3.56.98 \(\int \frac {17+12 x+x^2}{36+12 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ 5+e^5+x+\frac {7-\log \left (e^{2 x}\right )}{6+x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.41, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {27, 683} \begin {gather*} x+\frac {19}{x+6} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(17 + 12*x + x^2)/(36 + 12*x + x^2),x]

[Out]

x + 19/(6 + x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {17+12 x+x^2}{(6+x)^2} \, dx\\ &=\int \left (1-\frac {19}{(6+x)^2}\right ) \, dx\\ &=x+\frac {19}{6+x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 9, normalized size = 0.41 \begin {gather*} x+\frac {19}{6+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(17 + 12*x + x^2)/(36 + 12*x + x^2),x]

[Out]

x + 19/(6 + x)

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 14, normalized size = 0.64 \begin {gather*} \frac {x^{2} + 6 \, x + 19}{x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+12*x+17)/(x^2+12*x+36),x, algorithm="fricas")

[Out]

(x^2 + 6*x + 19)/(x + 6)

________________________________________________________________________________________

giac [A]  time = 0.22, size = 9, normalized size = 0.41 \begin {gather*} x + \frac {19}{x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+12*x+17)/(x^2+12*x+36),x, algorithm="giac")

[Out]

x + 19/(x + 6)

________________________________________________________________________________________

maple [A]  time = 0.31, size = 10, normalized size = 0.45

method	result	size
default	\(x +\frac {19}{x +6}\)	\(10\)
risch	\(x +\frac {19}{x +6}\)	\(10\)
gosper	\(\frac {x^{2}-17}{x +6}\)	\(12\)
norman	\(\frac {x^{2}-17}{x +6}\)	\(12\)
meijerg	\(-\frac {55 x}{36 \left (1+\frac {x}{6}\right )}+\frac {x \left (\frac {x}{2}+6\right )}{\frac {x}{2}+3}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+12*x+17)/(x^2+12*x+36),x,method=_RETURNVERBOSE)

[Out]

x+19/(x+6)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 9, normalized size = 0.41 \begin {gather*} x + \frac {19}{x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+12*x+17)/(x^2+12*x+36),x, algorithm="maxima")

[Out]

x + 19/(x + 6)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 9, normalized size = 0.41 \begin {gather*} x+\frac {19}{x+6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + x^2 + 17)/(12*x + x^2 + 36),x)

[Out]

x + 19/(x + 6)

________________________________________________________________________________________

sympy [A]  time = 0.07, size = 5, normalized size = 0.23 \begin {gather*} x + \frac {19}{x + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+12*x+17)/(x**2+12*x+36),x)

[Out]

x + 19/(x + 6)

________________________________________________________________________________________