3.2390 \(\int \frac{(5-x) (3+2 x)}{(2+5 x+3 x^2)^2} \, dx\)

Optimal. Leaf size=36 \[ -\frac{139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \]

[Out]

-(121 + 139*x)/(3*(2 + 5*x + 3*x^2)) + 47*Log[1 + x] - 47*Log[2 + 3*x]

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Rubi [A]  time = 0.0145328, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {777, 616, 31} \[ -\frac{139 x+121}{3 \left (3 x^2+5 x+2\right )}+47 \log (x+1)-47 \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(121 + 139*x)/(3*(2 + 5*x + 3*x^2)) + 47*Log[1 + x] - 47*Log[2 + 3*x]

Rule 777

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x)*(a + b*x + c*x^2)^
(p + 1))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p
+ 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && N
eQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

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{(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^2} \, dx &=-\frac{121+139 x}{3 \left (2+5 x+3 x^2\right )}-47 \int \frac{1}{2+5 x+3 x^2} \, dx\\ &=-\frac{121+139 x}{3 \left (2+5 x+3 x^2\right )}-141 \int \frac{1}{2+3 x} \, dx+141 \int \frac{1}{3+3 x} \, dx\\ &=-\frac{121+139 x}{3 \left (2+5 x+3 x^2\right )}+47 \log (1+x)-47 \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0143276, size = 34, normalized size = 0.94 \[ -\frac{139 x+121}{9 x^2+15 x+6}+47 \log (x+1)-47 \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((121 + 139*x)/(6 + 15*x + 9*x^2)) + 47*Log[1 + x] - 47*Log[2 + 3*x]

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Maple [A]  time = 0.009, size = 32, normalized size = 0.9 \begin{align*} -6\, \left ( 1+x \right ) ^{-1}+47\,\ln \left ( 1+x \right ) -{\frac{85}{6+9\,x}}-47\,\ln \left ( 2+3\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6/(1+x)+47*ln(1+x)-85/3/(2+3*x)-47*ln(2+3*x)

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Maxima [A]  time = 1.21612, size = 46, normalized size = 1.28 \begin{align*} -\frac{139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - 47 \, \log \left (3 \, x + 2\right ) + 47 \, \log \left (x + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(139*x + 121)/(3*x^2 + 5*x + 2) - 47*log(3*x + 2) + 47*log(x + 1)

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Fricas [A]  time = 1.21872, size = 149, normalized size = 4.14 \begin{align*} -\frac{141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 141 \,{\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (x + 1\right ) + 139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(141*(3*x^2 + 5*x + 2)*log(3*x + 2) - 141*(3*x^2 + 5*x + 2)*log(x + 1) + 139*x + 121)/(3*x^2 + 5*x + 2)

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Sympy [A]  time = 0.142315, size = 29, normalized size = 0.81 \begin{align*} - \frac{139 x + 121}{9 x^{2} + 15 x + 6} - 47 \log{\left (x + \frac{2}{3} \right )} + 47 \log{\left (x + 1 \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(139*x + 121)/(9*x**2 + 15*x + 6) - 47*log(x + 2/3) + 47*log(x + 1)

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Giac [A]  time = 1.13793, size = 49, normalized size = 1.36 \begin{align*} -\frac{139 \, x + 121}{3 \,{\left (3 \, x^{2} + 5 \, x + 2\right )}} - 47 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + 47 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(139*x + 121)/(3*x^2 + 5*x + 2) - 47*log(abs(3*x + 2)) + 47*log(abs(x + 1))