\(\int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx\) [484]

Optimal result

Integrand size = 18, antiderivative size = 23 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=-\frac {5}{3 (1+x)^3}+\frac {3}{1+x}+2 \log (1+x) \]

[Out]

-5/3/(1+x)^3+3/(1+x)+2*ln(1+x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1864} \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=\frac {3}{x+1}-\frac {5}{3 (x+1)^3}+2 \log (x+1) \]

[In]

Int[(4 + 3*x^2 + 2*x^3)/(1 + x)^4,x]

[Out]

-5/(3*(1 + x)^3) + 3/(1 + x) + 2*Log[1 + x]

Rule 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {5}{(1+x)^4}-\frac {3}{(1+x)^2}+\frac {2}{1+x}\right ) \, dx \\ & = -\frac {5}{3 (1+x)^3}+\frac {3}{1+x}+2 \log (1+x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=-\frac {5}{3 (1+x)^3}+\frac {3}{1+x}+2 \log (1+x) \]

[In]

Integrate[(4 + 3*x^2 + 2*x^3)/(1 + x)^4,x]

[Out]

-5/(3*(1 + x)^3) + 3/(1 + x) + 2*Log[1 + x]

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96

method	result	size
default	\(-\frac {5}{3 \left (x +1\right )^{3}}+\frac {3}{x +1}+2 \ln \left (x +1\right )\)	\(22\)
norman	\(\frac {3 x^{2}+6 x +\frac {4}{3}}{\left (x +1\right )^{3}}+2 \ln \left (x +1\right )\)	\(24\)
risch	\(\frac {3 x^{2}+6 x +\frac {4}{3}}{\left (x +1\right )^{3}}+2 \ln \left (x +1\right )\)	\(24\)
parallelrisch	\(\frac {6 \ln \left (x +1\right ) x^{3}+4+18 \ln \left (x +1\right ) x^{2}+18 \ln \left (x +1\right ) x +9 x^{2}+6 \ln \left (x +1\right )+18 x}{3 \left (x +1\right )^{3}}\)	\(49\)
meijerg	\(\frac {4 x \left (x^{2}+3 x +3\right )}{3 \left (x +1\right )^{3}}-\frac {x \left (22 x^{2}+30 x +12\right )}{6 \left (x +1\right )^{3}}+2 \ln \left (x +1\right )+\frac {x^{3}}{\left (x +1\right )^{3}}\)	\(51\)

[In]

int((2*x^3+3*x^2+4)/(x+1)^4,x,method=_RETURNVERBOSE)

[Out]

-5/3/(x+1)^3+3/(x+1)+2*ln(x+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).

Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.00 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=\frac {9 \, x^{2} + 6 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 18 \, x + 4}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]

[In]

integrate((2*x^3+3*x^2+4)/(1+x)^4,x, algorithm="fricas")

[Out]

1/3*(9*x^2 + 6*(x^3 + 3*x^2 + 3*x + 1)*log(x + 1) + 18*x + 4)/(x^3 + 3*x^2 + 3*x + 1)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=\frac {9 x^{2} + 18 x + 4}{3 x^{3} + 9 x^{2} + 9 x + 3} + 2 \log {\left (x + 1 \right )} \]

[In]

integrate((2*x**3+3*x**2+4)/(1+x)**4,x)

[Out]

(9*x**2 + 18*x + 4)/(3*x**3 + 9*x**2 + 9*x + 3) + 2*log(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=\frac {9 \, x^{2} + 18 \, x + 4}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + 2 \, \log \left (x + 1\right ) \]

[In]

integrate((2*x^3+3*x^2+4)/(1+x)^4,x, algorithm="maxima")

[Out]

1/3*(9*x^2 + 18*x + 4)/(x^3 + 3*x^2 + 3*x + 1) + 2*log(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=\frac {9 \, x^{2} + 18 \, x + 4}{3 \, {\left (x + 1\right )}^{3}} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

integrate((2*x^3+3*x^2+4)/(1+x)^4,x, algorithm="giac")

[Out]

1/3*(9*x^2 + 18*x + 4)/(x + 1)^3 + 2*log(abs(x + 1))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx=2\,\ln \left (x+1\right )+\frac {3\,x^2+6\,x+\frac {4}{3}}{{\left (x+1\right )}^3} \]

[In]

int((3*x^2 + 2*x^3 + 4)/(x + 1)^4,x)

[Out]

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