3.93 \(\int \frac{1}{(2+5 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=34 \[ -\frac{6 x+5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \]

[Out]

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

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Rubi [A]  time = 0.0080393, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {614, 616, 31} \[ -\frac{6 x+5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x + 3*x^2)^(-2),x]

[Out]

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

Rule 614

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^(p + 1))/((p +
1)*(b^2 - 4*a*c)), x] - Dist[(2*c*(2*p + 3))/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2] && IntegerQ[4*p]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{5+6 x}{2+5 x+3 x^2}-6 \int \frac{1}{2+5 x+3 x^2} \, dx\\ &=-\frac{5+6 x}{2+5 x+3 x^2}-18 \int \frac{1}{2+3 x} \, dx+18 \int \frac{1}{3+3 x} \, dx\\ &=-\frac{5+6 x}{2+5 x+3 x^2}+6 \log (1+x)-6 \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0113213, size = 33, normalized size = 0.97 \[ \frac{-6 x-5}{3 x^2+5 x+2}+6 \log (x+1)-6 \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x + 3*x^2)^(-2),x]

[Out]

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

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Maple [A]  time = 0.056, size = 32, normalized size = 0.9 \begin{align*} - \left ( 1+x \right ) ^{-1}+6\,\ln \left ( 1+x \right ) -3\, \left ( 2+3\,x \right ) ^{-1}-6\,\ln \left ( 2+3\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2+5*x+2)^2,x)

[Out]

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

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Maxima [A]  time = 1.90578, size = 46, normalized size = 1.35 \begin{align*} -\frac{6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} - 6 \, \log \left (3 \, x + 2\right ) + 6 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+5*x+2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.15157, size = 132, normalized size = 3.88 \begin{align*} -\frac{6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 6 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+5*x+2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.266646, size = 29, normalized size = 0.85 \begin{align*} - \frac{6 x + 5}{3 x^{2} + 5 x + 2} - 6 \log{\left (x + \frac{2}{3} \right )} + 6 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2+5*x+2)**2,x)

[Out]

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

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Giac [A]  time = 1.27257, size = 49, normalized size = 1.44 \begin{align*} -\frac{6 \, x + 5}{3 \, x^{2} + 5 \, x + 2} - 6 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 6 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+5*x+2)^2,x, algorithm="giac")

[Out]

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