Integrand size = 40, antiderivative size = 194 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=-\frac {12568315 (17-22 x) \sqrt {3-x+2 x^2}}{23776267862016 (5+2 x)^2}-\frac {3667 \left (3-x+2 x^2\right )^{3/2}}{4032 (5+2 x)^7}+\frac {948341 \left (3-x+2 x^2\right )^{3/2}}{1741824 (5+2 x)^6}-\frac {1464037 \left (3-x+2 x^2\right )^{3/2}}{13934592 (5+2 x)^5}+\frac {19414831 \left (3-x+2 x^2\right )^{3/2}}{4013162496 (5+2 x)^4}+\frac {246159769 \left (3-x+2 x^2\right )^{3/2}}{866843099136 (5+2 x)^3}-\frac {289071245 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{285315214344192 \sqrt {2}} \]
-3667/4032*(2*x^2-x+3)^(3/2)/(5+2*x)^7+948341/1741824*(2*x^2-x+3)^(3/2)/(5 +2*x)^6-1464037/13934592*(2*x^2-x+3)^(3/2)/(5+2*x)^5+19414831/4013162496*( 2*x^2-x+3)^(3/2)/(5+2*x)^4+246159769/866843099136*(2*x^2-x+3)^(3/2)/(5+2*x )^3-289071245/570630428688384*arctanh(1/24*(17-22*x)*2^(1/2)/(2*x^2-x+3)^( 1/2))*2^(1/2)-12568315/23776267862016*(17-22*x)*(2*x^2-x+3)^(1/2)/(5+2*x)^ 2
Time = 1.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\frac {\frac {12 \sqrt {3-x+2 x^2} \left (-20465234808721+590492177460 x+14716683780036 x^2+41058010262368 x^3+4982916071952 x^4+27976951397184 x^5+1574342277056 x^6\right )}{(5+2 x)^7}+2023498715 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )}{1997206500409344} \]
((12*Sqrt[3 - x + 2*x^2]*(-20465234808721 + 590492177460*x + 1471668378003 6*x^2 + 41058010262368*x^3 + 4982916071952*x^4 + 27976951397184*x^5 + 1574 342277056*x^6))/(5 + 2*x)^7 + 2023498715*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6])/1997206500409344
Time = 0.56 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.13, 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, 27, 2181, 27, 1237, 25, 1228, 1152, 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.
| \(\displaystyle \int \frac {\sqrt {2 x^2-x+3} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^8} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {1}{504} \int \frac {\sqrt {2 x^2-x+3} \left (-20160 x^3+54432 x^2-89504 x+69381\right )}{16 (2 x+5)^7}dx-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\sqrt {2 x^2-x+3} \left (-20160 x^3+54432 x^2-89504 x+69381\right )}{(2 x+5)^7}dx}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {1}{432} \int \frac {15 \sqrt {2 x^2-x+3} \left (290304 x^2-750908 x+700441\right )}{(2 x+5)^6}dx+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5}{144} \int \frac {\sqrt {2 x^2-x+3} \left (290304 x^2-750908 x+700441\right )}{(2 x+5)^6}dx+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle \frac {\frac {5}{144} \left (-\frac {1}{360} \int \frac {3 (5149971-5705944 x) \sqrt {2 x^2-x+3}}{(2 x+5)^5}dx-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {5}{144} \left (-\frac {1}{120} \int \frac {(5149971-5705944 x) \sqrt {2 x^2-x+3}}{(2 x+5)^5}dx-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {5}{144} \left (\frac {1}{120} \left (\frac {1}{288} \int -\frac {(52011459-77659324 x) \sqrt {2 x^2-x+3}}{(2 x+5)^4}dx+\frac {19414831 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}\right )-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {5}{144} \left (\frac {1}{120} \left (\frac {19414831 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}-\frac {1}{288} \int \frac {(52011459-77659324 x) \sqrt {2 x^2-x+3}}{(2 x+5)^4}dx\right )-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {\frac {5}{144} \left (\frac {1}{120} \left (\frac {1}{288} \left (\frac {87978205}{72} \int \frac {\sqrt {2 x^2-x+3}}{(2 x+5)^3}dx+\frac {246159769 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {19414831 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}\right )-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {\frac {5}{144} \left (\frac {1}{120} \left (\frac {1}{288} \left (\frac {87978205}{72} \left (\frac {23}{576} \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {(17-22 x) \sqrt {2 x^2-x+3}}{288 (2 x+5)^2}\right )+\frac {246159769 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {19414831 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}\right )-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {5}{144} \left (\frac {1}{120} \left (\frac {1}{288} \left (\frac {87978205}{72} \left (-\frac {23}{288} \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {\sqrt {2 x^2-x+3} (17-22 x)}{288 (2 x+5)^2}\right )+\frac {246159769 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {19414831 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}\right )-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {5}{144} \left (\frac {1}{120} \left (\frac {1}{288} \left (\frac {87978205}{72} \left (-\frac {23 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )}{3456 \sqrt {2}}-\frac {\sqrt {2 x^2-x+3} (17-22 x)}{288 (2 x+5)^2}\right )+\frac {246159769 \left (2 x^2-x+3\right )^{3/2}}{108 (2 x+5)^3}\right )+\frac {19414831 \left (2 x^2-x+3\right )^{3/2}}{144 (2 x+5)^4}\right )-\frac {1464037 \left (2 x^2-x+3\right )^{3/2}}{60 (2 x+5)^5}\right )+\frac {948341 \left (2 x^2-x+3\right )^{3/2}}{216 (2 x+5)^6}}{8064}-\frac {3667 \left (2 x^2-x+3\right )^{3/2}}{4032 (2 x+5)^7}\) |
(-3667*(3 - x + 2*x^2)^(3/2))/(4032*(5 + 2*x)^7) + ((948341*(3 - x + 2*x^2 )^(3/2))/(216*(5 + 2*x)^6) + (5*((-1464037*(3 - x + 2*x^2)^(3/2))/(60*(5 + 2*x)^5) + ((19414831*(3 - x + 2*x^2)^(3/2))/(144*(5 + 2*x)^4) + ((2461597 69*(3 - x + 2*x^2)^(3/2))/(108*(5 + 2*x)^3) + (87978205*(-1/288*((17 - 22* x)*Sqrt[3 - x + 2*x^2])/(5 + 2*x)^2 - (23*ArcTanh[(17 - 22*x)/(12*Sqrt[2]* Sqrt[3 - x + 2*x^2])])/(3456*Sqrt[2])))/72)/288)/120))/144)/8064
3.4.33.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(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*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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]
Time = 0.41 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.48
| method | result | size |
| risch | \(\frac {3148684554112 x^{8}+54379560517312 x^{7}-13288092422112 x^{6}+161063958644336 x^{5}+3324105513560 x^{4}+109638331361988 x^{3}+2629089545206 x^{2}+22236711341101 x -61395704426163}{166433875034112 \left (5+2 x \right )^{7} \sqrt {2 x^{2}-x +3}}-\frac {289071245 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{570630428688384}\) | \(93\) |
| trager | \(\frac {\left (1574342277056 x^{6}+27976951397184 x^{5}+4982916071952 x^{4}+41058010262368 x^{3}+14716683780036 x^{2}+590492177460 x -20465234808721\right ) \sqrt {2 x^{2}-x +3}}{166433875034112 \left (5+2 x \right )^{7}}+\frac {289071245 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {22 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )+24 \sqrt {2 x^{2}-x +3}}{5+2 x}\right )}{570630428688384}\) | \(102\) |
| default | \(\frac {19414831 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{64210599936 \left (x +\frac {5}{2}\right )^{4}}+\frac {246159769 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{6934744793088 \left (x +\frac {5}{2}\right )^{3}}-\frac {12568315 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{23776267862016 \left (x +\frac {5}{2}\right )^{2}}-\frac {138251465 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{855945643032576 \left (x +\frac {5}{2}\right )}+\frac {289071245 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{1711891286065152}-\frac {289071245 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (\frac {17}{2}-11 x \right ) \sqrt {2}}{12 \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}\right )}{570630428688384}+\frac {138251465 \left (4 x -1\right ) \sqrt {2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}}}{1711891286065152}-\frac {1464037 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{445906944 \left (x +\frac {5}{2}\right )^{5}}+\frac {948341 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{111476736 \left (x +\frac {5}{2}\right )^{6}}-\frac {3667 \left (2 \left (x +\frac {5}{2}\right )^{2}-11 x -\frac {19}{2}\right )^{\frac {3}{2}}}{516096 \left (x +\frac {5}{2}\right )^{7}}\) | \(216\) |
1/166433875034112*(3148684554112*x^8+54379560517312*x^7-13288092422112*x^6 +161063958644336*x^5+3324105513560*x^4+109638331361988*x^3+2629089545206*x ^2+22236711341101*x-61395704426163)/(5+2*x)^7/(2*x^2-x+3)^(1/2)-289071245/ 570630428688384*2^(1/2)*arctanh(1/12*(17/2-11*x)*2^(1/2)/(2*(x+5/2)^2-11*x -19/2)^(1/2))
Time = 0.26 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\frac {2023498715 \, \sqrt {2} {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )} \log \left (-\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} + 1060 \, x^{2} - 1036 \, x + 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 48 \, {\left (1574342277056 \, x^{6} + 27976951397184 \, x^{5} + 4982916071952 \, x^{4} + 41058010262368 \, x^{3} + 14716683780036 \, x^{2} + 590492177460 \, x - 20465234808721\right )} \sqrt {2 \, x^{2} - x + 3}}{7988826001637376 \, {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} \]
1/7988826001637376*(2023498715*sqrt(2)*(128*x^7 + 2240*x^6 + 16800*x^5 + 7 0000*x^4 + 175000*x^3 + 262500*x^2 + 218750*x + 78125)*log(-(24*sqrt(2)*sq rt(2*x^2 - x + 3)*(22*x - 17) + 1060*x^2 - 1036*x + 1153)/(4*x^2 + 20*x + 25)) + 48*(1574342277056*x^6 + 27976951397184*x^5 + 4982916071952*x^4 + 41 058010262368*x^3 + 14716683780036*x^2 + 590492177460*x - 20465234808721)*s qrt(2*x^2 - x + 3))/(128*x^7 + 2240*x^6 + 16800*x^5 + 70000*x^4 + 175000*x ^3 + 262500*x^2 + 218750*x + 78125)
\[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\int \frac {\sqrt {2 x^{2} - x + 3} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{8}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\frac {289071245}{570630428688384} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) + \frac {12568315}{11888133931008} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4032 \, {\left (128 \, x^{7} + 2240 \, x^{6} + 16800 \, x^{5} + 70000 \, x^{4} + 175000 \, x^{3} + 262500 \, x^{2} + 218750 \, x + 78125\right )}} + \frac {948341 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1741824 \, {\left (64 \, x^{6} + 960 \, x^{5} + 6000 \, x^{4} + 20000 \, x^{3} + 37500 \, x^{2} + 37500 \, x + 15625\right )}} - \frac {1464037 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{13934592 \, {\left (32 \, x^{5} + 400 \, x^{4} + 2000 \, x^{3} + 5000 \, x^{2} + 6250 \, x + 3125\right )}} + \frac {19414831 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{4013162496 \, {\left (16 \, x^{4} + 160 \, x^{3} + 600 \, x^{2} + 1000 \, x + 625\right )}} + \frac {246159769 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{866843099136 \, {\left (8 \, x^{3} + 60 \, x^{2} + 150 \, x + 125\right )}} - \frac {12568315 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{5944066965504 \, {\left (4 \, x^{2} + 20 \, x + 25\right )}} - \frac {138251465 \, \sqrt {2 \, x^{2} - x + 3}}{23776267862016 \, {\left (2 \, x + 5\right )}} \]
289071245/570630428688384*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) + 12568315/11888133931008*sqrt(2*x^2 - x + 3) - 3667/4032*(2*x^2 - x + 3)^(3/2)/(128*x^7 + 2240*x^6 + 16800*x^5 + 70000 *x^4 + 175000*x^3 + 262500*x^2 + 218750*x + 78125) + 948341/1741824*(2*x^2 - x + 3)^(3/2)/(64*x^6 + 960*x^5 + 6000*x^4 + 20000*x^3 + 37500*x^2 + 375 00*x + 15625) - 1464037/13934592*(2*x^2 - x + 3)^(3/2)/(32*x^5 + 400*x^4 + 2000*x^3 + 5000*x^2 + 6250*x + 3125) + 19414831/4013162496*(2*x^2 - x + 3 )^(3/2)/(16*x^4 + 160*x^3 + 600*x^2 + 1000*x + 625) + 246159769/8668430991 36*(2*x^2 - x + 3)^(3/2)/(8*x^3 + 60*x^2 + 150*x + 125) - 12568315/5944066 965504*(2*x^2 - x + 3)^(3/2)/(4*x^2 + 20*x + 25) - 138251465/2377626786201 6*sqrt(2*x^2 - x + 3)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (160) = 320\).
Time = 0.31 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.35 \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=-\frac {289071245}{570630428688384} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x + \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) + \frac {289071245}{570630428688384} \, \sqrt {2} \log \left ({\left | -2 \, \sqrt {2} x - 11 \, \sqrt {2} + 2 \, \sqrt {2 \, x^{2} - x + 3} \right |}\right ) - \frac {\sqrt {2} {\left (129503917760 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{13} - 3320259746027840 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{12} - 23966708071916736 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{11} - 186055342532355520 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{10} - 274256644494948976 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{9} + 796135370176031760 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{8} + 2531523139171005408 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{7} - 4610393811900786336 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{6} - 7997126854300052364 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{5} + 30842713619423538868 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{4} - 21873571601855032556 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{3} + 16204706960604668100 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} - 3196254593191113265 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} + 536799032216117911\right )}}{332867750068224 \, {\left (2 \, {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )}^{2} + 10 \, \sqrt {2} {\left (\sqrt {2} x - \sqrt {2 \, x^{2} - x + 3}\right )} - 11\right )}^{7}} \]
-289071245/570630428688384*sqrt(2)*log(abs(-2*sqrt(2)*x + sqrt(2) + 2*sqrt (2*x^2 - x + 3))) + 289071245/570630428688384*sqrt(2)*log(abs(-2*sqrt(2)*x - 11*sqrt(2) + 2*sqrt(2*x^2 - x + 3))) - 1/332867750068224*sqrt(2)*(12950 3917760*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^13 - 3320259746027840*(s qrt(2)*x - sqrt(2*x^2 - x + 3))^12 - 23966708071916736*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^11 - 186055342532355520*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^10 - 274256644494948976*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^9 + 796135370176031760*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^8 + 25315231391710 05408*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^7 - 4610393811900786336*(s qrt(2)*x - sqrt(2*x^2 - x + 3))^6 - 7997126854300052364*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^5 + 30842713619423538868*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^4 - 21873571601855032556*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3) )^3 + 16204706960604668100*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 - 319625459 3191113265*sqrt(2)*(sqrt(2)*x - sqrt(2*x^2 - x + 3)) + 536799032216117911) /(2*(sqrt(2)*x - sqrt(2*x^2 - x + 3))^2 + 10*sqrt(2)*(sqrt(2)*x - sqrt(2*x ^2 - x + 3)) - 11)^7
Timed out. \[ \int \frac {\sqrt {3-x+2 x^2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^8} \, dx=\int \frac {\sqrt {2\,x^2-x+3}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^8} \,d x \]