\(\int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx\) [2178]

Optimal result

Integrand size = 16, antiderivative size = 19 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=-\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {712} \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {1}{x+1}-\frac {1}{(x+1)^2}+2 \log (x+1) \]

[In]

Int[(3 + 3*x + 2*x^2)/(1 + x)^3,x]

[Out]

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

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=-\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 20.70 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79

method	result	size
norman	\(\frac {x}{\left (1+x \right )^{2}}+2 \ln \left (1+x \right )\)	\(15\)
risch	\(\frac {x}{\left (1+x \right )^{2}}+2 \ln \left (1+x \right )\)	\(15\)
default	\(-\frac {1}{\left (1+x \right )^{2}}+\frac {1}{1+x}+2 \ln \left (1+x \right )\)	\(20\)
parallelrisch	\(\frac {2 \ln \left (1+x \right ) x^{2}+4 \ln \left (1+x \right ) x +2 \ln \left (1+x \right )+x}{\left (1+x \right )^{2}}\)	\(31\)
meijerg	\(\frac {3 x \left (2+x \right )}{2 \left (1+x \right )^{2}}-\frac {x \left (9 x +6\right )}{3 \left (1+x \right )^{2}}+2 \ln \left (1+x \right )+\frac {3 x^{2}}{2 \left (1+x \right )^{2}}\)	\(42\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + x}{x^{2} + 2 \, x + 1} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {x}{x^{2} + 2 x + 1} + 2 \log {\left (x + 1 \right )} \]

[In]

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

[Out]

x/(x**2 + 2*x + 1) + 2*log(x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {x}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right ) \]

[In]

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

[Out]

x/(x^2 + 2*x + 1) + 2*log(x + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=\frac {x}{{\left (x + 1\right )}^{2}} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \]

[In]

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

[Out]

x/(x + 1)^2 + 2*log(abs(x + 1))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \frac {3+3 x+2 x^2}{(1+x)^3} \, dx=2\,\ln \left (x+1\right )+\frac {x}{{\left (x+1\right )}^2} \]

[In]

int((3*x + 2*x^2 + 3)/(x + 1)^3,x)

[Out]

2*log(x + 1) + x/(x + 1)^2