### 3.2263 $$\int \frac{x^3}{2-3 x+x^2} \, dx$$

Optimal. Leaf size=27 $\frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x)$

[Out]

3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]

________________________________________________________________________________________

Rubi [A]  time = 0.0121766, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.214, Rules used = {701, 632, 31} $\frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3/(2 - 3*x + x^2),x]

[Out]

3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]

Rule 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, 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] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

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^3}{2-3 x+x^2} \, dx &=\int \left (3+x-\frac{6-7 x}{2-3 x+x^2}\right ) \, dx\\ &=3 x+\frac{x^2}{2}-\int \frac{6-7 x}{2-3 x+x^2} \, dx\\ &=3 x+\frac{x^2}{2}+8 \int \frac{1}{-2+x} \, dx-\int \frac{1}{-1+x} \, dx\\ &=3 x+\frac{x^2}{2}-\log (1-x)+8 \log (2-x)\\ \end{align*}

Mathematica [A]  time = 0.0035987, size = 27, normalized size = 1. $\frac{x^2}{2}+3 x-\log (1-x)+8 \log (2-x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3/(2 - 3*x + x^2),x]

[Out]

3*x + x^2/2 - Log[1 - x] + 8*Log[2 - x]

________________________________________________________________________________________

Maple [A]  time = 0.043, size = 22, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{2}}+3\,x-\ln \left ( -1+x \right ) +8\,\ln \left ( -2+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2+3*x-ln(-1+x)+8*ln(-2+x)

________________________________________________________________________________________

Maxima [A]  time = 0.974202, size = 28, normalized size = 1.04 \begin{align*} \frac{1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 3*x - log(x - 1) + 8*log(x - 2)

________________________________________________________________________________________

Fricas [A]  time = 2.2626, size = 58, normalized size = 2.15 \begin{align*} \frac{1}{2} \, x^{2} + 3 \, x - \log \left (x - 1\right ) + 8 \, \log \left (x - 2\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 3*x - log(x - 1) + 8*log(x - 2)

________________________________________________________________________________________

Sympy [A]  time = 0.095846, size = 19, normalized size = 0.7 \begin{align*} \frac{x^{2}}{2} + 3 x + 8 \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**3/(x**2-3*x+2),x)

[Out]

x**2/2 + 3*x + 8*log(x - 2) - log(x - 1)

________________________________________________________________________________________

Giac [A]  time = 1.09396, size = 31, normalized size = 1.15 \begin{align*} \frac{1}{2} \, x^{2} + 3 \, x - \log \left ({\left | x - 1 \right |}\right ) + 8 \, \log \left ({\left | x - 2 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^2 + 3*x - log(abs(x - 1)) + 8*log(abs(x - 2))