### 3.2176 $$\int \frac{5+4 x+x^2}{-2+x} \, dx$$

Optimal. Leaf size=19 $\frac{x^2}{2}+6 x+17 \log (2-x)$

[Out]

6*x + x^2/2 + 17*Log[2 - x]

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Rubi [A]  time = 0.009546, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.071, Rules used = {698} $\frac{x^2}{2}+6 x+17 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Int[(5 + 4*x + x^2)/(-2 + x),x]

[Out]

6*x + x^2/2 + 17*Log[2 - x]

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{5+4 x+x^2}{-2+x} \, dx &=\int \left (6+\frac{17}{-2+x}+x\right ) \, dx\\ &=6 x+\frac{x^2}{2}+17 \log (2-x)\\ \end{align*}

Mathematica [A]  time = 0.0033058, size = 18, normalized size = 0.95 $\frac{x^2}{2}+6 x+17 \log (x-2)-14$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(5 + 4*x + x^2)/(-2 + x),x]

[Out]

-14 + 6*x + x^2/2 + 17*Log[-2 + x]

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Maple [A]  time = 0.04, size = 16, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+6\,x+17\,\ln \left ( -2+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+4*x+5)/(-2+x),x)

[Out]

1/2*x^2+6*x+17*ln(-2+x)

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Maxima [A]  time = 0.959847, size = 20, normalized size = 1.05 \begin{align*} \frac{1}{2} \, x^{2} + 6 \, x + 17 \, \log \left (x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x+5)/(-2+x),x, algorithm="maxima")

[Out]

1/2*x^2 + 6*x + 17*log(x - 2)

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Fricas [A]  time = 1.92843, size = 42, normalized size = 2.21 \begin{align*} \frac{1}{2} \, x^{2} + 6 \, x + 17 \, \log \left (x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x+5)/(-2+x),x, algorithm="fricas")

[Out]

1/2*x^2 + 6*x + 17*log(x - 2)

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Sympy [A]  time = 0.076976, size = 14, normalized size = 0.74 \begin{align*} \frac{x^{2}}{2} + 6 x + 17 \log{\left (x - 2 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+4*x+5)/(-2+x),x)

[Out]

x**2/2 + 6*x + 17*log(x - 2)

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Giac [A]  time = 1.13047, size = 22, normalized size = 1.16 \begin{align*} \frac{1}{2} \, x^{2} + 6 \, x + 17 \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x+5)/(-2+x),x, algorithm="giac")

[Out]

1/2*x^2 + 6*x + 17*log(abs(x - 2))