3.14.42 \(\int \frac {1960+336 x+12 x^2}{196+28 x+x^2} \, dx\)

Optimal. Leaf size=20 \[ \frac {13}{2}+2 x \left (5+\frac {x}{14+x}\right )-\log (2) \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {27, 683} \begin {gather*} 12 x+\frac {392}{x+14} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1960 + 336*x + 12*x^2)/(196 + 28*x + x^2),x]

[Out]

12*x + 392/(14 + 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 {1960+336 x+12 x^2}{(14+x)^2} \, dx\\ &=\int \left (12-\frac {392}{(14+x)^2}\right ) \, dx\\ &=12 x+\frac {392}{14+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 0.75 \begin {gather*} 4 \left (\frac {98}{14+x}+3 (14+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1960 + 336*x + 12*x^2)/(196 + 28*x + x^2),x]

[Out]

4*(98/(14 + x) + 3*(14 + x))

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fricas [A]  time = 1.23, size = 17, normalized size = 0.85 \begin {gather*} \frac {4 \, {\left (3 \, x^{2} + 42 \, x + 98\right )}}{x + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2+336*x+1960)/(x^2+28*x+196),x, algorithm="fricas")

[Out]

4*(3*x^2 + 42*x + 98)/(x + 14)

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giac [A]  time = 0.30, size = 11, normalized size = 0.55 \begin {gather*} 12 \, x + \frac {392}{x + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2+336*x+1960)/(x^2+28*x+196),x, algorithm="giac")

[Out]

12*x + 392/(x + 14)

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maple [A]  time = 0.26, size = 12, normalized size = 0.60

method	result	size
default	\(12 x +\frac {392}{x +14}\)	\(12\)
risch	\(12 x +\frac {392}{x +14}\)	\(12\)
norman	\(\frac {12 x^{2}-1960}{x +14}\)	\(14\)
gosper	\(\frac {12 x^{2}-1960}{x +14}\)	\(15\)
meijerg	\(-\frac {14 x}{1+\frac {x}{14}}+\frac {4 x \left (\frac {3 x}{14}+6\right )}{1+\frac {x}{14}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^2+336*x+1960)/(x^2+28*x+196),x,method=_RETURNVERBOSE)

[Out]

12*x+392/(x+14)

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maxima [A]  time = 0.33, size = 11, normalized size = 0.55 \begin {gather*} 12 \, x + \frac {392}{x + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^2+336*x+1960)/(x^2+28*x+196),x, algorithm="maxima")

[Out]

12*x + 392/(x + 14)

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mupad [B]  time = 0.03, size = 11, normalized size = 0.55 \begin {gather*} 12\,x+\frac {392}{x+14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((336*x + 12*x^2 + 1960)/(28*x + x^2 + 196),x)

[Out]

12*x + 392/(x + 14)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.35 \begin {gather*} 12 x + \frac {392}{x + 14} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**2+336*x+1960)/(x**2+28*x+196),x)

[Out]

12*x + 392/(x + 14)

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