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*x^2)/2 + x^3/3

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Rubi [A]  time = 0.0103565, antiderivative size = 18, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0006199, size = 18, normalized size = 1. \[ \frac{x^3}{3}+\frac{3 x^2}{2}-7 x \]

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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Maple [A]  time = 0., size = 28, normalized size = 1.6 \begin{align*}{\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 \end{align*}

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 = 1.64185, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{3} \, x^{3} + \frac{3}{2} \, x^{2} - 7 \, x \end{align*}

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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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

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: UnboundLocalError

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Sympy [A]  time = 0.051822, size = 14, normalized size = 0.78 \begin{align*} \frac{x^{3}}{3} + \frac{3 x^{2}}{2} - 7 x \end{align*}

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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Giac [A]  time = 1.12858, size = 19, normalized size = 1.06 \begin{align*} \frac{1}{3} \, x^{3} + \frac{3}{2} \, x^{2} - 7 \, x \end{align*}

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