\(\int (2 x+\sqrt {2} x^2) \, dx\) [174]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=x^2+\frac {\sqrt {2} x^3}{3} \]

[Out]

x^2+1/3*x^3*2^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, 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 = {} \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=\frac {\sqrt {2} x^3}{3}+x^2 \]

[In]

Int[2*x + Sqrt[2]*x^2,x]

[Out]

x^2 + (Sqrt[2]*x^3)/3

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=x^2+\frac {\sqrt {2} x^3}{3} \]

[In]

Integrate[2*x + Sqrt[2]*x^2,x]

[Out]

x^2 + (Sqrt[2]*x^3)/3

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
norman	\(x^{2}+\frac {x^{3} \sqrt {2}}{3}\)	\(13\)
risch	\(x^{2}+\frac {x^{3} \sqrt {2}}{3}\)	\(13\)
parallelrisch	\(x^{2}+\frac {x^{3} \sqrt {2}}{3}\)	\(13\)
parts	\(x^{2}+\frac {x^{3} \sqrt {2}}{3}\)	\(13\)
default	\(\sqrt {2}\, \left (\frac {x^{3}}{3}+\frac {x^{2} \sqrt {2}}{2}\right )\)	\(19\)
gosper	\(\frac {x^{2} \left (2 x +3 \sqrt {2}\right ) \left (x \sqrt {2}+2\right )}{6 x +6 \sqrt {2}}\)	\(29\)

[In]

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

[Out]

x^2+1/3*x^3*2^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=\frac {1}{3} \, \sqrt {2} x^{3} + x^{2} \]

[In]

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

[Out]

1/3*sqrt(2)*x^3 + x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=\frac {\sqrt {2} x^{3}}{3} + x^{2} \]

[In]

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

[Out]

sqrt(2)*x**3/3 + x**2

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=\frac {1}{3} \, \sqrt {2} x^{3} + x^{2} \]

[In]

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

[Out]

1/3*sqrt(2)*x^3 + x^2

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=\frac {1}{3} \, \sqrt {2} x^{3} + x^{2} \]

[In]

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

[Out]

1/3*sqrt(2)*x^3 + x^2

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \left (2 x+\sqrt {2} x^2\right ) \, dx=\frac {x^2\,\left (\sqrt {2}\,x+3\right )}{3} \]

[In]

int(2*x + 2^(1/2)*x^2,x)

[Out]

(x^2*(2^(1/2)*x + 3))/3