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

Optimal. Leaf size=23 \[ \frac {3}{x+1}-\frac {5}{3 (x+1)^3}+2 \log (x+1) \]

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {1850} \[ \frac {3}{x+1}-\frac {5}{3 (x+1)^3}+2 \log (x+1) \]

Antiderivative was successfully verified.

[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 1850

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*} \int \frac {4+3 x^2+2 x^3}{(1+x)^4} \, dx &=\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*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.00 \[ \frac {3}{x+1}-\frac {5}{3 (x+1)^3}+2 \log (x+1) \]

Antiderivative was successfully verified.

[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]

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fricas [B]  time = 0.56, size = 46, normalized size = 2.00 \[ \frac {9 \, x^{2} + 6 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (x + 1\right ) + 18 \, x + 4}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.37, size = 25, normalized size = 1.09 \[ \frac {9 \, x^{2} + 18 \, x + 4}{3 \, {\left (x + 1\right )}^{3}} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.01, size = 22, normalized size = 0.96 \[ 2 \ln \left (x +1\right )-\frac {5}{3 \left (x +1\right )^{3}}+\frac {3}{x +1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.74, size = 34, normalized size = 1.48 \[ \frac {9 \, x^{2} + 18 \, x + 4}{3 \, {\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} + 2 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.03, size = 23, normalized size = 1.00 \[ 2\,\ln \left (x+1\right )+\frac {3\,x^2+6\,x+\frac {4}{3}}{{\left (x+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.11, size = 31, normalized size = 1.35 \[ \frac {9 x^{2} + 18 x + 4}{3 x^{3} + 9 x^{2} + 9 x + 3} + 2 \log {\left (x + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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