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

Optimal. Leaf size=27 \[ -6 \log (x+1)+\frac {13}{5} \log (2 x+3)+\frac {17}{5} \log (3 x+2) \]

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {800} \begin {gather*} -6 \log (x+1)+\frac {13}{5} \log (2 x+3)+\frac {17}{5} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {5-x}{(3+2 x) \left (2+5 x+3 x^2\right )} \, dx &=\int \left (-\frac {6}{1+x}+\frac {26}{5 (3+2 x)}+\frac {51}{5 (2+3 x)}\right ) \, dx\\ &=-6 \log (1+x)+\frac {13}{5} \log (3+2 x)+\frac {17}{5} \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \begin {gather*} -6 \log (x+1)+\frac {13}{5} \log (2 x+3)+\frac {17}{5} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-6*Log[1 + x] + (13*Log[3 + 2*x])/5 + (17*Log[2 + 3*x])/5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {5-x}{(3+2 x) \left (2+5 x+3 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.38, size = 23, normalized size = 0.85 \begin {gather*} \frac {17}{5} \, \log \left (3 \, x + 2\right ) + \frac {13}{5} \, \log \left (2 \, x + 3\right ) - 6 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17/5*log(3*x + 2) + 13/5*log(2*x + 3) - 6*log(x + 1)

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giac [A]  time = 0.15, size = 26, normalized size = 0.96 \begin {gather*} \frac {17}{5} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {13}{5} \, \log \left ({\left | 2 \, x + 3 \right |}\right ) - 6 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17/5*log(abs(3*x + 2)) + 13/5*log(abs(2*x + 3)) - 6*log(abs(x + 1))

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maple [A]  time = 0.05, size = 24, normalized size = 0.89 \begin {gather*} \frac {17 \ln \left (3 x +2\right )}{5}+\frac {13 \ln \left (2 x +3\right )}{5}-6 \ln \left (x +1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*ln(x+1)+13/5*ln(2*x+3)+17/5*ln(3*x+2)

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maxima [A]  time = 0.55, size = 23, normalized size = 0.85 \begin {gather*} \frac {17}{5} \, \log \left (3 \, x + 2\right ) + \frac {13}{5} \, \log \left (2 \, x + 3\right ) - 6 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17/5*log(3*x + 2) + 13/5*log(2*x + 3) - 6*log(x + 1)

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mupad [B]  time = 0.05, size = 19, normalized size = 0.70 \begin {gather*} \frac {17\,\ln \left (x+\frac {2}{3}\right )}{5}-6\,\ln \left (x+1\right )+\frac {13\,\ln \left (x+\frac {3}{2}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(17*log(x + 2/3))/5 - 6*log(x + 1) + (13*log(x + 3/2))/5

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sympy [A]  time = 0.14, size = 26, normalized size = 0.96 \begin {gather*} \frac {17 \log {\left (x + \frac {2}{3} \right )}}{5} - 6 \log {\left (x + 1 \right )} + \frac {13 \log {\left (x + \frac {3}{2} \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17*log(x + 2/3)/5 - 6*log(x + 1) + 13*log(x + 3/2)/5

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