3.1.64 \(\int \frac {-84-576 x-400 x^2+2560 x^3}{9+24 x-12 x^2+80 x^3+320 x^4} \, dx\) [64]

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

[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 = 0.04, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2125, 2115} \begin {gather*} -2 \sqrt {11} \text {ArcTan}\left (\frac {800 x^3-40 x^2+30 x+57}{6 \sqrt {11}}\right )+2 \sqrt {11} \text {ArcTan}\left (\frac {7-40 x}{5 \sqrt {11}}\right )+2 \log \left (320 x^4+80 x^3-12 x^2+24 x+9\right ) \end {gather*}

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 2115

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/q)*ArcTan[(C*d - B*e + 2*C*
e*x)/q], x] - Simp[2*(C^2/q)*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)))], 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 2125

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coef
f[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] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.01, size = 99, normalized size = 1.27 \begin {gather*} \frac {1}{2} \text {RootSum}\left [9+24 \text {$\#$1}-12 \text {$\#$1}^2+80 \text {$\#$1}^3+320 \text {$\#$1}^4\&,\frac {-21 \log (x-\text {$\#$1})-144 \log (x-\text {$\#$1}) \text {$\#$1}-100 \log (x-\text {$\#$1}) \text {$\#$1}^2+640 \log (x-\text {$\#$1}) \text {$\#$1}^3}{3-3 \text {$\#$1}+30 \text {$\#$1}^2+160 \text {$\#$1}^3}\&\right ] \end {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.03, size = 70, normalized size = 0.90

method	result	size
default	\(4 \left (\frac {i \sqrt {11}}{4}+\frac {1}{2}\right ) \ln \left (80 x^{2}+\left (-10 i \sqrt {11}+10\right ) x -3 i \sqrt {11}-9\right )+4 \left (\frac {1}{2}-\frac {i \sqrt {11}}{4}\right ) \ln \left (80 x^{2}+\left (10 i \sqrt {11}+10\right ) x +3 i \sqrt {11}-9\right )\)	\(70\)
risch	\(2 \ln \left (6400 x^{4}+1600 x^{3}-240 x^{2}+480 x +180\right )-2 \sqrt {11}\, \arctan \left (-\frac {20 \sqrt {11}\, x^{2}}{33}+\frac {5 \sqrt {11}\, x}{11}+\frac {19 \sqrt {11}}{22}+\frac {400 \sqrt {11}\, x^{3}}{33}\right )-2 \sqrt {11}\, \arctan \left (\frac {\left (40 x -7\right ) \sqrt {11}}{55}\right )\)	\(75\)

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,method=_RETURNVERBOSE)

[Out]

4*(1/4*I*11^(1/2)+1/2)*ln(80*x^2+(-10*I*11^(1/2)+10)*x-3*I*11^(1/2)-9)+4*(1/2-1/4*I*11^(1/2))*ln(80*x^2+(10*I*
11^(1/2)+10)*x+3*I*11^(1/2)-9)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.76, size = 66, normalized size = 0.85 \begin {gather*} -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 {gather*}

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 = 0.07, size = 100, normalized size = 1.28 \begin {gather*} \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 {gather*}

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 [A]
time = 1.66, size = 64, normalized size = 0.82 \begin {gather*} -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 ) \end {gather*}

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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Mupad [B]
time = 0.09, size = 76, normalized size = 0.97 \begin {gather*} 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 ) \end {gather*}

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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