\(\int (2+2 x+e (20 x-4 x^3)) \, dx\) [7766]

Optimal result

Integrand size = 16, antiderivative size = 22 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=3-2 x+x (4+x)-e \left (5-x^2\right )^2 \]

[Out]

3-2*x+(4+x)*x-exp(1)*(-x^2+5)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=-e x^4+10 e x^2+x^2+2 x \]

[In]

Int[2 + 2*x + E*(20*x - 4*x^3),x]

[Out]

2*x + x^2 + 10*E*x^2 - E*x^4

Rubi steps \begin{align*} \text {integral}& = 2 x+x^2+e \int \left (20 x-4 x^3\right ) \, dx \\ & = 2 x+x^2+10 e x^2-e x^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=2 x+(1+10 e) x^2-e x^4 \]

[In]

Integrate[2 + 2*x + E*(20*x - 4*x^3),x]

[Out]

2*x + (1 + 10*E)*x^2 - E*x^4

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

method	result	size
gosper	\(-x \left (x^{3} {\mathrm e}-10 x \,{\mathrm e}-x -2\right )\)	\(20\)
default	\(-x^{4} {\mathrm e}+10 x^{2} {\mathrm e}+x^{2}+2 x\)	\(22\)
norman	\(\left (10 \,{\mathrm e}+1\right ) x^{2}+2 x -x^{4} {\mathrm e}\)	\(22\)
parallelrisch	\(-x^{4} {\mathrm e}+10 x^{2} {\mathrm e}+x^{2}+2 x\)	\(22\)
parts	\(-x^{4} {\mathrm e}+10 x^{2} {\mathrm e}+x^{2}+2 x\)	\(22\)
risch	\(-x^{4} {\mathrm e}+10 x^{2} {\mathrm e}+x^{2}-25 \,{\mathrm e}+2 x\)	\(26\)

[In]

int((-4*x^3+20*x)*exp(1)+2*x+2,x,method=_RETURNVERBOSE)

[Out]

-x*(x^3*exp(1)-10*x*exp(1)-x-2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=x^{2} - {\left (x^{4} - 10 \, x^{2}\right )} e + 2 \, x \]

[In]

integrate((-4*x^3+20*x)*exp(1)+2*x+2,x, algorithm="fricas")

[Out]

x^2 - (x^4 - 10*x^2)*e + 2*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=- e x^{4} + x^{2} \cdot \left (1 + 10 e\right ) + 2 x \]

[In]

integrate((-4*x**3+20*x)*exp(1)+2*x+2,x)

[Out]

-E*x**4 + x**2*(1 + 10*E) + 2*x

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=x^{2} - {\left (x^{4} - 10 \, x^{2}\right )} e + 2 \, x \]

[In]

integrate((-4*x^3+20*x)*exp(1)+2*x+2,x, algorithm="maxima")

[Out]

x^2 - (x^4 - 10*x^2)*e + 2*x

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=x^{2} - {\left (x^{4} - 10 \, x^{2}\right )} e + 2 \, x \]

[In]

integrate((-4*x^3+20*x)*exp(1)+2*x+2,x, algorithm="giac")

[Out]

x^2 - (x^4 - 10*x^2)*e + 2*x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \left (2+2 x+e \left (20 x-4 x^3\right )\right ) \, dx=-\mathrm {e}\,x^4+\left (10\,\mathrm {e}+1\right )\,x^2+2\,x \]

[In]

int(2*x + exp(1)*(20*x - 4*x^3) + 2,x)

[Out]

2*x + x^2*(10*exp(1) + 1) - x^4*exp(1)