivative size = 79 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {a^2 x^2}{2}+\frac {16 a x^3}{3}+4 (4-a) x^4-\frac {8}{5} (16-a) x^5+\frac {1}{3} (64-a) x^6-\frac {80 x^7}{7}+4 x^8-\frac {8 x^9}{9}+\frac {x^{10}}{10} \]

[Out]

1/2*a^2*x^2+16/3*a*x^3+4*(4-a)*x^4-8/5*(16-a)*x^5+1/3*(64-a)*x^6-80/7*x^7+4*x^8-8/9*x^9+1/10*x^10

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6874} \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {a^2 x^2}{2}+\frac {1}{3} (64-a) x^6-\frac {8}{5} (16-a) x^5+4 (4-a) x^4+\frac {16 a x^3}{3}+\frac {x^{10}}{10}-\frac {8 x^9}{9}+4 x^8-\frac {80 x^7}{7} \]

[In]

Int[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

(a^2*x^2)/2 + (16*a*x^3)/3 + 4*(4 - a)*x^4 - (8*(16 - a)*x^5)/5 + ((64 - a)*x^6)/3 - (80*x^7)/7 + 4*x^8 - (8*x
^9)/9 + x^10/10

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 x+16 a x^2-16 (-4+a) x^3+8 (-16+a) x^4-2 (-64+a) x^5-80 x^6+32 x^7-8 x^8+x^9\right ) \, dx \\ & = \frac {a^2 x^2}{2}+\frac {16 a x^3}{3}+4 (4-a) x^4-\frac {8}{5} (16-a) x^5+\frac {1}{3} (64-a) x^6-\frac {80 x^7}{7}+4 x^8-\frac {8 x^9}{9}+\frac {x^{10}}{10} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.95 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {a^2 x^2}{2}+\frac {16 a x^3}{3}-4 (-4+a) x^4+\frac {8}{5} (-16+a) x^5+\frac {1}{3} (64-a) x^6-\frac {80 x^7}{7}+4 x^8-\frac {8 x^9}{9}+\frac {x^{10}}{10} \]

[In]

Integrate[x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x]

[Out]

(a^2*x^2)/2 + (16*a*x^3)/3 - 4*(-4 + a)*x^4 + (8*(-16 + a)*x^5)/5 + ((64 - a)*x^6)/3 - (80*x^7)/7 + 4*x^8 - (8
*x^9)/9 + x^10/10

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.80

method	result	size
norman	\(\frac {x^{10}}{10}-\frac {8 x^{9}}{9}+4 x^{8}-\frac {80 x^{7}}{7}+\left (\frac {64}{3}-\frac {a}{3}\right ) x^{6}+\left (\frac {8 a}{5}-\frac {128}{5}\right ) x^{5}+\left (-4 a +16\right ) x^{4}+\frac {16 a \,x^{3}}{3}+\frac {a^{2} x^{2}}{2}\)	\(63\)
default	\(\frac {x^{10}}{10}-\frac {8 x^{9}}{9}+4 x^{8}-\frac {80 x^{7}}{7}+\frac {\left (-2 a +128\right ) x^{6}}{6}+\frac {\left (8 a -128\right ) x^{5}}{5}+\frac {\left (-16 a +64\right ) x^{4}}{4}+\frac {16 a \,x^{3}}{3}+\frac {a^{2} x^{2}}{2}\)	\(66\)
gosper	\(\frac {1}{10} x^{10}-\frac {8}{9} x^{9}+4 x^{8}-\frac {80}{7} x^{7}+\frac {64}{3} x^{6}-\frac {1}{3} a \,x^{6}+\frac {8}{5} a \,x^{5}-\frac {128}{5} x^{5}-4 a \,x^{4}+16 x^{4}+\frac {16}{3} a \,x^{3}+\frac {1}{2} a^{2} x^{2}\)	\(69\)
risch	\(\frac {1}{10} x^{10}-\frac {8}{9} x^{9}+4 x^{8}-\frac {80}{7} x^{7}+\frac {64}{3} x^{6}-\frac {1}{3} a \,x^{6}+\frac {8}{5} a \,x^{5}-\frac {128}{5} x^{5}-4 a \,x^{4}+16 x^{4}+\frac {16}{3} a \,x^{3}+\frac {1}{2} a^{2} x^{2}\)	\(69\)
parallelrisch	\(\frac {1}{10} x^{10}-\frac {8}{9} x^{9}+4 x^{8}-\frac {80}{7} x^{7}+\frac {64}{3} x^{6}-\frac {1}{3} a \,x^{6}+\frac {8}{5} a \,x^{5}-\frac {128}{5} x^{5}-4 a \,x^{4}+16 x^{4}+\frac {16}{3} a \,x^{3}+\frac {1}{2} a^{2} x^{2}\)	\(69\)

[In]

int(x*(-x^4+4*x^3-8*x^2+a+8*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/10*x^10-8/9*x^9+4*x^8-80/7*x^7+(64/3-1/3*a)*x^6+(8/5*a-128/5)*x^5+(-4*a+16)*x^4+16/3*a*x^3+1/2*a^2*x^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {1}{10} \, x^{10} - \frac {8}{9} \, x^{9} + 4 \, x^{8} - \frac {1}{3} \, {\left (a - 64\right )} x^{6} - \frac {80}{7} \, x^{7} + \frac {8}{5} \, {\left (a - 16\right )} x^{5} - 4 \, {\left (a - 4\right )} x^{4} + \frac {1}{2} \, a^{2} x^{2} + \frac {16}{3} \, a x^{3} \]

[In]

integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="fricas")

[Out]

1/10*x^10 - 8/9*x^9 + 4*x^8 - 1/3*(a - 64)*x^6 - 80/7*x^7 + 8/5*(a - 16)*x^5 - 4*(a - 4)*x^4 + 1/2*a^2*x^2 + 1
6/3*a*x^3

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {a^{2} x^{2}}{2} + \frac {16 a x^{3}}{3} + \frac {x^{10}}{10} - \frac {8 x^{9}}{9} + 4 x^{8} - \frac {80 x^{7}}{7} + x^{6} \cdot \left (\frac {64}{3} - \frac {a}{3}\right ) + x^{5} \cdot \left (\frac {8 a}{5} - \frac {128}{5}\right ) + x^{4} \cdot \left (16 - 4 a\right ) \]

[In]

integrate(x*(-x**4+4*x**3-8*x**2+a+8*x)**2,x)

[Out]

a**2*x**2/2 + 16*a*x**3/3 + x**10/10 - 8*x**9/9 + 4*x**8 - 80*x**7/7 + x**6*(64/3 - a/3) + x**5*(8*a/5 - 128/5
) + x**4*(16 - 4*a)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.75 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {1}{10} \, x^{10} - \frac {8}{9} \, x^{9} + 4 \, x^{8} - \frac {1}{3} \, {\left (a - 64\right )} x^{6} - \frac {80}{7} \, x^{7} + \frac {8}{5} \, {\left (a - 16\right )} x^{5} - 4 \, {\left (a - 4\right )} x^{4} + \frac {1}{2} \, a^{2} x^{2} + \frac {16}{3} \, a x^{3} \]

[In]

integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="maxima")

[Out]

1/10*x^10 - 8/9*x^9 + 4*x^8 - 1/3*(a - 64)*x^6 - 80/7*x^7 + 8/5*(a - 16)*x^5 - 4*(a - 4)*x^4 + 1/2*a^2*x^2 + 1
6/3*a*x^3

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.86 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=\frac {1}{10} \, x^{10} - \frac {8}{9} \, x^{9} + 4 \, x^{8} - \frac {1}{3} \, a x^{6} - \frac {80}{7} \, x^{7} + \frac {8}{5} \, a x^{5} + \frac {64}{3} \, x^{6} - 4 \, a x^{4} - \frac {128}{5} \, x^{5} + \frac {1}{2} \, a^{2} x^{2} + \frac {16}{3} \, a x^{3} + 16 \, x^{4} \]

[In]

integrate(x*(-x^4+4*x^3-8*x^2+a+8*x)^2,x, algorithm="giac")

[Out]

1/10*x^10 - 8/9*x^9 + 4*x^8 - 1/3*a*x^6 - 80/7*x^7 + 8/5*a*x^5 + 64/3*x^6 - 4*a*x^4 - 128/5*x^5 + 1/2*a^2*x^2
+ 16/3*a*x^3 + 16*x^4

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.81 \[ \int x \left (a+8 x-8 x^2+4 x^3-x^4\right )^2 \, dx=x^5\,\left (\frac {8\,a}{5}-\frac {128}{5}\right )-x^6\,\left (\frac {a}{3}-\frac {64}{3}\right )-x^4\,\left (4\,a-16\right )+\frac {16\,a\,x^3}{3}-\frac {80\,x^7}{7}+4\,x^8-\frac {8\,x^9}{9}+\frac {x^{10}}{10}+\frac {a^2\,x^2}{2} \]

[In]

int(x*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^2,x)

[Out]

x^5*((8*a)/5 - 128/5) - x^6*(a/3 - 64/3) - x^4*(4*a - 16) + (16*a*x^3)/3 - (80*x^7)/7 + 4*x^8 - (8*x^9)/9 + x^
10/10 + (a^2*x^2)/2