\(\int (1+4 x+4 x^2+4 x^4) \, dx\) [54]

Optimal result

Integrand size = 15, antiderivative size = 21 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=x+2 x^2+\frac {4 x^3}{3}+\frac {4 x^5}{5} \]

[Out]

x+2*x^2+4/3*x^3+4/5*x^5

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, 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 = {} \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=\frac {4 x^5}{5}+\frac {4 x^3}{3}+2 x^2+x \]

[In]

Int[1 + 4*x + 4*x^2 + 4*x^4,x]

[Out]

x + 2*x^2 + (4*x^3)/3 + (4*x^5)/5

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=x+2 x^2+\frac {4 x^3}{3}+\frac {4 x^5}{5} \]

[In]

Integrate[1 + 4*x + 4*x^2 + 4*x^4,x]

[Out]

x + 2*x^2 + (4*x^3)/3 + (4*x^5)/5

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
gosper	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)
default	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)
norman	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)
risch	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)
parallelrisch	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)
parts	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)

[In]

int(4*x^4+4*x^2+4*x+1,x,method=_RETURNVERBOSE)

[Out]

x+2*x^2+4/3*x^3+4/5*x^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=\frac {4}{5} \, x^{5} + \frac {4}{3} \, x^{3} + 2 \, x^{2} + x \]

[In]

integrate(4*x^4+4*x^2+4*x+1,x, algorithm="fricas")

[Out]

4/5*x^5 + 4/3*x^3 + 2*x^2 + x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=\frac {4 x^{5}}{5} + \frac {4 x^{3}}{3} + 2 x^{2} + x \]

[In]

integrate(4*x**4+4*x**2+4*x+1,x)

[Out]

4*x**5/5 + 4*x**3/3 + 2*x**2 + x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=\frac {4}{5} \, x^{5} + \frac {4}{3} \, x^{3} + 2 \, x^{2} + x \]

[In]

integrate(4*x^4+4*x^2+4*x+1,x, algorithm="maxima")

[Out]

4/5*x^5 + 4/3*x^3 + 2*x^2 + x

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=\frac {4}{5} \, x^{5} + \frac {4}{3} \, x^{3} + 2 \, x^{2} + x \]

[In]

integrate(4*x^4+4*x^2+4*x+1,x, algorithm="giac")

[Out]

4/5*x^5 + 4/3*x^3 + 2*x^2 + x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \left (1+4 x+4 x^2+4 x^4\right ) \, dx=\frac {4\,x^5}{5}+\frac {4\,x^3}{3}+2\,x^2+x \]

[In]

int(4*x + 4*x^2 + 4*x^4 + 1,x)

[Out]

x + 2*x^2 + (4*x^3)/3 + (4*x^5)/5