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

Optimal. Leaf size=84 \[ \frac{121 (21193-12828 x)}{33856 \left (2 x^2-x+3\right )}-\frac{1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}+\frac{825}{32} \log \left (2 x^2-x+3\right )+\frac{125 x}{8}+\frac{165099 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8464 \sqrt{23}} \]

[Out]

(125*x)/8 - (1331*(17 - 45*x))/(1472*(3 - x + 2*x^2)^2) + (121*(21193 - 12828*x)
)/(33856*(3 - x + 2*x^2)) + (165099*ArcTan[(1 - 4*x)/Sqrt[23]])/(8464*Sqrt[23])
+ (825*Log[3 - x + 2*x^2])/32

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Rubi [A]  time = 0.149614, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{121 (21193-12828 x)}{33856 \left (2 x^2-x+3\right )}-\frac{1331 (17-45 x)}{1472 \left (2 x^2-x+3\right )^2}+\frac{825}{32} \log \left (2 x^2-x+3\right )+\frac{125 x}{8}+\frac{165099 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{8464 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(125*x)/8 - (1331*(17 - 45*x))/(1472*(3 - x + 2*x^2)^2) + (121*(21193 - 12828*x)
)/(33856*(3 - x + 2*x^2)) + (165099*ArcTan[(1 - 4*x)/Sqrt[23]])/(8464*Sqrt[23])
+ (825*Log[3 - x + 2*x^2])/32

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{125 x^{3}}{46} + \frac{14883 \left (- 7 x + 19\right )}{8464 \left (2 x^{2} - x + 3\right )} - \frac{\left (- 4 x + 1\right ) \left (5 x^{2} + 3 x + 2\right )^{3}}{46 \left (2 x^{2} - x + 3\right )^{2}} + \frac{825 \log{\left (2 x^{2} - x + 3 \right )}}{32} - \frac{165099 \sqrt{23} \operatorname{atan}{\left (\sqrt{23} \left (\frac{4 x}{23} - \frac{1}{23}\right ) \right )}}{194672} - \frac{\int \left (- \frac{1155}{2}\right )\, dx}{46} - \frac{1275 \int x\, dx}{92} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*x**3/46 + 14883*(-7*x + 19)/(8464*(2*x**2 - x + 3)) - (-4*x + 1)*(5*x**2 +
3*x + 2)**3/(46*(2*x**2 - x + 3)**2) + 825*log(2*x**2 - x + 3)/32 - 165099*sqrt(
23)*atan(sqrt(23)*(4*x/23 - 1/23))/194672 - Integral(-1155/2, x)/46 - 1275*Integ
ral(x, x)/92

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Mathematica [A]  time = 0.0696539, size = 84, normalized size = 1. \[ -\frac{121 (12828 x-21193)}{33856 \left (2 x^2-x+3\right )}+\frac{1331 (45 x-17)}{1472 \left (2 x^2-x+3\right )^2}+\frac{825}{32} \log \left (2 x^2-x+3\right )+\frac{125 x}{8}-\frac{165099 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{8464 \sqrt{23}} \]

Antiderivative was successfully verified.

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

[Out]

(125*x)/8 + (1331*(-17 + 45*x))/(1472*(3 - x + 2*x^2)^2) - (121*(-21193 + 12828*
x))/(33856*(3 - x + 2*x^2)) - (165099*ArcTan[(-1 + 4*x)/Sqrt[23]])/(8464*Sqrt[23
]) + (825*Log[3 - x + 2*x^2])/32

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Maple [A]  time = 0.01, size = 63, normalized size = 0.8 \[{\frac{125\,x}{8}}+{\frac{11}{2\, \left ( 2\,{x}^{2}-x+3 \right ) ^{2}} \left ( -{\frac{35277\,{x}^{3}}{2116}}+{\frac{303677\,{x}^{2}}{8464}}-{\frac{132803\,x}{4232}}+{\frac{326029}{8464}} \right ) }+{\frac{825\,\ln \left ( 8\,{x}^{2}-4\,x+12 \right ) }{32}}-{\frac{165099\,\sqrt{23}}{194672}\arctan \left ({\frac{ \left ( 16\,x-4 \right ) \sqrt{23}}{92}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125/8*x+11/2*(-35277/2116*x^3+303677/8464*x^2-132803/4232*x+326029/8464)/(2*x^2-
x+3)^2+825/32*ln(8*x^2-4*x+12)-165099/194672*23^(1/2)*arctan(1/92*(16*x-4)*23^(1
/2))

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Maxima [A]  time = 0.769468, size = 97, normalized size = 1.15 \[ -\frac{165099}{194672} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{125}{8} \, x - \frac{121 \,{\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} + \frac{825}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-165099/194672*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 125/8*x - 121/16928*(1
2828*x^3 - 27607*x^2 + 24146*x - 29639)/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9) + 825
/32*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.265633, size = 170, normalized size = 2.02 \[ \frac{\sqrt{23}{\left (436425 \, \sqrt{23}{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x^{2} - x + 3\right ) - 330198 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \sqrt{23}{\left (1058000 \, x^{5} - 1058000 \, x^{4} + 1886312 \, x^{3} + 1753447 \, x^{2} - 541166 \, x + 3586319\right )}\right )}}{389344 \,{\left (4 \, x^{4} - 4 \, x^{3} + 13 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/389344*sqrt(23)*(436425*sqrt(23)*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*log(2*x^2
- x + 3) - 330198*(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)*arctan(1/23*sqrt(23)*(4*x -
1)) + sqrt(23)*(1058000*x^5 - 1058000*x^4 + 1886312*x^3 + 1753447*x^2 - 541166*
x + 3586319))/(4*x^4 - 4*x^3 + 13*x^2 - 6*x + 9)

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Sympy [A]  time = 0.281622, size = 82, normalized size = 0.98 \[ \frac{125 x}{8} - \frac{1552188 x^{3} - 3340447 x^{2} + 2921666 x - 3586319}{67712 x^{4} - 67712 x^{3} + 220064 x^{2} - 101568 x + 152352} + \frac{825 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{32} - \frac{165099 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{194672} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

125*x/8 - (1552188*x**3 - 3340447*x**2 + 2921666*x - 3586319)/(67712*x**4 - 6771
2*x**3 + 220064*x**2 - 101568*x + 152352) + 825*log(x**2 - x/2 + 3/2)/32 - 16509
9*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/194672

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GIAC/XCAS [A]  time = 0.265513, size = 84, normalized size = 1. \[ -\frac{165099}{194672} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{125}{8} \, x - \frac{121 \,{\left (12828 \, x^{3} - 27607 \, x^{2} + 24146 \, x - 29639\right )}}{16928 \,{\left (2 \, x^{2} - x + 3\right )}^{2}} + \frac{825}{32} \,{\rm ln}\left (2 \, x^{2} - x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-165099/194672*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 125/8*x - 121/16928*(1
2828*x^3 - 27607*x^2 + 24146*x - 29639)/(2*x^2 - x + 3)^2 + 825/32*ln(2*x^2 - x
+ 3)