\(\int (a+8 x-8 x^2+4 x^3-x^4) \, dx\) [119]

Optimal result

Integrand size = 20, antiderivative size = 26 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=a x+4 x^2-\frac {8 x^3}{3}+x^4-\frac {x^5}{5} \]

[Out]

a*x+4*x^2-8/3*x^3+x^4-1/5*x^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=a x-\frac {x^5}{5}+x^4-\frac {8 x^3}{3}+4 x^2 \]

[In]

Int[a + 8*x - 8*x^2 + 4*x^3 - x^4,x]

[Out]

a*x + 4*x^2 - (8*x^3)/3 + x^4 - x^5/5

Rubi steps \begin{align*} \text {integral}& = a x+4 x^2-\frac {8 x^3}{3}+x^4-\frac {x^5}{5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=a x+4 x^2-\frac {8 x^3}{3}+x^4-\frac {x^5}{5} \]

[In]

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

[Out]

a*x + 4*x^2 - (8*x^3)/3 + x^4 - x^5/5

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
gosper	\(a x +4 x^{2}-\frac {8}{3} x^{3}+x^{4}-\frac {1}{5} x^{5}\)	\(23\)
default	\(a x +4 x^{2}-\frac {8}{3} x^{3}+x^{4}-\frac {1}{5} x^{5}\)	\(23\)
norman	\(a x +4 x^{2}-\frac {8}{3} x^{3}+x^{4}-\frac {1}{5} x^{5}\)	\(23\)
risch	\(a x +4 x^{2}-\frac {8}{3} x^{3}+x^{4}-\frac {1}{5} x^{5}\)	\(23\)
parallelrisch	\(a x +4 x^{2}-\frac {8}{3} x^{3}+x^{4}-\frac {1}{5} x^{5}\)	\(23\)
parts	\(a x +4 x^{2}-\frac {8}{3} x^{3}+x^{4}-\frac {1}{5} x^{5}\)	\(23\)

[In]

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

[Out]

a*x+4*x^2-8/3*x^3+x^4-1/5*x^5

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {1}{5} \, x^{5} + x^{4} - \frac {8}{3} \, x^{3} + a x + 4 \, x^{2} \]

[In]

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

[Out]

-1/5*x^5 + x^4 - 8/3*x^3 + a*x + 4*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=a x - \frac {x^{5}}{5} + x^{4} - \frac {8 x^{3}}{3} + 4 x^{2} \]

[In]

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

[Out]

a*x - x**5/5 + x**4 - 8*x**3/3 + 4*x**2

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {1}{5} \, x^{5} + x^{4} - \frac {8}{3} \, x^{3} + a x + 4 \, x^{2} \]

[In]

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

[Out]

-1/5*x^5 + x^4 - 8/3*x^3 + a*x + 4*x^2

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {1}{5} \, x^{5} + x^{4} - \frac {8}{3} \, x^{3} + a x + 4 \, x^{2} \]

[In]

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

[Out]

-1/5*x^5 + x^4 - 8/3*x^3 + a*x + 4*x^2

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx=-\frac {x^5}{5}+x^4-\frac {8\,x^3}{3}+4\,x^2+a\,x \]

[In]

int(a + 8*x - 8*x^2 + 4*x^3 - x^4,x)

[Out]

a*x + 4*x^2 - (8*x^3)/3 + x^4 - x^5/5