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

Optimal. Leaf size=60 \[ \frac{3 (2 x+3)}{121 \left (x^2+3 x+5\right )}+\frac{2 x+3}{22 \left (x^2+3 x+5\right )^2}+\frac{12 \tan ^{-1}\left (\frac{2 x+3}{\sqrt{11}}\right )}{121 \sqrt{11}} \]

[Out]

(3 + 2*x)/(22*(5 + 3*x + x^2)^2) + (3*(3 + 2*x))/(121*(5 + 3*x + x^2)) + (12*ArcTan[(3 + 2*x)/Sqrt[11]])/(121*
Sqrt[11])

________________________________________________________________________________________

Rubi [A]  time = 0.02187, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {614, 618, 204} \[ \frac{3 (2 x+3)}{121 \left (x^2+3 x+5\right )}+\frac{2 x+3}{22 \left (x^2+3 x+5\right )^2}+\frac{12 \tan ^{-1}\left (\frac{2 x+3}{\sqrt{11}}\right )}{121 \sqrt{11}} \]

Antiderivative was successfully verified.

[In]

Int[(5 + 3*x + x^2)^(-3),x]

[Out]

(3 + 2*x)/(22*(5 + 3*x + x^2)^2) + (3*(3 + 2*x))/(121*(5 + 3*x + x^2)) + (12*ArcTan[(3 + 2*x)/Sqrt[11]])/(121*
Sqrt[11])

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 618

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

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (5+3 x+x^2\right )^3} \, dx &=\frac{3+2 x}{22 \left (5+3 x+x^2\right )^2}+\frac{3}{11} \int \frac{1}{\left (5+3 x+x^2\right )^2} \, dx\\ &=\frac{3+2 x}{22 \left (5+3 x+x^2\right )^2}+\frac{3 (3+2 x)}{121 \left (5+3 x+x^2\right )}+\frac{6}{121} \int \frac{1}{5+3 x+x^2} \, dx\\ &=\frac{3+2 x}{22 \left (5+3 x+x^2\right )^2}+\frac{3 (3+2 x)}{121 \left (5+3 x+x^2\right )}-\frac{12}{121} \operatorname{Subst}\left (\int \frac{1}{-11-x^2} \, dx,x,3+2 x\right )\\ &=\frac{3+2 x}{22 \left (5+3 x+x^2\right )^2}+\frac{3 (3+2 x)}{121 \left (5+3 x+x^2\right )}+\frac{12 \tan ^{-1}\left (\frac{3+2 x}{\sqrt{11}}\right )}{121 \sqrt{11}}\\ \end{align*}

Mathematica [A]  time = 0.0277894, size = 51, normalized size = 0.85 \[ \frac{\frac{11 (2 x+3) \left (6 x^2+18 x+41\right )}{\left (x^2+3 x+5\right )^2}+24 \sqrt{11} \tan ^{-1}\left (\frac{2 x+3}{\sqrt{11}}\right )}{2662} \]

Antiderivative was successfully verified.

[In]

Integrate[(5 + 3*x + x^2)^(-3),x]

[Out]

((11*(3 + 2*x)*(41 + 18*x + 6*x^2))/(5 + 3*x + x^2)^2 + 24*Sqrt[11]*ArcTan[(3 + 2*x)/Sqrt[11]])/2662

________________________________________________________________________________________

Maple [A]  time = 0.003, size = 52, normalized size = 0.9 \begin{align*}{\frac{3+2\,x}{22\, \left ({x}^{2}+3\,x+5 \right ) ^{2}}}+{\frac{9+6\,x}{121\,{x}^{2}+363\,x+605}}+{\frac{12\,\sqrt{11}}{1331}\arctan \left ({\frac{ \left ( 3+2\,x \right ) \sqrt{11}}{11}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/22*(3+2*x)/(x^2+3*x+5)^2+3/121*(3+2*x)/(x^2+3*x+5)+12/1331*arctan(1/11*(3+2*x)*11^(1/2))*11^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.41801, size = 73, normalized size = 1.22 \begin{align*} \frac{12}{1331} \, \sqrt{11} \arctan \left (\frac{1}{11} \, \sqrt{11}{\left (2 \, x + 3\right )}\right ) + \frac{12 \, x^{3} + 54 \, x^{2} + 136 \, x + 123}{242 \,{\left (x^{4} + 6 \, x^{3} + 19 \, x^{2} + 30 \, x + 25\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12/1331*sqrt(11)*arctan(1/11*sqrt(11)*(2*x + 3)) + 1/242*(12*x^3 + 54*x^2 + 136*x + 123)/(x^4 + 6*x^3 + 19*x^2
+ 30*x + 25)

________________________________________________________________________________________

Fricas [A]  time = 1.97968, size = 216, normalized size = 3.6 \begin{align*} \frac{132 \, x^{3} + 24 \, \sqrt{11}{\left (x^{4} + 6 \, x^{3} + 19 \, x^{2} + 30 \, x + 25\right )} \arctan \left (\frac{1}{11} \, \sqrt{11}{\left (2 \, x + 3\right )}\right ) + 594 \, x^{2} + 1496 \, x + 1353}{2662 \,{\left (x^{4} + 6 \, x^{3} + 19 \, x^{2} + 30 \, x + 25\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2662*(132*x^3 + 24*sqrt(11)*(x^4 + 6*x^3 + 19*x^2 + 30*x + 25)*arctan(1/11*sqrt(11)*(2*x + 3)) + 594*x^2 + 1
496*x + 1353)/(x^4 + 6*x^3 + 19*x^2 + 30*x + 25)

________________________________________________________________________________________

Sympy [A]  time = 0.152225, size = 63, normalized size = 1.05 \begin{align*} \frac{12 x^{3} + 54 x^{2} + 136 x + 123}{242 x^{4} + 1452 x^{3} + 4598 x^{2} + 7260 x + 6050} + \frac{12 \sqrt{11} \operatorname{atan}{\left (\frac{2 \sqrt{11} x}{11} + \frac{3 \sqrt{11}}{11} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(12*x**3 + 54*x**2 + 136*x + 123)/(242*x**4 + 1452*x**3 + 4598*x**2 + 7260*x + 6050) + 12*sqrt(11)*atan(2*sqrt
(11)*x/11 + 3*sqrt(11)/11)/1331

________________________________________________________________________________________

Giac [A]  time = 1.06634, size = 59, normalized size = 0.98 \begin{align*} \frac{12}{1331} \, \sqrt{11} \arctan \left (\frac{1}{11} \, \sqrt{11}{\left (2 \, x + 3\right )}\right ) + \frac{12 \, x^{3} + 54 \, x^{2} + 136 \, x + 123}{242 \,{\left (x^{2} + 3 \, x + 5\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12/1331*sqrt(11)*arctan(1/11*sqrt(11)*(2*x + 3)) + 1/242*(12*x^3 + 54*x^2 + 136*x + 123)/(x^2 + 3*x + 5)^2