3.63 \(\int \frac {3+12 x+20 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)

Optimal. Leaf size=59 \[ \frac {\tan ^{-1}\left (\frac {800 x^3-40 x^2+30 x+57}{6 \sqrt {11}}\right )}{2 \sqrt {11}}-\frac {\tan ^{-1}\left (\frac {7-40 x}{5 \sqrt {11}}\right )}{2 \sqrt {11}} \]

[Out]

-1/22*arctan(1/55*(7-40*x)*11^(1/2))*11^(1/2)+1/22*arctan(1/66*(800*x^3-40*x^2+30*x+57)*11^(1/2))*11^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {2090} \[ \frac {\tan ^{-1}\left (\frac {800 x^3-40 x^2+30 x+57}{6 \sqrt {11}}\right )}{2 \sqrt {11}}-\frac {\tan ^{-1}\left (\frac {7-40 x}{5 \sqrt {11}}\right )}{2 \sqrt {11}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 12*x + 20*x^2)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4),x]

[Out]

-ArcTan[(7 - 40*x)/(5*Sqrt[11])]/(2*Sqrt[11]) + ArcTan[(57 + 30*x - 40*x^2 + 800*x^3)/(6*Sqrt[11])]/(2*Sqrt[11
])

Rule 2090

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4), x_Sy
mbol] :> With[{q = Rt[-(C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))), 2]}, Simp[(2*C^2*ArcTan[(C*d - B*e + 2*C*e*x
)/q])/q, x] - Simp[(2*C^2*ArcTan[(C*(4*B*c*C - 3*B^2*d - 4*A*C*d + 12*A*B*e + 4*C*(2*c*C - B*d + 2*A*e)*x + 4*
C*(2*C*d - B*e)*x^2 + 8*C^2*e*x^3))/(q*(B^2 - 4*A*C))])/q, x]] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && EqQ[B^
2*d + 2*C*(b*C + A*d) - 2*B*(c*C + 2*A*e), 0] && EqQ[2*B^2*c*C - 8*a*C^3 - B^3*d - 4*A*B*C*d + 4*A*(B^2 + 2*A*
C)*e, 0] && NegQ[C*(2*e*(B*d - 4*A*e) + C*(d^2 - 4*c*e))]

Rubi steps

\begin {align*} \int \frac {3+12 x+20 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx &=-\frac {\tan ^{-1}\left (\frac {7-40 x}{5 \sqrt {11}}\right )}{2 \sqrt {11}}+\frac {\tan ^{-1}\left (\frac {57+30 x-40 x^2+800 x^3}{6 \sqrt {11}}\right )}{2 \sqrt {11}}\\ \end {align*}

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Mathematica [C]  time = 0.02, size = 86, normalized size = 1.46 \[ \frac {1}{8} \text {RootSum}\left [320 \text {$\#$1}^4+80 \text {$\#$1}^3-12 \text {$\#$1}^2+24 \text {$\#$1}+9\& ,\frac {20 \text {$\#$1}^2 \log (x-\text {$\#$1})+12 \text {$\#$1} \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1})}{160 \text {$\#$1}^3+30 \text {$\#$1}^2-3 \text {$\#$1}+3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 12*x + 20*x^2)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4),x]

[Out]

RootSum[9 + 24*#1 - 12*#1^2 + 80*#1^3 + 320*#1^4 & , (3*Log[x - #1] + 12*Log[x - #1]*#1 + 20*Log[x - #1]*#1^2)
/(3 - 3*#1 + 30*#1^2 + 160*#1^3) & ]/8

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fricas [A]  time = 0.85, size = 43, normalized size = 0.73 \[ \frac {1}{22} \, \sqrt {11} \arctan \left (\frac {1}{66} \, \sqrt {11} {\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) + \frac {1}{22} \, \sqrt {11} \arctan \left (\frac {1}{55} \, \sqrt {11} {\left (40 \, x - 7\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^2+12*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="fricas")

[Out]

1/22*sqrt(11)*arctan(1/66*sqrt(11)*(800*x^3 - 40*x^2 + 30*x + 57)) + 1/22*sqrt(11)*arctan(1/55*sqrt(11)*(40*x
- 7))

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giac [A]  time = 0.85, size = 40, normalized size = 0.68 \[ \frac {1}{22} \, \sqrt {11} {\left (\arctan \left (\frac {1}{66} \, \sqrt {11} {\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) - \arctan \left (-\frac {1}{55} \, \sqrt {11} {\left (40 \, x - 7\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^2+12*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="giac")

[Out]

1/22*sqrt(11)*(arctan(1/66*sqrt(11)*(800*x^3 - 40*x^2 + 30*x + 57)) - arctan(-1/55*sqrt(11)*(40*x - 7)))

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maple [A]  time = 0.03, size = 52, normalized size = 0.88 \[ \frac {\sqrt {11}\, \arctan \left (\frac {\left (40 x -7\right ) \sqrt {11}}{55}\right )}{22}+\frac {\sqrt {11}\, \arctan \left (\frac {400 \sqrt {11}\, x^{3}}{33}-\frac {20 \sqrt {11}\, x^{2}}{33}+\frac {5 \sqrt {11}\, x}{11}+\frac {19 \sqrt {11}}{22}\right )}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((20*x^2+12*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x)

[Out]

1/22*11^(1/2)*arctan(1/55*(40*x-7)*11^(1/2))+1/22*11^(1/2)*arctan(-20/33*11^(1/2)*x^2+5/11*11^(1/2)*x+19/22*11
^(1/2)+400/33*11^(1/2)*x^3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {20 \, x^{2} + 12 \, x + 3}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x^2+12*x+3)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="maxima")

[Out]

integrate((20*x^2 + 12*x + 3)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9), x)

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mupad [B]  time = 0.34, size = 53, normalized size = 0.90 \[ \frac {\sqrt {11}\,\mathrm {atan}\left (\frac {8\,\sqrt {11}\,x}{11}-\frac {7\,\sqrt {11}}{55}\right )}{22}+\frac {\sqrt {11}\,\mathrm {atan}\left (\frac {400\,\sqrt {11}\,x^3}{33}-\frac {20\,\sqrt {11}\,x^2}{33}+\frac {5\,\sqrt {11}\,x}{11}+\frac {19\,\sqrt {11}}{22}\right )}{22} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x + 20*x^2 + 3)/(24*x - 12*x^2 + 80*x^3 + 320*x^4 + 9),x)

[Out]

(11^(1/2)*atan((8*11^(1/2)*x)/11 - (7*11^(1/2))/55))/22 + (11^(1/2)*atan((5*11^(1/2)*x)/11 + (19*11^(1/2))/22
- (20*11^(1/2)*x^2)/33 + (400*11^(1/2)*x^3)/33))/22

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sympy [A]  time = 0.17, size = 73, normalized size = 1.24 \[ \frac {\sqrt {11} \left (2 \operatorname {atan}{\left (\frac {8 \sqrt {11} x}{11} - \frac {7 \sqrt {11}}{55} \right )} + 2 \operatorname {atan}{\left (\frac {400 \sqrt {11} x^{3}}{33} - \frac {20 \sqrt {11} x^{2}}{33} + \frac {5 \sqrt {11} x}{11} + \frac {19 \sqrt {11}}{22} \right )}\right )}{44} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((20*x**2+12*x+3)/(320*x**4+80*x**3-12*x**2+24*x+9),x)

[Out]

sqrt(11)*(2*atan(8*sqrt(11)*x/11 - 7*sqrt(11)/55) + 2*atan(400*sqrt(11)*x**3/33 - 20*sqrt(11)*x**2/33 + 5*sqrt
(11)*x/11 + 19*sqrt(11)/22))/44

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