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

Optimal. Leaf size=18 \[ \frac {x^3}{3}+\frac {3 x^2}{2}-7 x \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {43} \[ \frac {x^3}{3}+\frac {3 x^2}{2}-7 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7*x + (3*x^2)/2 + x^3/3

Rule 43

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*} \int \left (\frac {1}{2} \left (3-\sqrt {37}\right )+x\right ) \left (\frac {1}{2} \left (3+\sqrt {37}\right )+x\right ) \, dx &=\int \left (-7+3 x+x^2\right ) \, dx\\ &=-7 x+\frac {3 x^2}{2}+\frac {x^3}{3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {x^3}{3}+\frac {3 x^2}{2}-7 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

-7*x + (3*x^2)/2 + x^3/3

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: SyntaxError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SyntaxError >> Malformed expression

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giac [A]  time = 0.35, size = 14, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} - 7 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.00, size = 28, normalized size = 1.56 \[ \frac {x^{3}}{3}+\frac {3 x^{2}}{2}+\left (\frac {3}{2}-\frac {\sqrt {37}}{2}\right ) \left (\frac {3}{2}+\frac {\sqrt {37}}{2}\right ) x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.71, size = 14, normalized size = 0.78 \[ \frac {1}{3} \, x^{3} + \frac {3}{2} \, x^{2} - 7 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 0.02, size = 13, normalized size = 0.72 \[ \frac {x\,\left (2\,x^2+9\,x-42\right )}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.06, size = 14, normalized size = 0.78 \[ \frac {x^{3}}{3} + \frac {3 x^{2}}{2} - 7 x \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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