3.190 \(\int \frac{-2+3 x+5 x^2}{2 x^2+x^3} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]

[Out]

x^(-1) + 2*Log[x] + 3*Log[2 + x]

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Rubi [A]  time = 0.0233708, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1593, 893} \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[(-2 + 3*x + 5*x^2)/(2*x^2 + x^3),x]

[Out]

x^(-1) + 2*Log[x] + 3*Log[2 + x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

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

Mathematica [A]  time = 0.0036714, size = 14, normalized size = 1. \[ \frac{1}{x}+2 \log (x)+3 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(-2 + 3*x + 5*x^2)/(2*x^2 + x^3),x]

[Out]

x^(-1) + 2*Log[x] + 3*Log[2 + x]

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Maple [A]  time = 0.006, size = 15, normalized size = 1.1 \begin{align*}{x}^{-1}+2\,\ln \left ( x \right ) +3\,\ln \left ( 2+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2+3*x-2)/(x^3+2*x^2),x)

[Out]

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

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Maxima [A]  time = 0.927164, size = 19, normalized size = 1.36 \begin{align*} \frac{1}{x} + 3 \, \log \left (x + 2\right ) + 2 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x-2)/(x^3+2*x^2),x, algorithm="maxima")

[Out]

1/x + 3*log(x + 2) + 2*log(x)

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Fricas [A]  time = 1.74643, size = 50, normalized size = 3.57 \begin{align*} \frac{3 \, x \log \left (x + 2\right ) + 2 \, x \log \left (x\right ) + 1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x-2)/(x^3+2*x^2),x, algorithm="fricas")

[Out]

(3*x*log(x + 2) + 2*x*log(x) + 1)/x

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Sympy [A]  time = 0.100243, size = 14, normalized size = 1. \begin{align*} 2 \log{\left (x \right )} + 3 \log{\left (x + 2 \right )} + \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**2+3*x-2)/(x**3+2*x**2),x)

[Out]

2*log(x) + 3*log(x + 2) + 1/x

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Giac [A]  time = 1.05714, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{x} + 3 \, \log \left ({\left | x + 2 \right |}\right ) + 2 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^2+3*x-2)/(x^3+2*x^2),x, algorithm="giac")

[Out]

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