3.20.33 \(\int \frac {x^3}{1+2 x+x^2} \, dx\)

Optimal. Leaf size=22 \[ \frac {x^2}{2}-2 x+\frac {1}{x+1}+3 \log (x+1) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43} \begin {gather*} \frac {x^2}{2}-2 x+\frac {1}{x+1}+3 \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + 2*x + x^2),x]

[Out]

-2*x + x^2/2 + (1 + x)^(-1) + 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 43

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 26, normalized size = 1.18 \begin {gather*} \frac {1}{2} (x+1)^2-3 (x+1)+\frac {1}{x+1}+3 \log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + 2*x + x^2),x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{1+2 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3/(1 + 2*x + x^2),x]

[Out]

IntegrateAlgebraic[x^3/(1 + 2*x + x^2), x]

________________________________________________________________________________________

fricas [A]  time = 0.41, size = 29, normalized size = 1.32 \begin {gather*} \frac {x^{3} - 3 \, x^{2} + 6 \, {\left (x + 1\right )} \log \left (x + 1\right ) - 4 \, x + 2}{2 \, {\left (x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 21, normalized size = 0.95 \begin {gather*} \frac {1}{2} \, x^{2} - 2 \, x + \frac {1}{x + 1} + 3 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.05, size = 21, normalized size = 0.95 \begin {gather*} \frac {x^{2}}{2}-2 x +3 \ln \left (x +1\right )+\frac {1}{x +1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^2+2*x+1),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.76, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, x^{2} - 2 \, x + \frac {1}{x + 1} + 3 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 20, normalized size = 0.91 \begin {gather*} 3\,\ln \left (x+1\right )-2\,x+\frac {1}{x+1}+\frac {x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(2*x + x^2 + 1),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.08, size = 19, normalized size = 0.86 \begin {gather*} \frac {x^{2}}{2} - 2 x + 3 \log {\left (x + 1 \right )} + \frac {1}{x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________