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

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 = 0.0529201, 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, Rules used = {1657, 634, 618, 204, 628} \[ \frac{8 x^5}{25}-\frac{21 x^4}{25}+\frac{1222 x^3}{375}-\frac{7451 x^2}{1250}-\frac{158389 \log \left (5 x^2+3 x+2\right )}{31250}+\frac{49508 x}{3125}+\frac{328757 \tan ^{-1}\left (\frac{10 x+3}{\sqrt{31}}\right )}{15625 \sqrt{31}} \]

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

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{\left (3-x+2 x^2\right )^3}{2+3 x+5 x^2} \, dx &=\int \left (\frac{49508}{3125}-\frac{7451 x}{625}+\frac{1222 x^2}{125}-\frac{84 x^3}{25}+\frac{8 x^4}{5}-\frac{1331 (11+119 x)}{3125 \left (2+3 x+5 x^2\right )}\right ) \, dx\\ &=\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{1331 \int \frac{11+119 x}{2+3 x+5 x^2} \, dx}{3125}\\ &=\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 \int \frac{3+10 x}{2+3 x+5 x^2} \, dx}{31250}+\frac{328757 \int \frac{1}{2+3 x+5 x^2} \, dx}{31250}\\ &=\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 \log \left (2+3 x+5 x^2\right )}{31250}-\frac{328757 \operatorname{Subst}\left (\int \frac{1}{-31-x^2} \, dx,x,3+10 x\right )}{15625}\\ &=\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{328757 \tan ^{-1}\left (\frac{3+10 x}{\sqrt{31}}\right )}{15625 \sqrt{31}}-\frac{158389 \log \left (2+3 x+5 x^2\right )}{31250}\\ \end{align*}

Mathematica [A]  time = 0.0224648, 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 + 61100*x^2 - 15750*x^3 + 6000*x^4)
- 475167*Log[2 + 3*x + 5*x^2]))/2906250

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Maple [A]  time = 0.046, size = 54, normalized size = 0.8 \begin{align*}{\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 ) } \end{align*}

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 = 1.46638, size = 72, normalized size = 1.03 \begin{align*} \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 ) \end{align*}

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)*arctan(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 = 0.991387, size = 219, normalized size = 3.13 \begin{align*} \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 ) \end{align*}

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]

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

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Sympy [A]  time = 0.149317, size = 76, normalized size = 1.09 \begin{align*} \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} \end{align*}

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(31)/31)/484375

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Giac [A]  time = 1.187, size = 72, normalized size = 1.03 \begin{align*} \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 ) \end{align*}

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)*arctan(1/31*sqrt(31)*(10*x + 3))
+ 49508/3125*x - 158389/31250*log(5*x^2 + 3*x + 2)