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

Optimal. Leaf size=77 \[ \frac{125 x^3}{12}+\frac{175 x^2}{4}-\frac{1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}-\frac{2057}{32} \log \left (2 x^2-x+3\right )+\frac{915 x}{16}+\frac{223971 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{368 \sqrt{23}} \]

[Out]

(915*x)/16 + (175*x^2)/4 + (125*x^3)/12 - (1331*(17 - 45*x))/(736*(3 - x + 2*x^2)) + (223971*ArcTan[(1 - 4*x)/
Sqrt[23]])/(368*Sqrt[23]) - (2057*Log[3 - x + 2*x^2])/32

________________________________________________________________________________________

Rubi [A]  time = 0.0725653, 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, Rules used = {1660, 1657, 634, 618, 204, 628} \[ \frac{125 x^3}{12}+\frac{175 x^2}{4}-\frac{1331 (17-45 x)}{736 \left (2 x^2-x+3\right )}-\frac{2057}{32} \log \left (2 x^2-x+3\right )+\frac{915 x}{16}+\frac{223971 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{368 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(915*x)/16 + (175*x^2)/4 + (125*x^3)/12 - (1331*(17 - 45*x))/(736*(3 - x + 2*x^2)) + (223971*ArcTan[(1 - 4*x)/
Sqrt[23]])/(368*Sqrt[23]) - (2057*Log[3 - x + 2*x^2])/32

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

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 (2+3 x+5 x^2\right )^3}{\left (3-x+2 x^2\right )^2} \, dx &=-\frac{1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac{1}{23} \int \frac{-\frac{25195}{16}-\frac{19067 x}{16}+\frac{22195 x^2}{8}+\frac{13225 x^3}{4}+\frac{2875 x^4}{2}}{3-x+2 x^2} \, dx\\ &=-\frac{1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac{1}{23} \int \left (\frac{21045}{16}+\frac{4025 x}{2}+\frac{2875 x^2}{4}-\frac{121 (365+391 x)}{8 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=\frac{915 x}{16}+\frac{175 x^2}{4}+\frac{125 x^3}{12}-\frac{1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}-\frac{121}{184} \int \frac{365+391 x}{3-x+2 x^2} \, dx\\ &=\frac{915 x}{16}+\frac{175 x^2}{4}+\frac{125 x^3}{12}-\frac{1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}-\frac{2057}{32} \int \frac{-1+4 x}{3-x+2 x^2} \, dx-\frac{223971}{736} \int \frac{1}{3-x+2 x^2} \, dx\\ &=\frac{915 x}{16}+\frac{175 x^2}{4}+\frac{125 x^3}{12}-\frac{1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}-\frac{2057}{32} \log \left (3-x+2 x^2\right )+\frac{223971}{368} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac{915 x}{16}+\frac{175 x^2}{4}+\frac{125 x^3}{12}-\frac{1331 (17-45 x)}{736 \left (3-x+2 x^2\right )}+\frac{223971 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{368 \sqrt{23}}-\frac{2057}{32} \log \left (3-x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0274481, size = 77, normalized size = 1. \[ \frac{125 x^3}{12}+\frac{175 x^2}{4}+\frac{1331 (45 x-17)}{736 \left (2 x^2-x+3\right )}-\frac{2057}{32} \log \left (2 x^2-x+3\right )+\frac{915 x}{16}-\frac{223971 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{368 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(915*x)/16 + (175*x^2)/4 + (125*x^3)/12 + (1331*(-17 + 45*x))/(736*(3 - x + 2*x^2)) - (223971*ArcTan[(-1 + 4*x
)/Sqrt[23]])/(368*Sqrt[23]) - (2057*Log[3 - x + 2*x^2])/32

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Maple [A]  time = 0.047, size = 61, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{3}}{12}}+{\frac{175\,{x}^{2}}{4}}+{\frac{915\,x}{16}}-{\frac{121}{16} \left ( -{\frac{495\,x}{92}}+{\frac{187}{92}} \right ) \left ({x}^{2}-{\frac{x}{2}}+{\frac{3}{2}} \right ) ^{-1}}-{\frac{2057\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{32}}-{\frac{223971\,\sqrt{23}}{8464}\arctan \left ({\frac{ \left ( -1+4\,x \right ) \sqrt{23}}{23}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/12*x^3+175/4*x^2+915/16*x-121/16*(-495/92*x+187/92)/(x^2-1/2*x+3/2)-2057/32*ln(2*x^2-x+3)-223971/8464*23^(
1/2)*arctan(1/23*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.43452, size = 84, normalized size = 1.09 \begin{align*} \frac{125}{12} \, x^{3} + \frac{175}{4} \, x^{2} - \frac{223971}{8464} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{915}{16} \, x + \frac{1331 \,{\left (45 \, x - 17\right )}}{736 \,{\left (2 \, x^{2} - x + 3\right )}} - \frac{2057}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/12*x^3 + 175/4*x^2 - 223971/8464*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 915/16*x + 1331/736*(45*x - 17
)/(2*x^2 - x + 3) - 2057/32*log(2*x^2 - x + 3)

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Fricas [A]  time = 1.03317, size = 292, normalized size = 3.79 \begin{align*} \frac{1058000 \, x^{5} + 3914600 \, x^{4} + 5173620 \, x^{3} - 1343826 \, \sqrt{23}{\left (2 \, x^{2} - x + 3\right )} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + 3761190 \, x^{2} - 3264459 \,{\left (2 \, x^{2} - x + 3\right )} \log \left (2 \, x^{2} - x + 3\right ) + 12845385 \, x - 1561263}{50784 \,{\left (2 \, x^{2} - x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50784*(1058000*x^5 + 3914600*x^4 + 5173620*x^3 - 1343826*sqrt(23)*(2*x^2 - x + 3)*arctan(1/23*sqrt(23)*(4*x
- 1)) + 3761190*x^2 - 3264459*(2*x^2 - x + 3)*log(2*x^2 - x + 3) + 12845385*x - 1561263)/(2*x^2 - x + 3)

________________________________________________________________________________________

Sympy [A]  time = 0.254475, size = 75, normalized size = 0.97 \begin{align*} \frac{125 x^{3}}{12} + \frac{175 x^{2}}{4} + \frac{915 x}{16} + \frac{59895 x - 22627}{1472 x^{2} - 736 x + 2208} - \frac{2057 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{32} - \frac{223971 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{8464} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*x**3/12 + 175*x**2/4 + 915*x/16 + (59895*x - 22627)/(1472*x**2 - 736*x + 2208) - 2057*log(x**2 - x/2 + 3/2
)/32 - 223971*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/8464

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Giac [A]  time = 1.18254, size = 84, normalized size = 1.09 \begin{align*} \frac{125}{12} \, x^{3} + \frac{175}{4} \, x^{2} - \frac{223971}{8464} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{915}{16} \, x + \frac{1331 \,{\left (45 \, x - 17\right )}}{736 \,{\left (2 \, x^{2} - x + 3\right )}} - \frac{2057}{32} \, \log \left (2 \, x^{2} - x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/12*x^3 + 175/4*x^2 - 223971/8464*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 915/16*x + 1331/736*(45*x - 17
)/(2*x^2 - x + 3) - 2057/32*log(2*x^2 - x + 3)