3.73 \(\int (3-x+2 x^2)^{5/2} (2+3 x+5 x^2)^3 \, dx\)

Optimal. Leaf size=212 \[ \frac{125}{24} \left (2 x^2-x+3\right )^{7/2} x^5+\frac{1175}{96} \left (2 x^2-x+3\right )^{7/2} x^4+\frac{3823}{256} \left (2 x^2-x+3\right )^{7/2} x^3+\frac{80483 \left (2 x^2-x+3\right )^{7/2} x^2}{9216}+\frac{509257 \left (2 x^2-x+3\right )^{7/2} x}{294912}-\frac{1696165 \left (2 x^2-x+3\right )^{7/2}}{2752512}-\frac{57915 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{2097152}-\frac{6660225 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{67108864}-\frac{459555525 (1-4 x) \sqrt{2 x^2-x+3}}{1073741824}-\frac{10569777075 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2147483648 \sqrt{2}} \]

[Out]

(-459555525*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1073741824 - (6660225*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/67108864 - (
57915*(1 - 4*x)*(3 - x + 2*x^2)^(5/2))/2097152 - (1696165*(3 - x + 2*x^2)^(7/2))/2752512 + (509257*x*(3 - x +
2*x^2)^(7/2))/294912 + (80483*x^2*(3 - x + 2*x^2)^(7/2))/9216 + (3823*x^3*(3 - x + 2*x^2)^(7/2))/256 + (1175*x
^4*(3 - x + 2*x^2)^(7/2))/96 + (125*x^5*(3 - x + 2*x^2)^(7/2))/24 - (10569777075*ArcSinh[(1 - 4*x)/Sqrt[23]])/
(2147483648*Sqrt[2])

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Rubi [A]  time = 0.219527, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {1661, 640, 612, 619, 215} \[ \frac{125}{24} \left (2 x^2-x+3\right )^{7/2} x^5+\frac{1175}{96} \left (2 x^2-x+3\right )^{7/2} x^4+\frac{3823}{256} \left (2 x^2-x+3\right )^{7/2} x^3+\frac{80483 \left (2 x^2-x+3\right )^{7/2} x^2}{9216}+\frac{509257 \left (2 x^2-x+3\right )^{7/2} x}{294912}-\frac{1696165 \left (2 x^2-x+3\right )^{7/2}}{2752512}-\frac{57915 (1-4 x) \left (2 x^2-x+3\right )^{5/2}}{2097152}-\frac{6660225 (1-4 x) \left (2 x^2-x+3\right )^{3/2}}{67108864}-\frac{459555525 (1-4 x) \sqrt{2 x^2-x+3}}{1073741824}-\frac{10569777075 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2147483648 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-459555525*(1 - 4*x)*Sqrt[3 - x + 2*x^2])/1073741824 - (6660225*(1 - 4*x)*(3 - x + 2*x^2)^(3/2))/67108864 - (
57915*(1 - 4*x)*(3 - x + 2*x^2)^(5/2))/2097152 - (1696165*(3 - x + 2*x^2)^(7/2))/2752512 + (509257*x*(3 - x +
2*x^2)^(7/2))/294912 + (80483*x^2*(3 - x + 2*x^2)^(7/2))/9216 + (3823*x^3*(3 - x + 2*x^2)^(7/2))/256 + (1175*x
^4*(3 - x + 2*x^2)^(7/2))/96 + (125*x^5*(3 - x + 2*x^2)^(7/2))/24 - (10569777075*ArcSinh[(1 - 4*x)/Sqrt[23]])/
(2147483648*Sqrt[2])

Rule 1661

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expo
n[Pq, x]]}, Simp[(e*x^(q - 1)*(a + b*x + c*x^2)^(p + 1))/(c*(q + 2*p + 1)), x] + Dist[1/(c*(q + 2*p + 1)), Int
[(a + b*x + c*x^2)^p*ExpandToSum[c*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + p)*x^(q - 1) - c*e*(q +
2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 640

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

Rule 612

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

Rule 619

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

Rule 215

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

Rubi steps

\begin{align*} \int \left (3-x+2 x^2\right )^{5/2} \left (2+3 x+5 x^2\right )^3 \, dx &=\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{24} \int \left (3-x+2 x^2\right )^{5/2} \left (192+864 x+2736 x^2+4968 x^3+4965 x^4+\frac{12925 x^5}{2}\right ) \, dx\\ &=\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{1}{528} \int \left (3-x+2 x^2\right )^{5/2} \left (4224+19008 x+60192 x^2+31746 x^3+\frac{630795 x^4}{4}\right ) \, dx\\ &=\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (84480+380160 x-\frac{861795 x^2}{4}+\frac{13279695 x^3}{8}\right ) \, dx}{10560}\\ &=\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (3-x+2 x^2\right )^{5/2} \left (1520640-\frac{12467565 x}{4}+\frac{84027405 x^2}{16}\right ) \, dx}{190080}\\ &=\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{\int \left (\frac{137201625}{16}-\frac{839601675 x}{32}\right ) \left (3-x+2 x^2\right )^{5/2} \, dx}{3041280}\\ &=-\frac{1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{173745 \int \left (3-x+2 x^2\right )^{5/2} \, dx}{262144}\\ &=-\frac{57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac{1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{6660225 \int \left (3-x+2 x^2\right )^{3/2} \, dx}{4194304}\\ &=-\frac{6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac{57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac{1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{459555525 \int \sqrt{3-x+2 x^2} \, dx}{134217728}\\ &=-\frac{459555525 (1-4 x) \sqrt{3-x+2 x^2}}{1073741824}-\frac{6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac{57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac{1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{10569777075 \int \frac{1}{\sqrt{3-x+2 x^2}} \, dx}{2147483648}\\ &=-\frac{459555525 (1-4 x) \sqrt{3-x+2 x^2}}{1073741824}-\frac{6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac{57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac{1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}+\frac{\left (459555525 \sqrt{\frac{23}{2}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{23}}} \, dx,x,-1+4 x\right )}{2147483648}\\ &=-\frac{459555525 (1-4 x) \sqrt{3-x+2 x^2}}{1073741824}-\frac{6660225 (1-4 x) \left (3-x+2 x^2\right )^{3/2}}{67108864}-\frac{57915 (1-4 x) \left (3-x+2 x^2\right )^{5/2}}{2097152}-\frac{1696165 \left (3-x+2 x^2\right )^{7/2}}{2752512}+\frac{509257 x \left (3-x+2 x^2\right )^{7/2}}{294912}+\frac{80483 x^2 \left (3-x+2 x^2\right )^{7/2}}{9216}+\frac{3823}{256} x^3 \left (3-x+2 x^2\right )^{7/2}+\frac{1175}{96} x^4 \left (3-x+2 x^2\right )^{7/2}+\frac{125}{24} x^5 \left (3-x+2 x^2\right )^{7/2}-\frac{10569777075 \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{2147483648 \sqrt{2}}\\ \end{align*}

Mathematica [A]  time = 0.287949, size = 95, normalized size = 0.45 \[ \frac{4 \sqrt{2 x^2-x+3} \left (2818572288000 x^{11}[phone] x^{10}[phone] x^9+14341894045696 x^8+27835561148416 x^7+28347538538496 x^6+34378613923840 x^5+26186527209472 x^4+20384824684416 x^3+10060731582048 x^2+4560943728924 x-1191399152715\right )-665895955725 \sqrt{2} \sinh ^{-1}\left (\frac{1-4 x}{\sqrt{23}}\right )}{270582939648} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[3 - x + 2*x^2]*(-1191399152715 + 4560943728924*x + 10060731582048*x^2 + 20384824684416*x^3 + 261865272
09472*x^4 + 34378613923840*x^5 + 28347538538496*x^6 + 27835561148416*x^7 + 14341894045696*x^8 + 12943588589568
*x^9 + 2395786444800*x^10 + 2818572288000*x^11) - 665895955725*Sqrt[2]*ArcSinh[(1 - 4*x)/Sqrt[23]])/2705829396
48

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Maple [A]  time = 0.061, size = 170, normalized size = 0.8 \begin{align*}{\frac{125\,{x}^{5}}{24} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{1175\,{x}^{4}}{96} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{3823\,{x}^{3}}{256} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{80483\,{x}^{2}}{9216} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{509257\,x}{294912} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}}+{\frac{-459555525+1838222100\,x}{1073741824}\sqrt{2\,{x}^{2}-x+3}}+{\frac{10569777075\,\sqrt{2}}{4294967296}{\it Arcsinh} \left ({\frac{4\,\sqrt{23}}{23} \left ( x-{\frac{1}{4}} \right ) } \right ) }+{\frac{-57915+231660\,x}{2097152} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{5}{2}}}}+{\frac{-6660225+26640900\,x}{67108864} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{3}{2}}}}-{\frac{1696165}{2752512} \left ( 2\,{x}^{2}-x+3 \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/24*x^5*(2*x^2-x+3)^(7/2)+1175/96*x^4*(2*x^2-x+3)^(7/2)+3823/256*x^3*(2*x^2-x+3)^(7/2)+80483/9216*x^2*(2*x^
2-x+3)^(7/2)+509257/294912*x*(2*x^2-x+3)^(7/2)[phone]/1073741824*(-1+4*x)*(2*x^2-x+3)^(1/2)[phone]/429
4967296*2^(1/2)*arcsinh(4/23*23^(1/2)*(x-1/4))+57915/2097152*(-1+4*x)*(2*x^2-x+3)^(5/2)+6660225/67108864*(-1+4
*x)*(2*x^2-x+3)^(3/2)-1696165/2752512*(2*x^2-x+3)^(7/2)

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Maxima [A]  time = 1.48024, size = 271, normalized size = 1.28 \begin{align*} \frac{125}{24} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{5} + \frac{1175}{96} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{4} + \frac{3823}{256} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{3} + \frac{80483}{9216} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} + \frac{509257}{294912} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x - \frac{1696165}{2752512} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{57915}{524288} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x - \frac{57915}{2097152} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{6660225}{16777216} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{6660225}{67108864} \,{\left (2 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{459555525}{268435456} \, \sqrt{2 \, x^{2} - x + 3} x + \frac{10569777075}{4294967296} \, \sqrt{2} \operatorname{arsinh}\left (\frac{1}{23} \, \sqrt{23}{\left (4 \, x - 1\right )}\right ) - \frac{459555525}{1073741824} \, \sqrt{2 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/24*(2*x^2 - x + 3)^(7/2)*x^5 + 1175/96*(2*x^2 - x + 3)^(7/2)*x^4 + 3823/256*(2*x^2 - x + 3)^(7/2)*x^3 + 80
483/9216*(2*x^2 - x + 3)^(7/2)*x^2 + 509257/294912*(2*x^2 - x + 3)^(7/2)*x - 1696165/2752512*(2*x^2 - x + 3)^(
7/2) + 57915/524288*(2*x^2 - x + 3)^(5/2)*x - 57915/2097152*(2*x^2 - x + 3)^(5/2) + 6660225/16777216*(2*x^2 -
x + 3)^(3/2)*x - 6660225/67108864*(2*x^2 - x + 3)^(3/2) + 459555525/268435456*sqrt(2*x^2 - x + 3)*x + 10569777
075/4294967296*sqrt(2)*arcsinh(1/23*sqrt(23)*(4*x - 1)) - 459555525/1073741824*sqrt(2*x^2 - x + 3)

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Fricas [A]  time = 1.35757, size = 512, normalized size = 2.42 \begin{align*} \frac{1}{67645734912} \,{\left (2818572288000 \, x^{11} + 2395786444800 \, x^{10} + 12943588589568 \, x^{9} + 14341894045696 \, x^{8} + 27835561148416 \, x^{7} + 28347538538496 \, x^{6} + 34378613923840 \, x^{5} + 26186527209472 \, x^{4} + 20384824684416 \, x^{3} + 10060731582048 \, x^{2} + 4560943728924 \, x - 1191399152715\right )} \sqrt{2 \, x^{2} - x + 3} + \frac{10569777075}{8589934592} \, \sqrt{2} \log \left (-4 \, \sqrt{2} \sqrt{2 \, x^{2} - x + 3}{\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/67645734912*(2818572288000*x^11 + 2395786444800*x^10 + 12943588589568*x^9 + 14341894045696*x^8 + 27835561148
416*x^7 + 28347538538496*x^6 + 34378613923840*x^5 + 26186527209472*x^4 + 20384824684416*x^3 + 10060731582048*x
^2 + 4560943728924*x - 1191399152715)*sqrt(2*x^2 - x + 3) + 10569777075/8589934592*sqrt(2)*log(-4*sqrt(2)*sqrt
(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (2 x^{2} - x + 3\right )^{\frac{5}{2}} \left (5 x^{2} + 3 x + 2\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15239, size = 139, normalized size = 0.66 \begin{align*} \frac{1}{67645734912} \,{\left (4 \,{\left (8 \,{\left (4 \,{\left (16 \,{\left (4 \,{\left (8 \,{\left (28 \,{\left (32 \,{\left (12 \,{\left (200 \,{\left (20 \, x + 17\right )} x + 18369\right )} x + 244241\right )} x + 15169177\right )} x + 432549111\right )} x + 4196608145\right )} x + 12786390239\right )} x + 159256442847\right )} x + 314397861939\right )} x + 1140235932231\right )} x - 1191399152715\right )} \sqrt{2 \, x^{2} - x + 3} - \frac{10569777075}{4294967296} \, \sqrt{2} \log \left (-2 \, \sqrt{2}{\left (\sqrt{2} x - \sqrt{2 \, x^{2} - x + 3}\right )} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/67645734912*(4*(8*(4*(16*(4*(8*(28*(32*(12*(200*(20*x + 17)*x + 18369)*x + 244241)*x + 15169177)*x + 4325491
11)*x + 4196608145)*x + 12786390239)*x + 159256442847)*x + 314397861939)*x + 1140235932231)*x - 1191399152715)
*sqrt(2*x^2 - x + 3) - 10569777075/4294967296*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1)