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]

-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
])

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Rubi [A]  time = 0.031717, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, 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*}

Mathematica [C]  time = 0.0184698, 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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Maple [A]  time = 0.024, size = 52, normalized size = 0.9 \begin{align*}{\frac{\sqrt{11}}{22}\arctan \left ({\frac{ \left ( 40\,x-7 \right ) \sqrt{11}}{55}} \right ) }+{\frac{\sqrt{11}}{22}\arctan \left ( -{\frac{20\,\sqrt{11}{x}^{2}}{33}}+{\frac{5\,\sqrt{11}x}{11}}+{\frac{19\,\sqrt{11}}{22}}+{\frac{400\,\sqrt{11}{x}^{3}}{33}} \right ) } \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{20 \, x^{2} + 12 \, x + 3}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \end{align*}

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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Fricas [A]  time = 1.9788, size = 158, normalized size = 2.68 \begin{align*} \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 ) \end{align*}

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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Sympy [A]  time = 0.160272, size = 73, normalized size = 1.24 \begin{align*} \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} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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]

sage0*x