\(\int (56 x+8 e x+16 x^3) \, dx\) [2831]

Optimal result

Integrand size = 13, antiderivative size = 10 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=\left (7+e+2 x^2\right )^2 \]

[Out]

(2*x^2+7+exp(1))^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6} \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=4 x^4+4 (7+e) x^2 \]

[In]

Int[56*x + 8*E*x + 16*x^3,x]

[Out]

4*(7 + E)*x^2 + 4*x^4

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \int \left ((56+8 e) x+16 x^3\right ) \, dx \\ & = 4 (7+e) x^2+4 x^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.70 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=28 x^2+4 e x^2+4 x^4 \]

[In]

Integrate[56*x + 8*E*x + 16*x^3,x]

[Out]

28*x^2 + 4*E*x^2 + 4*x^4

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20

method	result	size
default	\(\left (2 x^{2}+7+{\mathrm e}\right )^{2}\)	\(12\)
gosper	\(4 x^{2} \left (x^{2}+{\mathrm e}+7\right )\)	\(13\)
norman	\(\left (4 \,{\mathrm e}+28\right ) x^{2}+4 x^{4}\)	\(17\)
risch	\(4 x^{2} {\mathrm e}+4 x^{4}+28 x^{2}\)	\(19\)
parallelrisch	\(4 x^{2} {\mathrm e}+4 x^{4}+28 x^{2}\)	\(19\)
parts	\(4 x^{2} {\mathrm e}+4 x^{4}+28 x^{2}\)	\(19\)

[In]

int(8*x*exp(1)+16*x^3+56*x,x,method=_RETURNVERBOSE)

[Out]

(2*x^2+7+exp(1))^2

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=4 \, x^{4} + 4 \, x^{2} e + 28 \, x^{2} \]

[In]

integrate(8*x*exp(1)+16*x^3+56*x,x, algorithm="fricas")

[Out]

4*x^4 + 4*x^2*e + 28*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.40 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=4 x^{4} + x^{2} \cdot \left (4 e + 28\right ) \]

[In]

integrate(8*x*exp(1)+16*x**3+56*x,x)

[Out]

4*x**4 + x**2*(4*E + 28)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=4 \, x^{4} + 4 \, x^{2} e + 28 \, x^{2} \]

[In]

integrate(8*x*exp(1)+16*x^3+56*x,x, algorithm="maxima")

[Out]

4*x^4 + 4*x^2*e + 28*x^2

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=4 \, x^{4} + 4 \, x^{2} e + 28 \, x^{2} \]

[In]

integrate(8*x*exp(1)+16*x^3+56*x,x, algorithm="giac")

[Out]

4*x^4 + 4*x^2*e + 28*x^2

Mupad [B] (verification not implemented)

Time = 9.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \left (56 x+8 e x+16 x^3\right ) \, dx=4\,x^4+\left (4\,\mathrm {e}+28\right )\,x^2 \]

[In]

int(56*x + 8*x*exp(1) + 16*x^3,x)

[Out]

x^2*(4*exp(1) + 28) + 4*x^4