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

Optimal. Leaf size=70 \[ \frac{8 x^5}{25}-\frac{21 x^4}{25}+\frac{1222 x^3}{375}-\frac{7451 x^2}{1250}-\frac{158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac{49508 x}{3125}+\frac{328757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{15625 \sqrt{31}} \]

[Out]

(49508*x)/3125 - (7451*x^2)/1250 + (1222*x^3)/375 - (21*x^4)/25 + (8*x^5)/25 + (
328757*ArcTan[(3 + 10*x)/Sqrt[31]])/(15625*Sqrt[31]) - (158389*Log[2 + 3*x + 5*x
^2])/31250

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Rubi [A]  time = 0.102539, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{8 x^5}{25}-\frac{21 x^4}{25}+\frac{1222 x^3}{375}-\frac{7451 x^2}{1250}-\frac{158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac{49508 x}{3125}+\frac{328757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{15625 \sqrt{31}} \]

Antiderivative was successfully verified.

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

[Out]

(49508*x)/3125 - (7451*x^2)/1250 + (1222*x^3)/375 - (21*x^4)/25 + (8*x^5)/25 + (
328757*ArcTan[(3 + 10*x)/Sqrt[31]])/(15625*Sqrt[31]) - (158389*Log[2 + 3*x + 5*x
^2])/31250

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{637 x^{3}}{375} - \frac{\left (- 40 x + 65\right ) \left (2 x^{2} - x + 3\right )^{2}}{500} - \frac{158389 \log{\left (5 x^{2} + 3 x + 2 \right )}}{31250} + \frac{328757 \sqrt{31} \operatorname{atan}{\left (\sqrt{31} \left (\frac{10 x}{31} + \frac{3}{31}\right ) \right )}}{484375} - \frac{\int \left (- \frac{179282}{25}\right )\, dx}{500} - \frac{9477 \int x\, dx}{1250} \]

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),x)

[Out]

637*x**3/375 - (-40*x + 65)*(2*x**2 - x + 3)**2/500 - 158389*log(5*x**2 + 3*x +
2)/31250 + 328757*sqrt(31)*atan(sqrt(31)*(10*x/31 + 3/31))/484375 - Integral(-17
9282/25, x)/500 - 9477*Integral(x, x)/1250

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Mathematica [A]  time = 0.040395, size = 63, normalized size = 0.9 \[ \frac{31 \left (5 x \left (6000 x^4-15750 x^3+61100 x^2-111765 x+297048\right )-475167 \log \left (5 x^2+3 x+2\right )\right )+1972542 \sqrt{31} \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{2906250} \]

Antiderivative was successfully verified.

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

[Out]

(1972542*Sqrt[31]*ArcTan[(3 + 10*x)/Sqrt[31]] + 31*(5*x*(297048 - 111765*x + 611
00*x^2 - 15750*x^3 + 6000*x^4) - 475167*Log[2 + 3*x + 5*x^2]))/2906250

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Maple [A]  time = 0.005, size = 54, normalized size = 0.8 \[{\frac{49508\,x}{3125}}-{\frac{7451\,{x}^{2}}{1250}}+{\frac{1222\,{x}^{3}}{375}}-{\frac{21\,{x}^{4}}{25}}+{\frac{8\,{x}^{5}}{25}}-{\frac{158389\,\ln \left ( 5\,{x}^{2}+3\,x+2 \right ) }{31250}}+{\frac{328757\,\sqrt{31}}{484375}\arctan \left ({\frac{ \left ( 3+10\,x \right ) \sqrt{31}}{31}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

49508/3125*x-7451/1250*x^2+1222/375*x^3-21/25*x^4+8/25*x^5-158389/31250*ln(5*x^2
+3*x+2)+328757/484375*arctan(1/31*(3+10*x)*31^(1/2))*31^(1/2)

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Maxima [A]  time = 0.768738, size = 72, normalized size = 1.03 \[ \frac{8}{25} \, x^{5} - \frac{21}{25} \, x^{4} + \frac{1222}{375} \, x^{3} - \frac{7451}{1250} \, x^{2} + \frac{328757}{484375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{49508}{3125} \, x - \frac{158389}{31250} \, \log \left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/25*x^5 - 21/25*x^4 + 1222/375*x^3 - 7451/1250*x^2 + 328757/484375*sqrt(31)*arc
tan(1/31*sqrt(31)*(10*x + 3)) + 49508/3125*x - 158389/31250*log(5*x^2 + 3*x + 2)

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Fricas [A]  time = 0.262724, size = 86, normalized size = 1.23 \[ \frac{1}{2906250} \, \sqrt{31}{\left (5 \, \sqrt{31}{\left (6000 \, x^{5} - 15750 \, x^{4} + 61100 \, x^{3} - 111765 \, x^{2} + 297048 \, x\right )} - 475167 \, \sqrt{31} \log \left (5 \, x^{2} + 3 \, x + 2\right ) + 1972542 \, \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2906250*sqrt(31)*(5*sqrt(31)*(6000*x^5 - 15750*x^4 + 61100*x^3 - 111765*x^2 +
297048*x) - 475167*sqrt(31)*log(5*x^2 + 3*x + 2) + 1972542*arctan(1/31*sqrt(31)*
(10*x + 3)))

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Sympy [A]  time = 0.158155, size = 76, normalized size = 1.09 \[ \frac{8 x^{5}}{25} - \frac{21 x^{4}}{25} + \frac{1222 x^{3}}{375} - \frac{7451 x^{2}}{1250} + \frac{49508 x}{3125} - \frac{158389 \log{\left (x^{2} + \frac{3 x}{5} + \frac{2}{5} \right )}}{31250} + \frac{328757 \sqrt{31} \operatorname{atan}{\left (\frac{10 \sqrt{31} x}{31} + \frac{3 \sqrt{31}}{31} \right )}}{484375} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*x**5/25 - 21*x**4/25 + 1222*x**3/375 - 7451*x**2/1250 + 49508*x/3125 - 158389*
log(x**2 + 3*x/5 + 2/5)/31250 + 328757*sqrt(31)*atan(10*sqrt(31)*x/31 + 3*sqrt(3
1)/31)/484375

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GIAC/XCAS [A]  time = 0.265556, size = 72, normalized size = 1.03 \[ \frac{8}{25} \, x^{5} - \frac{21}{25} \, x^{4} + \frac{1222}{375} \, x^{3} - \frac{7451}{1250} \, x^{2} + \frac{328757}{484375} \, \sqrt{31} \arctan \left (\frac{1}{31} \, \sqrt{31}{\left (10 \, x + 3\right )}\right ) + \frac{49508}{3125} \, x - \frac{158389}{31250} \,{\rm ln}\left (5 \, x^{2} + 3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8/25*x^5 - 21/25*x^4 + 1222/375*x^3 - 7451/1250*x^2 + 328757/484375*sqrt(31)*arc
tan(1/31*sqrt(31)*(10*x + 3)) + 49508/3125*x - 158389/31250*ln(5*x^2 + 3*x + 2)