3.298 \(\int \frac{-35+70 x-4 x^2+2 x^3}{\left (26-10 x+x^2\right ) \left (17-2 x+x^2\right )} \, dx\)

Optimal. Leaf size=49 \[ \frac{1003 \log \left (x^2-10 x+26\right )}{1025}+\frac{22 \log \left (x^2-2 x+17\right )}{1025}-\frac{15033 \tan ^{-1}(5-x)}{1025}-\frac{4607 \tan ^{-1}\left (\frac{x-1}{4}\right )}{4100} \]

[Out]

(-15033*ArcTan[5 - x])/1025 - (4607*ArcTan[(-1 + x)/4])/4100 + (1003*Log[26 - 10
*x + x^2])/1025 + (22*Log[17 - 2*x + x^2])/1025

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Rubi [A]  time = 0.290272, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.139 \[ \frac{1003 \log \left (x^2-10 x+26\right )}{1025}+\frac{22 \log \left (x^2-2 x+17\right )}{1025}-\frac{15033 \tan ^{-1}(5-x)}{1025}-\frac{4607 \tan ^{-1}\left (\frac{x-1}{4}\right )}{4100} \]

Antiderivative was successfully verified.

[In]  Int[(-35 + 70*x - 4*x^2 + 2*x^3)/((26 - 10*x + x^2)*(17 - 2*x + x^2)),x]

[Out]

(-15033*ArcTan[5 - x])/1025 - (4607*ArcTan[(-1 + x)/4])/4100 + (1003*Log[26 - 10
*x + x^2])/1025 + (22*Log[17 - 2*x + x^2])/1025

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Rubi in Sympy [A]  time = 82.3177, size = 46, normalized size = 0.94 \[ \frac{1003 \log{\left (x^{2} - 10 x + 26 \right )}}{1025} + \frac{22 \log{\left (x^{2} - 2 x + 17 \right )}}{1025} - \frac{4607 \operatorname{atan}{\left (\frac{x}{4} - \frac{1}{4} \right )}}{4100} + \frac{15033 \operatorname{atan}{\left (x - 5 \right )}}{1025} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**3-4*x**2+70*x-35)/(x**2-10*x+26)/(x**2-2*x+17),x)

[Out]

1003*log(x**2 - 10*x + 26)/1025 + 22*log(x**2 - 2*x + 17)/1025 - 4607*atan(x/4 -
1/4)/4100 + 15033*atan(x - 5)/1025

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Mathematica [A]  time = 0.0229319, size = 49, normalized size = 1. \[ \frac{1003 \log \left (x^2-10 x+26\right )}{1025}+\frac{22 \log \left (x^2-2 x+17\right )}{1025}-\frac{15033 \tan ^{-1}(5-x)}{1025}-\frac{4607 \tan ^{-1}\left (\frac{x-1}{4}\right )}{4100} \]

Antiderivative was successfully verified.

[In]  Integrate[(-35 + 70*x - 4*x^2 + 2*x^3)/((26 - 10*x + x^2)*(17 - 2*x + x^2)),x]

[Out]

(-15033*ArcTan[5 - x])/1025 - (4607*ArcTan[(-1 + x)/4])/4100 + (1003*Log[26 - 10
*x + x^2])/1025 + (22*Log[17 - 2*x + x^2])/1025

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Maple [A]  time = 0.009, size = 38, normalized size = 0.8 \[{\frac{15033\,\arctan \left ( -5+x \right ) }{1025}}-{\frac{4607}{4100}\arctan \left ( -{\frac{1}{4}}+{\frac{x}{4}} \right ) }+{\frac{1003\,\ln \left ({x}^{2}-10\,x+26 \right ) }{1025}}+{\frac{22\,\ln \left ({x}^{2}-2\,x+17 \right ) }{1025}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^3-4*x^2+70*x-35)/(x^2-10*x+26)/(x^2-2*x+17),x)

[Out]

15033/1025*arctan(-5+x)-4607/4100*arctan(-1/4+1/4*x)+1003/1025*ln(x^2-10*x+26)+2
2/1025*ln(x^2-2*x+17)

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Maxima [A]  time = 0.887453, size = 50, normalized size = 1.02 \[ \frac{15033}{1025} \, \arctan \left (x - 5\right ) - \frac{4607}{4100} \, \arctan \left (\frac{1}{4} \, x - \frac{1}{4}\right ) + \frac{22}{1025} \, \log \left (x^{2} - 2 \, x + 17\right ) + \frac{1003}{1025} \, \log \left (x^{2} - 10 \, x + 26\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 - 4*x^2 + 70*x - 35)/((x^2 - 2*x + 17)*(x^2 - 10*x + 26)),x, algorithm="maxima")

[Out]

15033/1025*arctan(x - 5) - 4607/4100*arctan(1/4*x - 1/4) + 22/1025*log(x^2 - 2*x
+ 17) + 1003/1025*log(x^2 - 10*x + 26)

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Fricas [A]  time = 0.255186, size = 50, normalized size = 1.02 \[ \frac{15033}{1025} \, \arctan \left (x - 5\right ) - \frac{4607}{4100} \, \arctan \left (\frac{1}{4} \, x - \frac{1}{4}\right ) + \frac{22}{1025} \, \log \left (x^{2} - 2 \, x + 17\right ) + \frac{1003}{1025} \, \log \left (x^{2} - 10 \, x + 26\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 - 4*x^2 + 70*x - 35)/((x^2 - 2*x + 17)*(x^2 - 10*x + 26)),x, algorithm="fricas")

[Out]

15033/1025*arctan(x - 5) - 4607/4100*arctan(1/4*x - 1/4) + 22/1025*log(x^2 - 2*x
+ 17) + 1003/1025*log(x^2 - 10*x + 26)

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Sympy [A]  time = 0.627449, size = 46, normalized size = 0.94 \[ \frac{1003 \log{\left (x^{2} - 10 x + 26 \right )}}{1025} + \frac{22 \log{\left (x^{2} - 2 x + 17 \right )}}{1025} - \frac{4607 \operatorname{atan}{\left (\frac{x}{4} - \frac{1}{4} \right )}}{4100} + \frac{15033 \operatorname{atan}{\left (x - 5 \right )}}{1025} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**3-4*x**2+70*x-35)/(x**2-10*x+26)/(x**2-2*x+17),x)

[Out]

1003*log(x**2 - 10*x + 26)/1025 + 22*log(x**2 - 2*x + 17)/1025 - 4607*atan(x/4 -
1/4)/4100 + 15033*atan(x - 5)/1025

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GIAC/XCAS [A]  time = 0.261597, size = 50, normalized size = 1.02 \[ \frac{15033}{1025} \, \arctan \left (x - 5\right ) - \frac{4607}{4100} \, \arctan \left (\frac{1}{4} \, x - \frac{1}{4}\right ) + \frac{22}{1025} \,{\rm ln}\left (x^{2} - 2 \, x + 17\right ) + \frac{1003}{1025} \,{\rm ln}\left (x^{2} - 10 \, x + 26\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 - 4*x^2 + 70*x - 35)/((x^2 - 2*x + 17)*(x^2 - 10*x + 26)),x, algorithm="giac")

[Out]

15033/1025*arctan(x - 5) - 4607/4100*arctan(1/4*x - 1/4) + 22/1025*ln(x^2 - 2*x
+ 17) + 1003/1025*ln(x^2 - 10*x + 26)