### 3.2260 $$\int \frac{x^2}{2-3 x+x^2} \, dx$$

Optimal. Leaf size=18 $x-\log (1-x)+4 \log (2-x)$

[Out]

x - Log[1 - x] + 4*Log[2 - x]

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Rubi [A]  time = 0.0079984, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.214, Rules used = {703, 632, 31} $x-\log (1-x)+4 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2/(2 - 3*x + x^2),x]

[Out]

x - Log[1 - x] + 4*Log[2 - x]

Rule 703

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1))/(c*
(m - 1)), x] + Dist[1/c, Int[((d + e*x)^(m - 2)*Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, x])/(a + b*x + c*x^2),
x], x] /; FreeQ[{a, b, c, d, e}, 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] && GtQ[m, 1]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^2}{2-3 x+x^2} \, dx &=x+\int \frac{-2+3 x}{2-3 x+x^2} \, dx\\ &=x+4 \int \frac{1}{-2+x} \, dx-\int \frac{1}{-1+x} \, dx\\ &=x-\log (1-x)+4 \log (2-x)\\ \end{align*}

Mathematica [A]  time = 0.0034947, size = 18, normalized size = 1. $x-\log (1-x)+4 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^2/(2 - 3*x + x^2),x]

[Out]

x - Log[1 - x] + 4*Log[2 - x]

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Maple [A]  time = 0.043, size = 15, normalized size = 0.8 \begin{align*} x-\ln \left ( -1+x \right ) +4\,\ln \left ( -2+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-ln(-1+x)+4*ln(-2+x)

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Maxima [A]  time = 0.955964, size = 19, normalized size = 1.06 \begin{align*} x - \log \left (x - 1\right ) + 4 \, \log \left (x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^2-3*x+2),x, algorithm="maxima")

[Out]

x - log(x - 1) + 4*log(x - 2)

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Fricas [A]  time = 2.25181, size = 42, normalized size = 2.33 \begin{align*} x - \log \left (x - 1\right ) + 4 \, \log \left (x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x - log(x - 1) + 4*log(x - 2)

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Sympy [A]  time = 0.094126, size = 12, normalized size = 0.67 \begin{align*} x + 4 \log{\left (x - 2 \right )} - \log{\left (x - 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**2-3*x+2),x)

[Out]

x + 4*log(x - 2) - log(x - 1)

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Giac [A]  time = 1.0922, size = 22, normalized size = 1.22 \begin{align*} x - \log \left ({\left | x - 1 \right |}\right ) + 4 \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^2-3*x+2),x, algorithm="giac")

[Out]

x - log(abs(x - 1)) + 4*log(abs(x - 2))