Integrand size = 40, antiderivative size = 165 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=-\frac {(4583087983+3174439702 x) \sqrt {3-x+2 x^2}}{6879707136 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{2880 (5+2 x)^5}+\frac {711961 \left (3-x+2 x^2\right )^{3/2}}{829440 (5+2 x)^4}-\frac {38732321 \left (3-x+2 x^2\right )^{3/2}}{179159040 (5+2 x)^3}-\frac {5 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{32 \sqrt {2}}+\frac {12895597463 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{82556485632 \sqrt {2}} \]
-3667/2880*(2*x^2-x+3)^(3/2)/(5+2*x)^5+711961/829440*(2*x^2-x+3)^(3/2)/(5+ 2*x)^4-38732321/179159040*(2*x^2-x+3)^(3/2)/(5+2*x)^3-5/64*arcsinh(1/23*(1 -4*x)*23^(1/2))*2^(1/2)+12895597463/165112971264*arctanh(1/24*(17-22*x)*2^ (1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-1/6879707136*(4583087983+3174439702*x)*(2 *x^2-x+3)^(1/2)/(5+2*x)^2
Time = 0.75 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=\frac {-\frac {12 \sqrt {3-x+2 x^2} \left (3110673952831+5608297138216 x+3919478861832 x^2+1285267446304 x^3+186470433136 x^4\right )}{(5+2 x)^5}-64477987315 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )-32248627200 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{412782428160} \]
((-12*Sqrt[3 - x + 2*x^2]*(3110673952831 + 5608297138216*x + 3919478861832 *x^2 + 1285267446304*x^3 + 186470433136*x^4))/(5 + 2*x)^5 - 64477987315*Sq rt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6] - 32248627200*Sqrt[2]*L og[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/412782428160
Time = 0.54 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2181, 27, 2181, 2181, 27, 1229, 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.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^6} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {1}{360} \int \frac {\sqrt {2 x^2-x+3} \left (-14400 x^3+38880 x^2-76504 x+52701\right )}{16 (2 x+5)^5}dx-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {2 x^2-x+3} \left (-14400 x^3+38880 x^2-76504 x+52701\right )}{(2 x+5)^5}dx}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {1}{288} \int \frac {\sqrt {2 x^2-x+3} \left (2073600 x^2-7934876 x+5935131\right )}{(2 x+5)^4}dx+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {1}{216} \int \frac {15 (9244801-14929920 x) \sqrt {2 x^2-x+3}}{(2 x+5)^3}dx-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \int \frac {(9244801-14929920 x) \sqrt {2 x^2-x+3}}{(2 x+5)^3}dx-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 1229 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {(3174439702 x+4583087983) \sqrt {2 x^2-x+3}}{288 (2 x+5)^2}-\frac {\int -\frac {2 (2146055063-4299816960 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx}{1152}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {1}{576} \int \frac {2146055063-4299816960 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {\sqrt {2 x^2-x+3} (3174439702 x+4583087983)}{288 (2 x+5)^2}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {1}{576} \left (12895597463 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-2149908480 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+\frac {\sqrt {2 x^2-x+3} (3174439702 x+4583087983)}{288 (2 x+5)^2}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {1}{576} \left (12895597463 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-1074954240 \sqrt {\frac {2}{23}} \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)\right )+\frac {\sqrt {2 x^2-x+3} (3174439702 x+4583087983)}{288 (2 x+5)^2}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {1}{576} \left (12895597463 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-1074954240 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (3174439702 x+4583087983)}{288 (2 x+5)^2}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {1}{576} \left (-25791194926 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-1074954240 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )\right )+\frac {\sqrt {2 x^2-x+3} (3174439702 x+4583087983)}{288 (2 x+5)^2}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{288} \left (-\frac {5}{72} \left (\frac {1}{576} \left (-1074954240 \sqrt {2} \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )-\frac {12895597463 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{6 \sqrt {2}}\right )+\frac {\sqrt {2 x^2-x+3} (3174439702 x+4583087983)}{288 (2 x+5)^2}\right )-\frac {38732321 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {711961 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}}{5760}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{2880 (2 x+5)^5}\) |
(-3667*(3 - x + 2*x^2)^(3/2))/(2880*(5 + 2*x)^5) + ((711961*(3 - x + 2*x^2 )^(3/2))/(144*(5 + 2*x)^4) + ((-38732321*(3 - x + 2*x^2)^(3/2))/(108*(5 + 2*x)^3) - (5*(((4583087983 + 3174439702*x)*Sqrt[3 - x + 2*x^2])/(288*(5 + 2*x)^2) + (-1074954240*Sqrt[2]*ArcSinh[(-1 + 4*x)/Sqrt[23]] - (12895597463 *ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/(6*Sqrt[2]))/576)) /72)/288)/5760
3.4.31.3.1 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))*((a + b*x + c*x^2 )^p/(e^2*(m + 1)*(m + 2)*(c*d^2 - b*d*e + a*e^2)))*((d*g - e*f*(m + 2))*(c* d^2 - b*d*e + a*e^2) - d*p*(2*c*d - b*e)*(e*f - d*g) - e*(g*(m + 1)*(c*d^2 - b*d*e + a*e^2) + p*(2*c*d - b*e)*(e*f - d*g))*x), x] - Simp[p/(e^2*(m + 1 )*(m + 2)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2 )^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) + b^2*e*(d*g*(p + 1) - e*f*(m + p + 2)) + b*(a*e^2*g*(m + 1) - c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2))) - c *(2*c*d*(d*g*(2*p + 1) - e*f*(m + 2*p + 2)) - e*(2*a*e*g*(m + 1) - b*(d*g*( m - 2*p) + e*f*(m + 2*p + 2))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g }, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3, 0]
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]
Timed out.
hanged
Time = 0.29 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=\frac {64497254400 \, \sqrt {2} {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 64477987315 \, \sqrt {2} {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) - 48 \, {\left (186470433136 \, x^{4} + 1285267446304 \, x^{3} + 3919478861832 \, x^{2} + 5608297138216 \, x + 3110673952831\right )} \sqrt {2 \, x^{2} - x + 3}}{1651129712640 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} \]
1/1651129712640*(64497254400*sqrt(2)*(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x ^2 + 6250*x + 3125)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3)*(4*x - 1) - 32*x^2 + 16*x - 25) + 64477987315*sqrt(2)*(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125)*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*(186470433136*x^4 + 12852674 46304*x^3 + 3919478861832*x^2 + 5608297138216*x + 3110673952831)*sqrt(2*x^ 2 - x + 3))/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125)
\[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{6}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.35 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=\frac {5}{64} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {12895597463}{165112971264} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {46569601}{3439853568} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2880 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac {711961 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{829440 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} - \frac {38732321 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{179159040 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} + \frac {46569601 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1719926784 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {562688629 \, \sqrt {2 \, x^{2} - x + 3}}{6879707136 \, {\left (2 \, x + 5\right )}} \]
5/64*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 12895597463/165112 971264*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs( 2*x + 5)) - 46569601/3439853568*sqrt(2*x^2 - x + 3) - 3667/2880*(2*x^2 - x + 3)^(3/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125) + 711 961/829440*(2*x^2 - x + 3)^(3/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 62 5) - 38732321/179159040*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2 + 150*x + 12 5) + 46569601/1719926784*(2*x^2 - x + 3)^(3/2)/(4*x^2 + 20*x + 25) - 56268 8629/6879707136*sqrt(2*x^2 - x + 3)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (134) = 268\).
Time = 0.30 (sec) , antiderivative size = 387, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=-\frac {5}{64} \, \sqrt {2} \log \left (-2 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 1\right ) + \frac {12895597463}{165112971264} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {12895597463}{165112971264} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (4368922304720 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 124570969998480 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 637804348664160 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} + 1828845222532320 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 3763189300187016 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} - 10794416351958120 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} + 25049834283305880 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} - 34708488692384520 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10654664764755165 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 2507056315485767\right )}}{68797071360 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{5}} \]
-5/64*sqrt(2)*log(-2*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 1) + 1289 5597463/165112971264*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 12895597463/165112971264*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*s qrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/68797071360*sqrt(2)*(4368922304720*sq rt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 124570969998480*(sqrt(2)*x - s qrt(2*x^2 - x + 3))^8 + 637804348664160*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 + 1828845222532320*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^6 - 3763189 300187016*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 - 10794416351958120* (sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 + 25049834283305880*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^3 - 34708488692384520*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10654664764755165*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) - 25 07056315485767)/(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt( 2)*x - sqrt(2*x^2 - x + 3)) - 11)^5
Timed out. \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^6} \, dx=\int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^6} \,d x \]