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

Optimal. Leaf size=77 \[ \frac{8 x^3}{75}-\frac{54 x^2}{125}+\frac{1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac{10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac{1466 x}{625}+\frac{3819607 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{96875 \sqrt{31}} \]

[Out]

(1466*x)/625 - (54*x^2)/125 + (8*x^3)/75 + (1331*(443 + 247*x))/(96875*(2 + 3*x
+ 5*x^2)) + (3819607*ArcTan[(3 + 10*x)/Sqrt[31]])/(96875*Sqrt[31]) - (10769*Log[
2 + 3*x + 5*x^2])/6250

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Rubi [A]  time = 0.125044, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{8 x^3}{75}-\frac{54 x^2}{125}+\frac{1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac{10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac{1466 x}{625}+\frac{3819607 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{96875 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(1466*x)/625 - (54*x^2)/125 + (8*x^3)/75 + (1331*(443 + 247*x))/(96875*(2 + 3*x
+ 5*x^2)) + (3819607*ArcTan[(3 + 10*x)/Sqrt[31]])/(96875*Sqrt[31]) - (10769*Log[
2 + 3*x + 5*x^2])/6250

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{16 x^{5}}{31} + \frac{144 x^{4}}{155} - \frac{6328 x^{3}}{2325} + \frac{\left (10 x + 3\right ) \left (2 x^{2} - x + 3\right )^{3}}{31 \left (5 x^{2} + 3 x + 2\right )} - \frac{10769 \log{\left (5 x^{2} + 3 x + 2 \right )}}{6250} + \frac{3819607 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{3003125} - \frac{\int \frac{31217}{625}\, dx}{31} + \frac{19124 \int x\, dx}{3875} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*x**5/31 + 144*x**4/155 - 6328*x**3/2325 + (10*x + 3)*(2*x**2 - x + 3)**3/(31
*(5*x**2 + 3*x + 2)) - 10769*log(5*x**2 + 3*x + 2)/6250 + 3819607*sqrt(31)*atan(
sqrt(31)*(10*x/31 + 3/31))/3003125 - Integral(31217/625, x)/31 + 19124*Integral(
x, x)/3875

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Mathematica [A]  time = 0.0539002, size = 77, normalized size = 1. \[ \frac{8 x^3}{75}-\frac{54 x^2}{125}+\frac{1331 (247 x+443)}{96875 \left (5 x^2+3 x+2\right )}-\frac{10769 \log \left (5 x^2+3 x+2\right )}{6250}+\frac{1466 x}{625}+\frac{3819607 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{96875 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(1466*x)/625 - (54*x^2)/125 + (8*x^3)/75 + (1331*(443 + 247*x))/(96875*(2 + 3*x
+ 5*x^2)) + (3819607*ArcTan[(3 + 10*x)/Sqrt[31]])/(96875*Sqrt[31]) - (10769*Log[
2 + 3*x + 5*x^2])/6250

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Maple [A]  time = 0.009, size = 61, normalized size = 0.8 \[{\frac{8\,{x}^{3}}{75}}-{\frac{54\,{x}^{2}}{125}}+{\frac{1466\,x}{625}}-{\frac{121}{625} \left ( -{\frac{2717\,x}{775}}-{\frac{4873}{775}} \right ) \left ({x}^{2}+{\frac{3\,x}{5}}+{\frac{2}{5}} \right ) ^{-1}}-{\frac{10769\,\ln \left ( 25\,{x}^{2}+15\,x+10 \right ) }{6250}}+{\frac{3819607\,\sqrt{31}}{3003125}\arctan \left ({\frac{ \left ( 50\,x+15 \right ) \sqrt{31}}{155}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/75*x^3-54/125*x^2+1466/625*x-121/625*(-2717/775*x-4873/775)/(x^2+3/5*x+2/5)-10
769/6250*ln(25*x^2+15*x+10)+3819607/3003125*31^(1/2)*arctan(1/155*(50*x+15)*31^(
1/2))

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Maxima [A]  time = 0.771164, size = 84, normalized size = 1.09 \[ \frac{8}{75} \, x^{3} - \frac{54}{125} \, x^{2} + \frac{3819607}{3003125} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{1466}{625} \, x + \frac{1331 \,{\left (247 \, x + 443\right )}}{96875 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac{10769}{6250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/75*x^3 - 54/125*x^2 + 3819607/3003125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)
) + 1466/625*x + 1331/96875*(247*x + 443)/(5*x^2 + 3*x + 2) - 10769/6250*log(5*x
^2 + 3*x + 2)

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Fricas [A]  time = 0.265656, size = 131, normalized size = 1.7 \[ -\frac{\sqrt{31}{\left (1001517 \, \sqrt{31}{\left (5 \, x^{2} + 3 \, x + 2\right )} \log \left (5 \, x^{2} + 3 \, x + 2\right ) - 22917642 \,{\left (5 \, x^{2} + 3 \, x + 2\right )} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) - 2 \, \sqrt{31}{\left (155000 \, x^{5} - 534750 \, x^{4} + 3093800 \, x^{3} + 1793970 \, x^{2} + 2349651 \, x + 1768899\right )}\right )}}{18018750 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18018750*sqrt(31)*(1001517*sqrt(31)*(5*x^2 + 3*x + 2)*log(5*x^2 + 3*x + 2) -
22917642*(5*x^2 + 3*x + 2)*arctan(1/31*sqrt(31)*(10*x + 3)) - 2*sqrt(31)*(155000
*x^5 - 534750*x^4 + 3093800*x^3 + 1793970*x^2 + 2349651*x + 1768899))/(5*x^2 + 3
*x + 2)

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Sympy [A]  time = 0.220291, size = 78, normalized size = 1.01 \[ \frac{8 x^{3}}{75} - \frac{54 x^{2}}{125} + \frac{1466 x}{625} + \frac{328757 x + 589633}{484375 x^{2} + 290625 x + 193750} - \frac{10769 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{6250} + \frac{3819607 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{3003125} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*x**3/75 - 54*x**2/125 + 1466*x/625 + (328757*x + 589633)/(484375*x**2 + 290625
*x + 193750) - 10769*log(x**2 + 3*x/5 + 2/5)/6250 + 3819607*sqrt(31)*atan(10*sqr
t(31)*x/31 + 3*sqrt(31)/31)/3003125

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GIAC/XCAS [A]  time = 0.264709, size = 84, normalized size = 1.09 \[ \frac{8}{75} \, x^{3} - \frac{54}{125} \, x^{2} + \frac{3819607}{3003125} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{1466}{625} \, x + \frac{1331 \,{\left (247 \, x + 443\right )}}{96875 \,{\left (5 \, x^{2} + 3 \, x + 2\right )}} - \frac{10769}{6250} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/75*x^3 - 54/125*x^2 + 3819607/3003125*sqrt(31)*arctan(1/31*sqrt(31)*(10*x + 3)
) + 1466/625*x + 1331/96875*(247*x + 443)/(5*x^2 + 3*x + 2) - 10769/6250*ln(5*x^
2 + 3*x + 2)