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

[Out]

2*ln(320*x^4+80*x^3-12*x^2+24*x+9)+2*arctan(1/55*(7-40*x)*11^(1/2))*11^(1/2)-2*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 = 0.08, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 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))]

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]

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*}

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Mathematica [C]  time = 0.02, 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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fricas [A]  time = 0.88, size = 66, normalized size = 0.85 \[ -2 \, \sqrt {11} \arctan \left (\frac {1}{66} \, \sqrt {11} {\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) - 2 \, \sqrt {11} \arctan \left (\frac {1}{55} \, \sqrt {11} {\left (40 \, x - 7\right )}\right ) + 2 \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \]

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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giac [A]  time = 0.97, size = 64, normalized size = 0.82 \[ -2 \, \sqrt {11} {\left (\arctan \left (\frac {1}{66} \, \sqrt {11} {\left (800 \, x^{3} - 40 \, x^{2} + 30 \, x + 57\right )}\right ) - \arctan \left (-\frac {1}{55} \, \sqrt {11} {\left (40 \, x - 7\right )}\right )\right )} + 2 \, \log \left (320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9\right ) \]

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]

-2*sqrt(11)*(arctan(1/66*sqrt(11)*(800*x^3 - 40*x^2 + 30*x + 57)) - 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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maple [A]  time = 0.03, size = 75, normalized size = 0.96 \[ -2 \sqrt {11}\, \arctan \left (\frac {\left (40 x -7\right ) \sqrt {11}}{55}\right )-2 \sqrt {11}\, \arctan \left (\frac {400 \sqrt {11}\, x^{3}}{33}-\frac {20 \sqrt {11}\, x^{2}}{33}+\frac {5 \sqrt {11}\, x}{11}+\frac {19 \sqrt {11}}{22}\right )+2 \ln \left (6400 x^{4}+1600 x^{3}-240 x^{2}+480 x +180\right ) \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ 4 \, \int \frac {640 \, x^{3} - 100 \, x^{2} - 144 \, x - 21}{320 \, x^{4} + 80 \, x^{3} - 12 \, x^{2} + 24 \, x + 9}\,{d x} \]

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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mupad [B]  time = 0.09, size = 76, normalized size = 0.97 \[ 2\,\ln \left (320\,x^4+80\,x^3-12\,x^2+24\,x+9\right )-2\,\sqrt {11}\,\mathrm {atan}\left (\frac {8\,\sqrt {11}\,x}{11}-\frac {7\,\sqrt {11}}{55}\right )-2\,\sqrt {11}\,\mathrm {atan}\left (\frac {400\,\sqrt {11}\,x^3}{33}-\frac {20\,\sqrt {11}\,x^2}{33}+\frac {5\,\sqrt {11}\,x}{11}+\frac {19\,\sqrt {11}}{22}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(24*x - 12*x^2 + 80*x^3 + 320*x^4 + 9) - 2*11^(1/2)*atan((8*11^(1/2)*x)/11 - (7*11^(1/2))/55) - 2*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)

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sympy [A]  time = 0.18, size = 100, normalized size = 1.28 \[ \sqrt {11} \left (- 2 \operatorname {atan}{\left (\frac {8 \sqrt {11} x}{11} - \frac {7 \sqrt {11}}{55} \right )} - 2 \operatorname {atan}{\left (\frac {400 \sqrt {11} x^{3}}{33} - \frac {20 \sqrt {11} x^{2}}{33} + \frac {5 \sqrt {11} x}{11} + \frac {19 \sqrt {11}}{22} \right )}\right ) + 2 \log {\left (x^{4} + \frac {x^{3}}{4} - \frac {3 x^{2}}{80} + \frac {3 x}{40} + \frac {9}{320} \right )} \]

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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