3.320 \(\int \frac{-32+5 x-27 x^2+4 x^3}{-70-299 x-286 x^2+50 x^3-13 x^4+30 x^5} \, dx\)

Optimal. Leaf size=63 \[ \frac{11049 \log \left (x^2+x+5\right )}{260015}-\frac{3146 \log (7-3 x)}{80155}-\frac{334}{323} \log (2 x+1)+\frac{4822 \log (5 x+2)}{4879}+\frac{3988 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{19}}\right )}{13685 \sqrt{19}} \]

[Out]

(3988*ArcTan[(1 + 2*x)/Sqrt[19]])/(13685*Sqrt[19]) - (3146*Log[7 - 3*x])/80155 -
(334*Log[1 + 2*x])/323 + (4822*Log[2 + 5*x])/4879 + (11049*Log[5 + x + x^2])/26
0015

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Rubi [A]  time = 0.166806, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.116 \[ \frac{11049 \log \left (x^2+x+5\right )}{260015}-\frac{3146 \log (7-3 x)}{80155}-\frac{334}{323} \log (2 x+1)+\frac{4822 \log (5 x+2)}{4879}+\frac{3988 \tan ^{-1}\left (\frac{2 x+1}{\sqrt{19}}\right )}{13685 \sqrt{19}} \]

Antiderivative was successfully verified.

[In]  Int[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 13*x^4 + 30*x^5),x]

[Out]

(3988*ArcTan[(1 + 2*x)/Sqrt[19]])/(13685*Sqrt[19]) - (3146*Log[7 - 3*x])/80155 -
(334*Log[1 + 2*x])/323 + (4822*Log[2 + 5*x])/4879 + (11049*Log[5 + x + x^2])/26
0015

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((4*x**3-27*x**2+5*x-32)/(30*x**5-13*x**4+50*x**3-286*x**2-299*x-70),x)

[Out]

Timed out

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Mathematica [A]  time = 0.0407911, size = 57, normalized size = 0.9 \[ \frac{453009 \log \left (x^2+x+5\right )-418418 \log (7-3 x)-11023670 \log (2 x+1)[phone] \log (5 x+2)+163508 \sqrt{19} \tan ^{-1}\left (\frac{2 x+1}{\sqrt{19}}\right )}{10660615} \]

Antiderivative was successfully verified.

[In]  Integrate[(-32 + 5*x - 27*x^2 + 4*x^3)/(-70 - 299*x - 286*x^2 + 50*x^3 - 13*x^4 + 30*x^5),x]

[Out]

(163508*Sqrt[19]*ArcTan[(1 + 2*x)/Sqrt[19]] - 418418*Log[7 - 3*x] - 11023670*Log
[1 + 2*x] + 10536070*Log[2 + 5*x] + 453009*Log[5 + x + x^2])/10660615

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Maple [A]  time = 0.016, size = 51, normalized size = 0.8 \[{\frac{4822\,\ln \left ( 2+5\,x \right ) }{4879}}-{\frac{334\,\ln \left ( 1+2\,x \right ) }{323}}+{\frac{11049\,\ln \left ({x}^{2}+x+5 \right ) }{260015}}+{\frac{3988\,\sqrt{19}}{260015}\arctan \left ({\frac{ \left ( 1+2\,x \right ) \sqrt{19}}{19}} \right ) }-{\frac{3146\,\ln \left ( 3\,x-7 \right ) }{80155}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((4*x^3-27*x^2+5*x-32)/(30*x^5-13*x^4+50*x^3-286*x^2-299*x-70),x)

[Out]

4822/4879*ln(2+5*x)-334/323*ln(1+2*x)+11049/260015*ln(x^2+x+5)+3988/260015*arcta
n(1/19*(1+2*x)*19^(1/2))*19^(1/2)-3146/80155*ln(3*x-7)

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Maxima [A]  time = 0.893196, size = 68, normalized size = 1.08 \[ \frac{3988}{260015} \, \sqrt{19} \arctan \left (\frac{1}{19} \, \sqrt{19}{\left (2 \, x + 1\right )}\right ) + \frac{11049}{260015} \, \log \left (x^{2} + x + 5\right ) + \frac{4822}{4879} \, \log \left (5 \, x + 2\right ) - \frac{3146}{80155} \, \log \left (3 \, x - 7\right ) - \frac{334}{323} \, \log \left (2 \, x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - 27*x^2 + 5*x - 32)/(30*x^5 - 13*x^4 + 50*x^3 - 286*x^2 - 299*x - 70),x, algorithm="maxima")

[Out]

3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*log(x^2 + x
+ 5) + 4822/4879*log(5*x + 2) - 3146/80155*log(3*x - 7) - 334/323*log(2*x + 1)

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Fricas [A]  time = 0.259556, size = 86, normalized size = 1.37 \[ \frac{1}{202551685} \, \sqrt{19}{\left (453009 \, \sqrt{19} \log \left (x^{2} + x + 5\right ) + 10536070 \, \sqrt{19} \log \left (5 \, x + 2\right ) - 418418 \, \sqrt{19} \log \left (3 \, x - 7\right ) - 11023670 \, \sqrt{19} \log \left (2 \, x + 1\right ) + 3106652 \, \arctan \left (\frac{1}{19} \, \sqrt{19}{\left (2 \, x + 1\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - 27*x^2 + 5*x - 32)/(30*x^5 - 13*x^4 + 50*x^3 - 286*x^2 - 299*x - 70),x, algorithm="fricas")

[Out]

1/202551685*sqrt(19)*(453009*sqrt(19)*log(x^2 + x + 5) + 10536070*sqrt(19)*log(5
*x + 2) - 418418*sqrt(19)*log(3*x - 7) - 11023670*sqrt(19)*log(2*x + 1) + 310665
2*arctan(1/19*sqrt(19)*(2*x + 1)))

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Sympy [A]  time = 1.08499, size = 68, normalized size = 1.08 \[ - \frac{3146 \log{\left (x - \frac{7}{3} \right )}}{80155} + \frac{4822 \log{\left (x + \frac{2}{5} \right )}}{4879} - \frac{334 \log{\left (x + \frac{1}{2} \right )}}{323} + \frac{11049 \log{\left (x^{2} + x + 5 \right )}}{260015} + \frac{3988 \sqrt{19} \operatorname{atan}{\left (\frac{2 \sqrt{19} x}{19} + \frac{\sqrt{19}}{19} \right )}}{260015} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x**3-27*x**2+5*x-32)/(30*x**5-13*x**4+50*x**3-286*x**2-299*x-70),x)

[Out]

-3146*log(x - 7/3)/80155 + 4822*log(x + 2/5)/4879 - 334*log(x + 1/2)/323 + 11049
*log(x**2 + x + 5)/260015 + 3988*sqrt(19)*atan(2*sqrt(19)*x/19 + sqrt(19)/19)/26
0015

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GIAC/XCAS [A]  time = 0.262111, size = 72, normalized size = 1.14 \[ \frac{3988}{260015} \, \sqrt{19} \arctan \left (\frac{1}{19} \, \sqrt{19}{\left (2 \, x + 1\right )}\right ) + \frac{11049}{260015} \,{\rm ln}\left (x^{2} + x + 5\right ) + \frac{4822}{4879} \,{\rm ln}\left ({\left | 5 \, x + 2 \right |}\right ) - \frac{3146}{80155} \,{\rm ln}\left ({\left | 3 \, x - 7 \right |}\right ) - \frac{334}{323} \,{\rm ln}\left ({\left | 2 \, x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((4*x^3 - 27*x^2 + 5*x - 32)/(30*x^5 - 13*x^4 + 50*x^3 - 286*x^2 - 299*x - 70),x, algorithm="giac")

[Out]

3988/260015*sqrt(19)*arctan(1/19*sqrt(19)*(2*x + 1)) + 11049/260015*ln(x^2 + x +
5) + 4822/4879*ln(abs(5*x + 2)) - 3146/80155*ln(abs(3*x - 7)) - 334/323*ln(abs(
2*x + 1))