\(\int \frac {x^3}{1-2 x+x^2} \, dx\) [2265]

Optimal result

Integrand size = 14, antiderivative size = 26 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{1-x}+2 x+\frac {x^2}{2}+3 \log (1-x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 45} \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {x^2}{2}+2 x+\frac {1}{1-x}+3 \log (1-x) \]

[In]

Int[x^3/(1 - 2*x + x^2),x]

[Out]

(1 - x)^(-1) + 2*x + x^2/2 + 3*Log[1 - 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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3}{(-1+x)^2} \, dx \\ & = \int \left (2+\frac {1}{(-1+x)^2}+\frac {3}{-1+x}+x\right ) \, dx \\ & = \frac {1}{1-x}+2 x+\frac {x^2}{2}+3 \log (1-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{2} \left (-5-\frac {2}{-1+x}+4 x+x^2+6 \log (-1+x)\right ) \]

[In]

Integrate[x^3/(1 - 2*x + x^2),x]

[Out]

(-5 - 2/(-1 + x) + 4*x + x^2 + 6*Log[-1 + x])/2

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88

method	result	size
default	\(2 x +\frac {x^{2}}{2}-\frac {1}{-1+x}+3 \ln \left (-1+x \right )\)	\(23\)
risch	\(2 x +\frac {x^{2}}{2}-\frac {1}{-1+x}+3 \ln \left (-1+x \right )\)	\(23\)
norman	\(\frac {\frac {3}{2} x^{2}+\frac {1}{2} x^{3}-3}{-1+x}+3 \ln \left (-1+x \right )\)	\(26\)
meijerg	\(\frac {x \left (-2 x^{2}-6 x +12\right )}{4-4 x}+3 \ln \left (1-x \right )\)	\(30\)
parallelrisch	\(\frac {x^{3}+6 \ln \left (-1+x \right ) x +3 x^{2}-6-6 \ln \left (-1+x \right )}{2 x -2}\)	\(31\)

[In]

int(x^3/(x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {x^{3} + 3 \, x^{2} + 6 \, {\left (x - 1\right )} \log \left (x - 1\right ) - 4 \, x - 2}{2 \, {\left (x - 1\right )}} \]

[In]

integrate(x^3/(x^2-2*x+1),x, algorithm="fricas")

[Out]

1/2*(x^3 + 3*x^2 + 6*(x - 1)*log(x - 1) - 4*x - 2)/(x - 1)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {x^{2}}{2} + 2 x + 3 \log {\left (x - 1 \right )} - \frac {1}{x - 1} \]

[In]

integrate(x**3/(x**2-2*x+1),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x - \frac {1}{x - 1} + 3 \, \log \left (x - 1\right ) \]

[In]

integrate(x^3/(x^2-2*x+1),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=\frac {1}{2} \, x^{2} + 2 \, x - \frac {1}{x - 1} + 3 \, \log \left ({\left | x - 1 \right |}\right ) \]

[In]

integrate(x^3/(x^2-2*x+1),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {x^3}{1-2 x+x^2} \, dx=2\,x+3\,\ln \left (x-1\right )-\frac {1}{x-1}+\frac {x^2}{2} \]

[In]

int(x^3/(x^2 - 2*x + 1),x)

[Out]

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