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

Optimal. Leaf size=42 \[ -\frac{5-6 x}{49 \left (-3 x^2+5 x+2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (3 x+1) \]

[Out]

-(5 - 6*x)/(49*(2 + 5*x - 3*x^2)) - (6*Log[2 - x])/343 + (6*Log[1 + 3*x])/343

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Rubi [A]  time = 0.0090168, antiderivative size = 42, 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} \[ -\frac{5-6 x}{49 \left (-3 x^2+5 x+2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (3 x+1) \]

Antiderivative was successfully verified.

[In]

Int[(2 + 5*x - 3*x^2)^(-2),x]

[Out]

-(5 - 6*x)/(49*(2 + 5*x - 3*x^2)) - (6*Log[2 - x])/343 + (6*Log[1 + 3*x])/343

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}{49 \left (2+5 x-3 x^2\right )}+\frac{6}{49} \int \frac{1}{2+5 x-3 x^2} \, dx\\ &=-\frac{5-6 x}{49 \left (2+5 x-3 x^2\right )}-\frac{18}{343} \int \frac{1}{-1-3 x} \, dx+\frac{18}{343} \int \frac{1}{6-3 x} \, dx\\ &=-\frac{5-6 x}{49 \left (2+5 x-3 x^2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (1+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0127719, size = 42, normalized size = 1. \[ \frac{5-6 x}{49 \left (3 x^2-5 x-2\right )}-\frac{6}{343} \log (2-x)+\frac{6}{343} \log (3 x+1) \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 5*x - 3*x^2)^(-2),x]

[Out]

(5 - 6*x)/(49*(-2 - 5*x + 3*x^2)) - (6*Log[2 - x])/343 + (6*Log[1 + 3*x])/343

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Maple [A]  time = 0.056, size = 32, normalized size = 0.8 \begin{align*} -{\frac{3}{49+147\,x}}+{\frac{6\,\ln \left ( 1+3\,x \right ) }{343}}-{\frac{1}{-98+49\,x}}-{\frac{6\,\ln \left ( -2+x \right ) }{343}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/49/(1+3*x)+6/343*ln(1+3*x)-1/49/(-2+x)-6/343*ln(-2+x)

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Maxima [A]  time = 1.18605, size = 46, normalized size = 1.1 \begin{align*} -\frac{6 \, x - 5}{49 \,{\left (3 \, x^{2} - 5 \, x - 2\right )}} + \frac{6}{343} \, \log \left (3 \, x + 1\right ) - \frac{6}{343} \, \log \left (x - 2\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]

-1/49*(6*x - 5)/(3*x^2 - 5*x - 2) + 6/343*log(3*x + 1) - 6/343*log(x - 2)

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Fricas [A]  time = 2.05154, size = 142, normalized size = 3.38 \begin{align*} \frac{6 \,{\left (3 \, x^{2} - 5 \, x - 2\right )} \log \left (3 \, x + 1\right ) - 6 \,{\left (3 \, x^{2} - 5 \, x - 2\right )} \log \left (x - 2\right ) - 42 \, x + 35}{343 \,{\left (3 \, x^{2} - 5 \, x - 2\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="fricas")

[Out]

1/343*(6*(3*x^2 - 5*x - 2)*log(3*x + 1) - 6*(3*x^2 - 5*x - 2)*log(x - 2) - 42*x + 35)/(3*x^2 - 5*x - 2)

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Sympy [A]  time = 0.191956, size = 32, normalized size = 0.76 \begin{align*} - \frac{6 x - 5}{147 x^{2} - 245 x - 98} - \frac{6 \log{\left (x - 2 \right )}}{343} + \frac{6 \log{\left (x + \frac{1}{3} \right )}}{343} \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)/(147*x**2 - 245*x - 98) - 6*log(x - 2)/343 + 6*log(x + 1/3)/343

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Giac [A]  time = 1.23187, size = 49, normalized size = 1.17 \begin{align*} -\frac{6 \, x - 5}{49 \,{\left (3 \, x^{2} - 5 \, x - 2\right )}} + \frac{6}{343} \, \log \left ({\left | 3 \, x + 1 \right |}\right ) - \frac{6}{343} \, \log \left ({\left | x - 2 \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]

-1/49*(6*x - 5)/(3*x^2 - 5*x - 2) + 6/343*log(abs(3*x + 1)) - 6/343*log(abs(x - 2))