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

Optimal. Leaf size=70 \[ \frac{25 x^5}{2}+\frac{575 x^4}{16}+\frac{965 x^3}{24}-\frac{829 x^2}{32}+\frac{1331}{128} \log \left (2 x^2-x+3\right )-\frac{4795 x}{32}-\frac{59895 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{23}} \]

[Out]

(-4795*x)/32 - (829*x^2)/32 + (965*x^3)/24 + (575*x^4)/16 + (25*x^5)/2 - (59895*ArcTan[(1 - 4*x)/Sqrt[23]])/(6
4*Sqrt[23]) + (1331*Log[3 - x + 2*x^2])/128

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Rubi [A]  time = 0.0560196, 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} \[ \frac{25 x^5}{2}+\frac{575 x^4}{16}+\frac{965 x^3}{24}-\frac{829 x^2}{32}+\frac{1331}{128} \log \left (2 x^2-x+3\right )-\frac{4795 x}{32}-\frac{59895 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4795*x)/32 - (829*x^2)/32 + (965*x^3)/24 + (575*x^4)/16 + (25*x^5)/2 - (59895*ArcTan[(1 - 4*x)/Sqrt[23]])/(6
4*Sqrt[23]) + (1331*Log[3 - x + 2*x^2])/128

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}{3-x+2 x^2} \, dx &=\int \left (-\frac{4795}{32}-\frac{829 x}{16}+\frac{965 x^2}{8}+\frac{575 x^3}{4}+\frac{125 x^4}{2}+\frac{1331 (11+x)}{32 \left (3-x+2 x^2\right )}\right ) \, dx\\ &=-\frac{4795 x}{32}-\frac{829 x^2}{32}+\frac{965 x^3}{24}+\frac{575 x^4}{16}+\frac{25 x^5}{2}+\frac{1331}{32} \int \frac{11+x}{3-x+2 x^2} \, dx\\ &=-\frac{4795 x}{32}-\frac{829 x^2}{32}+\frac{965 x^3}{24}+\frac{575 x^4}{16}+\frac{25 x^5}{2}+\frac{1331}{128} \int \frac{-1+4 x}{3-x+2 x^2} \, dx+\frac{59895}{128} \int \frac{1}{3-x+2 x^2} \, dx\\ &=-\frac{4795 x}{32}-\frac{829 x^2}{32}+\frac{965 x^3}{24}+\frac{575 x^4}{16}+\frac{25 x^5}{2}+\frac{1331}{128} \log \left (3-x+2 x^2\right )-\frac{59895}{64} \operatorname{Subst}\left (\int \frac{1}{-23-x^2} \, dx,x,-1+4 x\right )\\ &=-\frac{4795 x}{32}-\frac{829 x^2}{32}+\frac{965 x^3}{24}+\frac{575 x^4}{16}+\frac{25 x^5}{2}-\frac{59895 \tan ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{64 \sqrt{23}}+\frac{1331}{128} \log \left (3-x+2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0210563, size = 63, normalized size = 0.9 \[ \frac{1}{384} \left (4 x \left (1200 x^4+3450 x^3+3860 x^2-2487 x-14385\right )+3993 \log \left (2 x^2-x+3\right )\right )+\frac{59895 \tan ^{-1}\left (\frac{4 x-1}{\sqrt{23}}\right )}{64 \sqrt{23}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(59895*ArcTan[(-1 + 4*x)/Sqrt[23]])/(64*Sqrt[23]) + (4*x*(-14385 - 2487*x + 3860*x^2 + 3450*x^3 + 1200*x^4) +
3993*Log[3 - x + 2*x^2])/384

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Maple [A]  time = 0.047, size = 54, normalized size = 0.8 \begin{align*}{\frac{25\,{x}^{5}}{2}}+{\frac{575\,{x}^{4}}{16}}+{\frac{965\,{x}^{3}}{24}}-{\frac{829\,{x}^{2}}{32}}-{\frac{4795\,x}{32}}+{\frac{1331\,\ln \left ( 2\,{x}^{2}-x+3 \right ) }{128}}+{\frac{59895\,\sqrt{23}}{1472}\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),x)

[Out]

25/2*x^5+575/16*x^4+965/24*x^3-829/32*x^2-4795/32*x+1331/128*ln(2*x^2-x+3)+59895/1472*23^(1/2)*arctan(1/23*(-1
+4*x)*23^(1/2))

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Maxima [A]  time = 1.47194, size = 72, normalized size = 1.03 \begin{align*} \frac{25}{2} \, x^{5} + \frac{575}{16} \, x^{4} + \frac{965}{24} \, x^{3} - \frac{829}{32} \, x^{2} + \frac{59895}{1472} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4795}{32} \, x + \frac{1331}{128} \, \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),x, algorithm="maxima")

[Out]

25/2*x^5 + 575/16*x^4 + 965/24*x^3 - 829/32*x^2 + 59895/1472*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 4795/3
2*x + 1331/128*log(2*x^2 - x + 3)

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Fricas [A]  time = 0.935331, size = 196, normalized size = 2.8 \begin{align*} \frac{25}{2} \, x^{5} + \frac{575}{16} \, x^{4} + \frac{965}{24} \, x^{3} - \frac{829}{32} \, x^{2} + \frac{59895}{1472} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4795}{32} \, x + \frac{1331}{128} \, \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),x, algorithm="fricas")

[Out]

25/2*x^5 + 575/16*x^4 + 965/24*x^3 - 829/32*x^2 + 59895/1472*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 4795/3
2*x + 1331/128*log(2*x^2 - x + 3)

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Sympy [A]  time = 0.143687, size = 73, normalized size = 1.04 \begin{align*} \frac{25 x^{5}}{2} + \frac{575 x^{4}}{16} + \frac{965 x^{3}}{24} - \frac{829 x^{2}}{32} - \frac{4795 x}{32} + \frac{1331 \log{\left (x^{2} - \frac{x}{2} + \frac{3}{2} \right )}}{128} + \frac{59895 \sqrt{23} \operatorname{atan}{\left (\frac{4 \sqrt{23} x}{23} - \frac{\sqrt{23}}{23} \right )}}{1472} \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),x)

[Out]

25*x**5/2 + 575*x**4/16 + 965*x**3/24 - 829*x**2/32 - 4795*x/32 + 1331*log(x**2 - x/2 + 3/2)/128 + 59895*sqrt(
23)*atan(4*sqrt(23)*x/23 - sqrt(23)/23)/1472

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Giac [A]  time = 1.16731, size = 72, normalized size = 1.03 \begin{align*} \frac{25}{2} \, x^{5} + \frac{575}{16} \, x^{4} + \frac{965}{24} \, x^{3} - \frac{829}{32} \, x^{2} + \frac{59895}{1472} \, \sqrt{23} \arctan \left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{4795}{32} \, x + \frac{1331}{128} \, \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),x, algorithm="giac")

[Out]

25/2*x^5 + 575/16*x^4 + 965/24*x^3 - 829/32*x^2 + 59895/1472*sqrt(23)*arctan(1/23*sqrt(23)*(4*x - 1)) - 4795/3
2*x + 1331/128*log(2*x^2 - x + 3)