3.64 \(\int -\frac{84+576 x+400 x^2-2560 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\)

Optimal. Leaf size=78 \[ 2 \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right )-2 \sqrt{11} \tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )+2 \sqrt{11} \tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right ) \]

[Out]

2*Sqrt[11]*ArcTan[(7 - 40*x)/(5*Sqrt[11])] - 2*Sqrt[11]*ArcTan[(57 + 30*x - 40*x^2 + 800*x^3)/(6*Sqrt[11])] +
2*Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]

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Rubi [A]  time = 0.07667, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {2100, 2090} \[ 2 \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right )-2 \sqrt{11} \tan ^{-1}\left (\frac{800 x^3-40 x^2+30 x+57}{6 \sqrt{11}}\right )+2 \sqrt{11} \tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[-((84 + 576*x + 400*x^2 - 2560*x^3)/(9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4)),x]

[Out]

2*Sqrt[11]*ArcTan[(7 - 40*x)/(5*Sqrt[11])] - 2*Sqrt[11]*ArcTan[(57 + 30*x - 40*x^2 + 800*x^3)/(6*Sqrt[11])] +
2*Log[9 + 24*x - 12*x^2 + 80*x^3 + 320*x^4]

Rule 2100

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coe
ff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

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{84+576 x+400 x^2-2560 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx &=2 \log \left (9+24 x-12 x^2+80 x^3+320 x^4\right )-\frac{\int \frac{168960+675840 x+1126400 x^2}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx}{1280}\\ &=2 \sqrt{11} \tan ^{-1}\left (\frac{7-40 x}{5 \sqrt{11}}\right )-2 \sqrt{11} \tan ^{-1}\left (\frac{57+30 x-40 x^2+800 x^3}{6 \sqrt{11}}\right )+2 \log \left (9+24 x-12 x^2+80 x^3+320 x^4\right )\\ \end{align*}

Mathematica [C]  time = 0.0201547, size = 99, normalized size = 1.27 \[ \frac{1}{2} \text{RootSum}\left [320 \text{$\#$1}^4+80 \text{$\#$1}^3-12 \text{$\#$1}^2+24 \text{$\#$1}+9\& ,\frac{640 \text{$\#$1}^3 \log (x-\text{$\#$1})-100 \text{$\#$1}^2 \log (x-\text{$\#$1})-144 \text{$\#$1} \log (x-\text{$\#$1})-21 \log (x-\text{$\#$1})}{160 \text{$\#$1}^3+30 \text{$\#$1}^2-3 \text{$\#$1}+3}\& \right ] \]

Antiderivative was successfully verified.

[In]

Integrate[-((84 + 576*x + 400*x^2 - 2560*x^3)/(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 & , (-21*Log[x - #1] - 144*Log[x - #1]*#1 - 100*Log[x - #1]*#
1^2 + 640*Log[x - #1]*#1^3)/(3 - 3*#1 + 30*#1^2 + 160*#1^3) & ]/2

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Maple [A]  time = 0.018, size = 75, normalized size = 1. \begin{align*} 2\,\ln \left ( 6400\,{x}^{4}+1600\,{x}^{3}-240\,{x}^{2}+480\,x+180 \right ) -2\,\sqrt{11}\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 ) -2\,\sqrt{11}\arctan \left ({\frac{ \left ( 40\,x-7 \right ) \sqrt{11}}{55}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2560*x^3-400*x^2-576*x-84)/(320*x^4+80*x^3-12*x^2+24*x+9),x)

[Out]

2*ln(6400*x^4+1600*x^3-240*x^2+480*x+180)-2*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)-2*11^(1/2)*arctan(1/55*(40*x-7)*11^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, \int \frac{640 \, x^{3} - 100 \, x^{2} - 144 \, x - 21}{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((2560*x^3-400*x^2-576*x-84)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="maxima")

[Out]

4*integrate((640*x^3 - 100*x^2 - 144*x - 21)/(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9), x)

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Fricas [A]  time = 1.94244, size = 213, normalized size = 2.73 \begin{align*} -2 \, \sqrt{11} \arctan \left (\frac{1}{66} \, \sqrt{11}{\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) - 2 \, \sqrt{11} \arctan \left (\frac{1}{55} \, \sqrt{11}{\left (40 \, x - 7\right )}\right ) + 2 \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2560*x^3-400*x^2-576*x-84)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="fricas")

[Out]

-2*sqrt(11)*arctan(1/66*sqrt(11)*(800*x^3 - 40*x^2 + 30*x + 57)) - 2*sqrt(11)*arctan(1/55*sqrt(11)*(40*x - 7))
+ 2*log(320*x^4 + 80*x^3 - 12*x^2 + 24*x + 9)

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Sympy [A]  time = 0.16712, size = 100, normalized size = 1.28 \begin{align*} \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 ) + 2 \log{\left (x^{4} + \frac{x^{3}}{4} - \frac{3 x^{2}}{80} + \frac{3 x}{40} + \frac{9}{320} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2560*x**3-400*x**2-576*x-84)/(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*sqr
t(11)*x/11 + 19*sqrt(11)/22)) + 2*log(x**4 + x**3/4 - 3*x**2/80 + 3*x/40 + 9/320)

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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((2560*x^3-400*x^2-576*x-84)/(320*x^4+80*x^3-12*x^2+24*x+9),x, algorithm="giac")

[Out]

sage0*x