### 3.2178 $$\int \frac{3+3 x+2 x^2}{(1+x)^3} \, dx$$

Optimal. Leaf size=19 $\frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1)$

[Out]

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

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Rubi [A]  time = 0.0110944, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {698} $\frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 3*x + 2*x^2)/(1 + x)^3,x]

[Out]

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

Rule 698

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*} \int \frac{3+3 x+2 x^2}{(1+x)^3} \, dx &=\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]  time = 0.007844, size = 19, normalized size = 1. $\frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 20, normalized size = 1.1 \begin{align*} - \left ( 1+x \right ) ^{-2}+ \left ( 1+x \right ) ^{-1}+2\,\ln \left ( 1+x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01557, size = 26, normalized size = 1.37 \begin{align*} \frac{x}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.65808, size = 70, normalized size = 3.68 \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + x}{x^{2} + 2 \, x + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.091706, size = 15, normalized size = 0.79 \begin{align*} \frac{x}{x^{2} + 2 x + 1} + 2 \log{\left (x + 1 \right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x**2 + 2*x + 1) + 2*log(x + 1)

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Giac [A]  time = 1.07939, size = 20, normalized size = 1.05 \begin{align*} \frac{x}{{\left (x + 1\right )}^{2}} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/(x + 1)^2 + 2*log(abs(x + 1))