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

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

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Rubi [A]  time = 0.02, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 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])

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]

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*}

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Mathematica [A]  time = 0.03, 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

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IntegrateAlgebraic [A]  time = 0.04, size = 56, normalized size = 0.93 \[ \frac {(2 x+3) \left (6 x^2+18 x+41\right )}{242 \left (x^2+3 x+5\right )^2}+\frac {12 \tan ^{-1}\left (\frac {2 x}{\sqrt {11}}+\frac {3}{\sqrt {11}}\right )}{121 \sqrt {11}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 2*x)*(41 + 18*x + 6*x^2))/(242*(5 + 3*x + x^2)^2) + (12*ArcTan[3/Sqrt[11] + (2*x)/Sqrt[11]])/(121*Sqrt[1
1])

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fricas [A]  time = 1.08, size = 71, normalized size = 1.18 \[ \frac {132 \, x^{3} + 24 \, \sqrt {11} {\left (x^{4} + 6 \, x^{3} + 19 \, x^{2} + 30 \, x + 25\right )} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 3\right )}\right ) + 594 \, x^{2} + 1496 \, x + 1353}{2662 \, {\left (x^{4} + 6 \, x^{3} + 19 \, x^{2} + 30 \, x + 25\right )}} \]

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)

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giac [A]  time = 1.05, size = 44, normalized size = 0.73 \[ \frac {12}{1331} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 3\right )}\right ) + \frac {12 \, x^{3} + 54 \, x^{2} + 136 \, x + 123}{242 \, {\left (x^{2} + 3 \, x + 5\right )}^{2}} \]

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

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maple [A]  time = 0.43, size = 44, normalized size = 0.73

method	result	size
risch	\(\frac {\frac {6}{121} x^{3}+\frac {27}{121} x^{2}+\frac {68}{121} x +\frac {123}{242}}{\left (x^{2}+3 x +5\right )^{2}}+\frac {12 \arctan \left (\frac {\left (3+2 x \right ) \sqrt {11}}{11}\right ) \sqrt {11}}{1331}\)	\(44\)
default	\(\frac {3+2 x}{22 \left (x^{2}+3 x +5\right )^{2}}+\frac {\frac {9}{121}+\frac {6 x}{121}}{x^{2}+3 x +5}+\frac {12 \arctan \left (\frac {\left (3+2 x \right ) \sqrt {11}}{11}\right ) \sqrt {11}}{1331}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(6/121*x^3+27/121*x^2+68/121*x+123/242)/(x^2+3*x+5)^2+12/1331*arctan(1/11*(3+2*x)*11^(1/2))*11^(1/2)

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maxima [A]  time = 0.96, size = 54, normalized size = 0.90 \[ \frac {12}{1331} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (2 \, x + 3\right )}\right ) + \frac {12 \, x^{3} + 54 \, x^{2} + 136 \, x + 123}{242 \, {\left (x^{4} + 6 \, x^{3} + 19 \, x^{2} + 30 \, x + 25\right )}} \]

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)

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mupad [B]  time = 0.08, size = 45, normalized size = 0.75 \[ 6\,\left (x+\frac {3}{2}\right )\,\left (\frac {1}{121\,\left (x^2+3\,x+5\right )}+\frac {1}{66\,{\left (x^2+3\,x+5\right )}^2}\right )+\frac {12\,\sqrt {11}\,\mathrm {atan}\left (\frac {2\,\sqrt {11}\,\left (x+\frac {3}{2}\right )}{11}\right )}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x + x^2 + 5)^3,x)

[Out]

6*(x + 3/2)*(1/(121*(3*x + x^2 + 5)) + 1/(66*(3*x + x^2 + 5)^2)) + (12*11^(1/2)*atan((2*11^(1/2)*(x + 3/2))/11
))/1331

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sympy [A]  time = 0.18, size = 63, normalized size = 1.05 \[ \frac {12 x^{3} + 54 x^{2} + 136 x + 123}{242 x^{4} + 1452 x^{3} + 4598 x^{2} + 7260 x + 6050} + \frac {12 \sqrt {11} \operatorname {atan}{\left (\frac {2 \sqrt {11} x}{11} + \frac {3 \sqrt {11}}{11} \right )}}{1331} \]

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

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