Integrand size = 40, antiderivative size = 151 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\frac {5 (661065-110099 x) \sqrt {3-x+2 x^2}}{82944}+\frac {5}{48} \left (3-x+2 x^2\right )^{3/2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{1152 (5+2 x)^2}+\frac {357391 \left (3-x+2 x^2\right )^{3/2}}{82944 (5+2 x)}+\frac {117315 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{512 \sqrt {2}}-\frac {12670805 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{55296 \sqrt {2}} \]
5/48*(2*x^2-x+3)^(3/2)-3667/1152*(2*x^2-x+3)^(3/2)/(5+2*x)^2+357391/82944* (2*x^2-x+3)^(3/2)/(5+2*x)+117315/1024*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/ 2)-12670805/110592*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/ 2)+5/82944*(661065-110099*x)*(2*x^2-x+3)^(1/2)
Time = 0.59 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.73 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (4880551+2959330 x+272520 x^2-25632 x^3+3840 x^4\right )}{(5+2 x)^2}+12670805 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )+6335010 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{55296} \]
((12*Sqrt[3 - x + 2*x^2]*(4880551 + 2959330*x + 272520*x^2 - 25632*x^3 + 3 840*x^4))/(5 + 2*x)^2 + 12670805*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6] + 6335010*Sqrt[2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/5529 6
Time = 0.52 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2181, 27, 2181, 27, 2184, 27, 1231, 27, 1269, 1090, 222, 1154, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {2 x^2-x+3} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^3} \, dx\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle -\frac {1}{144} \int \frac {\sqrt {2 x^2-x+3} \left (-5760 x^3+15552 x^2-57004 x+27681\right )}{16 (2 x+5)^2}dx-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {\sqrt {2 x^2-x+3} \left (-5760 x^3+15552 x^2-57004 x+27681\right )}{(2 x+5)^2}dx}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 2181 |
\(\displaystyle \frac {\frac {1}{72} \int \frac {5 \sqrt {2 x^2-x+3} \left (41472 x^2-787480 x+306261\right )}{2 x+5}dx+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {5}{72} \int \frac {\sqrt {2 x^2-x+3} \left (41472 x^2-787480 x+306261\right )}{2 x+5}dx+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 2184 |
\(\displaystyle \frac {\frac {5}{72} \left (\frac {1}{24} \int \frac {24 (332181-880792 x) \sqrt {2 x^2-x+3}}{2 x+5}dx+3456 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {5}{72} \left (\int \frac {(332181-880792 x) \sqrt {2 x^2-x+3}}{2 x+5}dx+3456 \left (2 x^2-x+3\right )^{3/2}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 1231 |
\(\displaystyle \frac {\frac {5}{72} \left (-\frac {1}{32} \int -\frac {576 (422491-844668 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {5}{72} \left (18 \int \frac {422491-844668 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {\frac {5}{72} \left (18 \left (2534161 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-422334 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 1090 |
\(\displaystyle \frac {\frac {5}{72} \left (18 \left (2534161 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-211167 \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\right )+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {\frac {5}{72} \left (18 \left (2534161 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-211167 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {\frac {5}{72} \left (18 \left (-5068322 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-211167 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {5}{72} \left (18 \left (-211167 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-\frac {2534161 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{6 \sqrt {2}}\right )+3456 \left (2 x^2-x+3\right )^{3/2}+2 (661065-110099 x) \sqrt {2 x^2-x+3}\right )+\frac {357391 \left (2 x^2-x+3\right )^{3/2}}{36 (2 x+5)}}{2304}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{1152 (2 x+5)^2}\) |
(-3667*(3 - x + 2*x^2)^(3/2))/(1152*(5 + 2*x)^2) + ((357391*(3 - x + 2*x^2 )^(3/2))/(36*(5 + 2*x)) + (5*(2*(661065 - 110099*x)*Sqrt[3 - x + 2*x^2] + 3456*(3 - x + 2*x^2)^(3/2) + 18*(-211167*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[2 3]] - (2534161*ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(6*S qrt[2]))))/72)/2304
3.4.28.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[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]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Timed out.
hanged
Time = 0.26 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\frac {12670020 \, \sqrt {2} {\left (4 \, x^{2} + 20 \, x + 25\right )} \log \left (4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 12670805 \, \sqrt {2} {\left (4 \, x^{2} + 20 \, x + 25\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (3840 \, x^{4} - 25632 \, x^{3} + 272520 \, x^{2} + 2959330 \, x + 4880551\right )} \sqrt {2 \, x^{2} - x + 3}}{221184 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} \]
1/221184*(12670020*sqrt(2)*(4*x^2 + 20*x + 25)*log(4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 12670805*sqrt(2)*(4*x^2 + 20*x + 25)*log(-(24*sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) + 48*(3840*x^4 - 25632*x^3 + 272520*x^2 + 2959 330*x + 4880551)*sqrt(2*x^2 - x + 3))/(4*x^2 + 20*x + 25)
\[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{3}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\frac {5}{48} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {149}{64} \, \sqrt {2 \, x^{2} - x + 3} x - \frac {117315}{1024} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) + \frac {12670805}{110592} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {3877}{144} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1152 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} + \frac {357391 \, \sqrt {2 \, x^{2} - x + 3}}{4608 \, {\left (2 \, x + 5\right )}} \]
5/48*(2*x^2 - x + 3)^(3/2) - 149/64*sqrt(2*x^2 - x + 3)*x - 117315/1024*sq rt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) + 12670805/110592*sqrt(2)*a rcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 3877 /144*sqrt(2*x^2 - x + 3) - 3667/1152*(2*x^2 - x + 3)^(3/2)/(4*x^2 + 20*x + 25) + 357391/4608*sqrt(2*x^2 - x + 3)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (120) = 240\).
Time = 0.30 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\frac {1}{768} \, {\left (4 \, {\left (40 \, x - 467\right )} x + 19695\right )} \sqrt {2 \, x^{2} - x + 3} + \frac {117315}{1024} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) - \frac {12670805}{110592} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {12670805}{110592} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {\sqrt {2} {\left (10693526 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 79895946 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 124044603 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 80334011\right )}}{9216 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{2}} \]
1/768*(4*(40*x - 467)*x + 19695)*sqrt(2*x^2 - x + 3) + 117315/1024*sqrt(2) *log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) - 12670805/110592*s qrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) + 12670805 /110592*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3)) ) + 1/9216*sqrt(2)*(10693526*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 + 79895946*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 - 124044603*sqrt(2)*(sqrt(2) *x - sqrt(2*x^2 - x + 3)) + 80334011)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) ^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 11)^2
Timed out. \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^3} \, dx=\int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^3} \,d x \]