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

Optimal. Leaf size=57 \[ \frac {421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac {139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \]

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Rubi [A]  time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {777, 614, 616, 31} \begin {gather*} \frac {421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac {139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(121 + 139*x)/(6*(2 + 5*x + 3*x^2)^2) + (421*(5 + 6*x))/(6*(2 + 5*x + 3*x^2)) - 421*Log[1 + x] + 421*Log[2 +
3*x]

Rule 31

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

Rubi steps

\begin {align*} \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^3} \, dx &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}-\frac {421}{6} \int \frac {1}{\left (2+5 x+3 x^2\right )^2} \, dx\\ &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}+\frac {421 (5+6 x)}{6 \left (2+5 x+3 x^2\right )}+421 \int \frac {1}{2+5 x+3 x^2} \, dx\\ &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}+\frac {421 (5+6 x)}{6 \left (2+5 x+3 x^2\right )}+1263 \int \frac {1}{2+3 x} \, dx-1263 \int \frac {1}{3+3 x} \, dx\\ &=-\frac {121+139 x}{6 \left (2+5 x+3 x^2\right )^2}+\frac {421 (5+6 x)}{6 \left (2+5 x+3 x^2\right )}-421 \log (1+x)+421 \log (2+3 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 57, normalized size = 1.00 \begin {gather*} \frac {421 (6 x+5)}{6 \left (3 x^2+5 x+2\right )}-\frac {139 x+121}{6 \left (3 x^2+5 x+2\right )^2}-421 \log (x+1)+421 \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(121 + 139*x)/(2 + 5*x + 3*x^2)^2 + (421*(5 + 6*x))/(6*(2 + 5*x + 3*x^2)) - 421*Log[1 + x] + 421*Log[2 +
3*x]

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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 )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.37, size = 93, normalized size = 1.63 \begin {gather*} \frac {2526 \, x^{3} + 6315 \, x^{2} + 842 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 842 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (x + 1\right ) + 5146 \, x + 1363}{2 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\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)^3,x, algorithm="fricas")

[Out]

1/2*(2526*x^3 + 6315*x^2 + 842*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(3*x + 2) - 842*(9*x^4 + 30*x^3 + 37*x^
2 + 20*x + 4)*log(x + 1) + 5146*x + 1363)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)

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giac [A]  time = 0.20, size = 46, normalized size = 0.81 \begin {gather*} \frac {2526 \, x^{3} + 6315 \, x^{2} + 5146 \, x + 1363}{2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{2}} + 421 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - 421 \, \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)^3,x, algorithm="giac")

[Out]

1/2*(2526*x^3 + 6315*x^2 + 5146*x + 1363)/(3*x^2 + 5*x + 2)^2 + 421*log(abs(3*x + 2)) - 421*log(abs(x + 1))

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maple [A]  time = 0.05, size = 48, normalized size = 0.84 \begin {gather*} 421 \ln \left (3 x +2\right )-421 \ln \left (x +1\right )-\frac {85}{2 \left (3 x +2\right )^{2}}+\frac {226}{3 x +2}+\frac {3}{\left (x +1\right )^{2}}+\frac {65}{x +1} \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)^3,x)

[Out]

-85/2/(3*x+2)^2+226/(3*x+2)+421*ln(3*x+2)+3/(x+1)^2+65/(x+1)-421*ln(x+1)

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maxima [A]  time = 0.48, size = 54, normalized size = 0.95 \begin {gather*} \frac {2526 \, x^{3} + 6315 \, x^{2} + 5146 \, x + 1363}{2 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} + 421 \, \log \left (3 \, x + 2\right ) - 421 \, \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)^3,x, algorithm="maxima")

[Out]

1/2*(2526*x^3 + 6315*x^2 + 5146*x + 1363)/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4) + 421*log(3*x + 2) - 421*log(x
+ 1)

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mupad [B]  time = 2.30, size = 45, normalized size = 0.79 \begin {gather*} \frac {\frac {421\,x^3}{3}+\frac {2105\,x^2}{6}+\frac {2573\,x}{9}+\frac {1363}{18}}{x^4+\frac {10\,x^3}{3}+\frac {37\,x^2}{9}+\frac {20\,x}{9}+\frac {4}{9}}-842\,\mathrm {atanh}\left (6\,x+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2573*x)/9 + (2105*x^2)/6 + (421*x^3)/3 + 1363/18)/((20*x)/9 + (37*x^2)/9 + (10*x^3)/3 + x^4 + 4/9) - 842*ata
nh(6*x + 5)

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sympy [A]  time = 0.16, size = 51, normalized size = 0.89 \begin {gather*} - \frac {- 2526 x^{3} - 6315 x^{2} - 5146 x - 1363}{18 x^{4} + 60 x^{3} + 74 x^{2} + 40 x + 8} + 421 \log {\left (x + \frac {2}{3} \right )} - 421 \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)**3,x)

[Out]

-(-2526*x**3 - 6315*x**2 - 5146*x - 1363)/(18*x**4 + 60*x**3 + 74*x**2 + 40*x + 8) + 421*log(x + 2/3) - 421*lo
g(x + 1)

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