\(\int (\frac {1}{2} (3-\sqrt {37})+x) (\frac {1}{2} (3+\sqrt {37})+x) \, dx\) [483]

Optimal result

Integrand size = 29, antiderivative size = 18 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=-7 x+\frac {3 x^2}{2}+\frac {x^3}{3} \]

[Out]

-7*x+3/2*x^2+1/3*x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {45} \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=\frac {x^3}{3}+\frac {3 x^2}{2}-7 x \]

[In]

Int[((3 - Sqrt[37])/2 + x)*((3 + Sqrt[37])/2 + x),x]

[Out]

-7*x + (3*x^2)/2 + x^3/3

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (-7+3 x+x^2\right ) \, dx \\ & = -7 x+\frac {3 x^2}{2}+\frac {x^3}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=-7 x+\frac {3 x^2}{2}+\frac {x^3}{3} \]

[In]

Integrate[((3 - Sqrt[37])/2 + x)*((3 + Sqrt[37])/2 + x),x]

[Out]

-7*x + (3*x^2)/2 + x^3/3

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.83

method	result	size
norman	\(-7 x +\frac {3}{2} x^{2}+\frac {1}{3} x^{3}\)	\(15\)
risch	\(-7 x +\frac {3}{2} x^{2}+\frac {1}{3} x^{3}\)	\(15\)
parallelrisch	\(-7 x +\frac {3}{2} x^{2}+\frac {1}{3} x^{3}\)	\(15\)
default	\(\frac {x^{3}}{3}+\frac {3 x^{2}}{2}+\frac {\left (3-\sqrt {37}\right ) \left (3+\sqrt {37}\right ) x}{4}\)	\(27\)
gosper	\(-\frac {x \left (2 x^{2}+9 x -42\right ) \left (-2 x -3+\sqrt {37}\right ) \left (2 x +3+\sqrt {37}\right )}{24 \left (x^{2}+3 x -7\right )}\)	\(40\)

[In]

int((x+3/2-1/2*37^(1/2))*(x+3/2+1/2*37^(1/2)),x,method=_RETURNVERBOSE)

[Out]

-7*x+3/2*x^2+1/3*x^3

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=\frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} - 7 \, x \]

[In]

integrate((x+3/2-1/2*37^(1/2))*(x+3/2+1/2*37^(1/2)),x, algorithm="fricas")

[Out]

1/3*x^3 + 3/2*x^2 - 7*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=\frac {x^{3}}{3} + \frac {3 x^{2}}{2} - 7 x \]

[In]

integrate((x+3/2-1/2*37**(1/2))*(x+3/2+1/2*37**(1/2)),x)

[Out]

x**3/3 + 3*x**2/2 - 7*x

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=\frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} - 7 \, x \]

[In]

integrate((x+3/2-1/2*37^(1/2))*(x+3/2+1/2*37^(1/2)),x, algorithm="maxima")

[Out]

1/3*x^3 + 3/2*x^2 - 7*x

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=\frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} - 7 \, x \]

[In]

integrate((x+3/2-1/2*37^(1/2))*(x+3/2+1/2*37^(1/2)),x, algorithm="giac")

[Out]

1/3*x^3 + 3/2*x^2 - 7*x

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.72 \[ \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx=\frac {x\,\left (2\,x^2+9\,x-42\right )}{6} \]

[In]

int((x - 37^(1/2)/2 + 3/2)*(x + 37^(1/2)/2 + 3/2),x)

[Out]

(x*(9*x + 2*x^2 - 42))/6