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

Optimal. Leaf size=21 \[ x+2 x^2+\frac {4 x^3}{3}+\frac {4 x^5}{5} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \frac {4 x^5}{5}+\frac {4 x^3}{3}+2 x^2+x \end {gather*}

Antiderivative was successfully verified.

[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*} \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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} x+2 x^2+\frac {4 x^3}{3}+\frac {4 x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[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]
time = 0.01, size = 18, normalized size = 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	\(x +2 x^{2}+\frac {4}{3} x^{3}+\frac {4}{5} x^{5}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 17, normalized size = 0.81 \begin {gather*} \frac {4}{5} \, x^{5} + \frac {4}{3} \, x^{3} + 2 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 17, normalized size = 0.81 \begin {gather*} \frac {4}{5} \, x^{5} + \frac {4}{3} \, x^{3} + 2 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.01, size = 19, normalized size = 0.90 \begin {gather*} \frac {4 x^{5}}{5} + \frac {4 x^{3}}{3} + 2 x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Giac [A]
time = 3.47, size = 17, normalized size = 0.81 \begin {gather*} \frac {4}{5} \, x^{5} + \frac {4}{3} \, x^{3} + 2 \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.03, size = 17, normalized size = 0.81 \begin {gather*} \frac {4\,x^5}{5}+\frac {4\,x^3}{3}+2\,x^2+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________