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

Optimal. Leaf size=56 \[ \frac{25 x^3}{6}+\frac{85 x^2}{8}-\frac{363}{32} \log \left (2 x^2-x+3\right )+\frac{51 x}{8}+\frac{847 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{23}} \]

[Out]

(51*x)/8 + (85*x^2)/8 + (25*x^3)/6 + (847*ArcTan[(1 - 4*x)/Sqrt[23]])/(16*Sqrt[23]) - (363*Log[3 - x + 2*x^2])
/32

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Rubi [A]  time = 0.0503026, antiderivative size = 56, 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} \[ \frac{25 x^3}{6}+\frac{85 x^2}{8}-\frac{363}{32} \log \left (2 x^2-x+3\right )+\frac{51 x}{8}+\frac{847 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(51*x)/8 + (85*x^2)/8 + (25*x^3)/6 + (847*ArcTan[(1 - 4*x)/Sqrt[23]])/(16*Sqrt[23]) - (363*Log[3 - x + 2*x^2])
/32

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 )^2}{3-x+2 x^2} \, dx &=\int \left (\frac{51}{8}+\frac{85 x}{4}+\frac{25 x^2}{2}-\frac{121 (1+3 x)}{8 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=\frac{51 x}{8}+\frac{85 x^2}{8}+\frac{25 x^3}{6}-\frac{121}{8} \int \frac{1+3 x}{3-x+2 x^2} \, dx\\ &=\frac{51 x}{8}+\frac{85 x^2}{8}+\frac{25 x^3}{6}-\frac{363}{32} \int \frac{-1+4 x}{3-x+2 x^2} \, dx-\frac{847}{32} \int \frac{1}{3-x+2 x^2} \, dx\\ &=\frac{51 x}{8}+\frac{85 x^2}{8}+\frac{25 x^3}{6}-\frac{363}{32} \log \left (3-x+2 x^2\right )+\frac{847}{16} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=\frac{51 x}{8}+\frac{85 x^2}{8}+\frac{25 x^3}{6}+\frac{847 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{16 \sqrt{23}}-\frac{363}{32} \log \left (3-x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0168558, size = 52, normalized size = 0.93 \[ \frac{1}{24} x \left (100 x^2+255 x+153\right )-\frac{363}{32} \log \left (2 x^2-x+3\right )-\frac{847 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{16 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(153 + 255*x + 100*x^2))/24 - (847*ArcTan[(-1 + 4*x)/Sqrt[23]])/(16*Sqrt[23]) - (363*Log[3 - x + 2*x^2])/32

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Maple [A]  time = 0.049, size = 44, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{3}}{6}}+{\frac{85\,{x}^{2}}{8}}+{\frac{51\,x}{8}}-{\frac{363\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{32}}-{\frac{847\,\sqrt{23}}{368}\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)^2/(2*x^2-x+3),x)

[Out]

25/6*x^3+85/8*x^2+51/8*x-363/32*ln(2*x^2-x+3)-847/368*23^(1/2)*arctan(1/23*(-1+4*x)*23^(1/2))

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Maxima [A]  time = 1.43979, size = 58, normalized size = 1.04 \begin{align*} \frac{25}{6} \, x^{3} + \frac{85}{8} \, x^{2} - \frac{847}{368} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{51}{8} \, x - \frac{363}{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)^2/(2*x^2-x+3),x, algorithm="maxima")

[Out]

25/6*x^3 + 85/8*x^2 - 847/368*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 51/8*x - 363/32*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.993869, size = 147, normalized size = 2.62 \begin{align*} \frac{25}{6} \, x^{3} + \frac{85}{8} \, x^{2} - \frac{847}{368} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{51}{8} \, x - \frac{363}{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)^2/(2*x^2-x+3),x, algorithm="fricas")

[Out]

25/6*x^3 + 85/8*x^2 - 847/368*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 51/8*x - 363/32*log(2*x^2 - x + 3)

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Sympy [A]  time = 0.197072, size = 60, normalized size = 1.07 \begin{align*} \frac{25 x^{3}}{6} + \frac{85 x^{2}}{8} + \frac{51 x}{8} - \frac{363 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{32} - \frac{847 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{368} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*x**3/6 + 85*x**2/8 + 51*x/8 - 363*log(x**2 - x/2 + 3/2)/32 - 847*sqrt(23)*atan(4*sqrt(23)*x/23 - sqrt(23)/2
3)/368

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Giac [A]  time = 1.12605, size = 58, normalized size = 1.04 \begin{align*} \frac{25}{6} \, x^{3} + \frac{85}{8} \, x^{2} - \frac{847}{368} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) + \frac{51}{8} \, x - \frac{363}{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)^2/(2*x^2-x+3),x, algorithm="giac")

[Out]

25/6*x^3 + 85/8*x^2 - 847/368*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) + 51/8*x - 363/32*log(2*x^2 - x + 3)