3.524 \(\int \frac{x}{(-1+x)^3 \left (3+5 x+4 x^2\right )^2} \, dx\)

Optimal. Leaf size=97 \[ \frac{44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac{11 \log \left (4 x^2+5 x+3\right )}{4608}-\frac{97}{4416 (1-x)}-\frac{21}{736 (1-x)^2}+\frac{11 \log (1-x)}{2304}+\frac{6023 \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{52992 \sqrt{23}} \]

[Out]

-21/(736*(1 - x)^2) - 97/(4416*(1 - x)) + (39 + 44*x)/(276*(1 - x)^2*(3 + 5*x +
4*x^2)) + (6023*ArcTan[(5 + 8*x)/Sqrt[23]])/(52992*Sqrt[23]) + (11*Log[1 - x])/2
304 - (11*Log[3 + 5*x + 4*x^2])/4608

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Rubi [A]  time = 0.151769, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ \frac{44 x+39}{276 (1-x)^2 \left (4 x^2+5 x+3\right )}-\frac{11 \log \left (4 x^2+5 x+3\right )}{4608}-\frac{97}{4416 (1-x)}-\frac{21}{736 (1-x)^2}+\frac{11 \log (1-x)}{2304}+\frac{6023 \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{52992 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]  Int[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

-21/(736*(1 - x)^2) - 97/(4416*(1 - x)) + (39 + 44*x)/(276*(1 - x)^2*(3 + 5*x +
4*x^2)) + (6023*ArcTan[(5 + 8*x)/Sqrt[23]])/(52992*Sqrt[23]) + (11*Log[1 - x])/2
304 - (11*Log[3 + 5*x + 4*x^2])/4608

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Rubi in Sympy [A]  time = 21.3727, size = 83, normalized size = 0.86 \[ \frac{11 \log{\left (- x + 1 \right )}}{2304} - \frac{11 \log{\left (4 x^{2} + 5 x + 3 \right )}}{4608} + \frac{6023 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{8 x}{23} + \frac{5}{23}\right ) \right )}}{1218816} - \frac{97}{4416 \left (- x + 1\right )} + \frac{44 x + 39}{276 \left (- x + 1\right )^{2} \left (4 x^{2} + 5 x + 3\right )} - \frac{21}{736 \left (- x + 1\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(-1+x)**3/(4*x**2+5*x+3)**2,x)

[Out]

11*log(-x + 1)/2304 - 11*log(4*x**2 + 5*x + 3)/4608 + 6023*sqrt(23)*atan(sqrt(23
)*(8*x/23 + 5/23))/1218816 - 97/(4416*(-x + 1)) + (44*x + 39)/(276*(-x + 1)**2*(
4*x**2 + 5*x + 3)) - 21/(736*(-x + 1)**2)

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Mathematica [A]  time = 0.0697265, size = 78, normalized size = 0.8 \[ \frac{\frac{184 (2204 x+975)}{4 x^2+5 x+3}-17457 \log \left (4 x^2+5 x+3\right )+\frac{59248}{x-1}-\frac{25392}{(x-1)^2}+34914 \log (1-x)+36138 \sqrt{23} \tan ^{-1}\left (\frac{8 x+5}{\sqrt{23}}\right )}{7312896} \]

Antiderivative was successfully verified.

[In]  Integrate[x/((-1 + x)^3*(3 + 5*x + 4*x^2)^2),x]

[Out]

(-25392/(-1 + x)^2 + 59248/(-1 + x) + (184*(975 + 2204*x))/(3 + 5*x + 4*x^2) + 3
6138*Sqrt[23]*ArcTan[(5 + 8*x)/Sqrt[23]] + 34914*Log[1 - x] - 17457*Log[3 + 5*x
+ 4*x^2])/7312896

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Maple [A]  time = 0.017, size = 68, normalized size = 0.7 \[ -{\frac{1}{288\, \left ( -1+x \right ) ^{2}}}+{\frac{7}{-864+864\,x}}+{\frac{11\,\ln \left ( -1+x \right ) }{2304}}-{\frac{1}{6912} \left ( -{\frac{2204\,x}{23}}-{\frac{975}{23}} \right ) \left ({x}^{2}+{\frac{5\,x}{4}}+{\frac{3}{4}} \right ) ^{-1}}-{\frac{11\,\ln \left ( 16\,{x}^{2}+20\,x+12 \right ) }{4608}}+{\frac{6023\,\sqrt{23}}{1218816}\arctan \left ({\frac{ \left ( 32\,x+20 \right ) \sqrt{23}}{92}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(-1+x)^3/(4*x^2+5*x+3)^2,x)

[Out]

-1/288/(-1+x)^2+7/864/(-1+x)+11/2304*ln(-1+x)-1/6912*(-2204/23*x-975/23)/(x^2+5/
4*x+3/4)-11/4608*ln(16*x^2+20*x+12)+6023/1218816*23^(1/2)*arctan(1/92*(32*x+20)*
23^(1/2))

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Maxima [A]  time = 0.773856, size = 101, normalized size = 1.04 \[ \frac{6023}{1218816} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) + \frac{388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} - \frac{11}{4608} \, \log \left (4 \, x^{2} + 5 \, x + 3\right ) + \frac{11}{2304} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((4*x^2 + 5*x + 3)^2*(x - 1)^3),x, algorithm="maxima")

[Out]

6023/1218816*sqrt(23)*arctan(1/23*sqrt(23)*(8*x + 5)) + 1/4416*(388*x^3 - 407*x^
2 - 120*x - 45)/(4*x^4 - 3*x^3 - 3*x^2 - x + 3) - 11/4608*log(4*x^2 + 5*x + 3) +
11/2304*log(x - 1)

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Fricas [A]  time = 0.271904, size = 197, normalized size = 2.03 \[ -\frac{\sqrt{23}{\left (253 \, \sqrt{23}{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (4 \, x^{2} + 5 \, x + 3\right ) - 506 \, \sqrt{23}{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \log \left (x - 1\right ) - 12046 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) - 24 \, \sqrt{23}{\left (388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45\right )}\right )}}{2437632 \,{\left (4 \, x^{4} - 3 \, x^{3} - 3 \, x^{2} - x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((4*x^2 + 5*x + 3)^2*(x - 1)^3),x, algorithm="fricas")

[Out]

-1/2437632*sqrt(23)*(253*sqrt(23)*(4*x^4 - 3*x^3 - 3*x^2 - x + 3)*log(4*x^2 + 5*
x + 3) - 506*sqrt(23)*(4*x^4 - 3*x^3 - 3*x^2 - x + 3)*log(x - 1) - 12046*(4*x^4
- 3*x^3 - 3*x^2 - x + 3)*arctan(1/23*sqrt(23)*(8*x + 5)) - 24*sqrt(23)*(388*x^3
- 407*x^2 - 120*x - 45))/(4*x^4 - 3*x^3 - 3*x^2 - x + 3)

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Sympy [A]  time = 0.605728, size = 88, normalized size = 0.91 \[ \frac{388 x^{3} - 407 x^{2} - 120 x - 45}{17664 x^{4} - 13248 x^{3} - 13248 x^{2} - 4416 x + 13248} + \frac{11 \log{\left (x - 1 \right )}}{2304} - \frac{11 \log{\left (x^{2} + \frac{5 x}{4} + \frac{3}{4} \right )}}{4608} + \frac{6023 \sqrt{23} \operatorname{atan}{\left (\frac{8 \sqrt{23} x}{23} + \frac{5 \sqrt{23}}{23} \right )}}{1218816} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(-1+x)**3/(4*x**2+5*x+3)**2,x)

[Out]

(388*x**3 - 407*x**2 - 120*x - 45)/(17664*x**4 - 13248*x**3 - 13248*x**2 - 4416*
x + 13248) + 11*log(x - 1)/2304 - 11*log(x**2 + 5*x/4 + 3/4)/4608 + 6023*sqrt(23
)*atan(8*sqrt(23)*x/23 + 5*sqrt(23)/23)/1218816

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GIAC/XCAS [A]  time = 0.26709, size = 96, normalized size = 0.99 \[ \frac{6023}{1218816} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (8 \, x + 5\right )}\right ) + \frac{388 \, x^{3} - 407 \, x^{2} - 120 \, x - 45}{4416 \,{\left (4 \, x^{2} + 5 \, x + 3\right )}{\left (x - 1\right )}^{2}} - \frac{11}{4608} \,{\rm ln}\left (4 \, x^{2} + 5 \, x + 3\right ) + \frac{11}{2304} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/((4*x^2 + 5*x + 3)^2*(x - 1)^3),x, algorithm="giac")

[Out]

6023/1218816*sqrt(23)*arctan(1/23*sqrt(23)*(8*x + 5)) + 1/4416*(388*x^3 - 407*x^
2 - 120*x - 45)/((4*x^2 + 5*x + 3)*(x - 1)^2) - 11/4608*ln(4*x^2 + 5*x + 3) + 11
/2304*ln(abs(x - 1))