Integrand size = 24, antiderivative size = 154 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=\frac {4991}{12} x \sqrt {2+3 x^2}+\frac {4991}{36} x \left (2+3 x^2\right )^{3/2}+\frac {4991}{90} x \left (2+3 x^2\right )^{5/2}+\frac {6433 (3+2 x)^2 \left (2+3 x^2\right )^{7/2}}{4455}+\frac {49}{165} (3+2 x)^3 \left (2+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+3 x^2\right )^{7/2}+\frac {2 (181243+62244 x) \left (2+3 x^2\right )^{7/2}}{13365}+\frac {4991 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
4991/36*x*(3*x^2+2)^(3/2)+4991/90*x*(3*x^2+2)^(5/2)+6433/4455*(3+2*x)^2*(3 *x^2+2)^(7/2)+49/165*(3+2*x)^3*(3*x^2+2)^(7/2)-1/33*(3+2*x)^4*(3*x^2+2)^(7 /2)+2/13365*(181243+62244*x)*(3*x^2+2)^(7/2)+4991/18*arcsinh(1/2*x*6^(1/2) )*3^(1/2)+4991/12*x*(3*x^2+2)^(1/2)
Time = 0.39 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {\sqrt {2+3 x^2} \left (-19537120-64370295 x-92160240 x^2-127123425 x^3-150762600 x^4-129966606 x^5-93646260 x^6-50615928 x^7-12921120 x^8+769824 x^9+699840 x^{10}\right )}{53460}-\frac {4991 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{6 \sqrt {3}} \]
-1/53460*(Sqrt[2 + 3*x^2]*(-19537120 - 64370295*x - 92160240*x^2 - 1271234 25*x^3 - 150762600*x^4 - 129966606*x^5 - 93646260*x^6 - 50615928*x^7 - 129 21120*x^8 + 769824*x^9 + 699840*x^10)) - (4991*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(6*Sqrt[3])
Time = 0.29 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {687, 27, 687, 27, 687, 676, 211, 211, 211, 222}
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.
| \(\displaystyle \int (5-x) (2 x+3)^4 \left (3 x^2+2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {1}{33} \int 7 (2 x+3)^3 (42 x+73) \left (3 x^2+2\right )^{5/2}dx-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{33} \int (2 x+3)^3 (42 x+73) \left (3 x^2+2\right )^{5/2}dx-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{30} \int 6 (2 x+3)^2 (919 x+1011) \left (3 x^2+2\right )^{5/2}dx+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \int (2 x+3)^2 (919 x+1011) \left (3 x^2+2\right )^{5/2}dx+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 687 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \left (\frac {1}{27} \int (2 x+3) (71136 x+74539) \left (3 x^2+2\right )^{5/2}dx+\frac {919}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 676 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \left (\frac {1}{27} \left (211761 \int \left (3 x^2+2\right )^{5/2}dx+5928 x \left (3 x^2+2\right )^{7/2}+\frac {362486}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {919}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \left (\frac {1}{27} \left (211761 \left (\frac {5}{3} \int \left (3 x^2+2\right )^{3/2}dx+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+5928 x \left (3 x^2+2\right )^{7/2}+\frac {362486}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {919}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \left (\frac {1}{27} \left (211761 \left (\frac {5}{3} \left (\frac {3}{2} \int \sqrt {3 x^2+2}dx+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+5928 x \left (3 x^2+2\right )^{7/2}+\frac {362486}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {919}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \left (\frac {1}{27} \left (211761 \left (\frac {5}{3} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+5928 x \left (3 x^2+2\right )^{7/2}+\frac {362486}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {919}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {7}{33} \left (\frac {1}{5} \left (\frac {1}{27} \left (211761 \left (\frac {5}{3} \left (\frac {3}{2} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )+\frac {1}{6} x \left (3 x^2+2\right )^{5/2}\right )+5928 x \left (3 x^2+2\right )^{7/2}+\frac {362486}{21} \left (3 x^2+2\right )^{7/2}\right )+\frac {919}{27} (2 x+3)^2 \left (3 x^2+2\right )^{7/2}\right )+\frac {7}{5} (2 x+3)^3 \left (3 x^2+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+2\right )^{7/2}\) |
-1/33*((3 + 2*x)^4*(2 + 3*x^2)^(7/2)) + (7*((7*(3 + 2*x)^3*(2 + 3*x^2)^(7/ 2))/5 + ((919*(3 + 2*x)^2*(2 + 3*x^2)^(7/2))/27 + ((362486*(2 + 3*x^2)^(7/ 2))/21 + 5928*x*(2 + 3*x^2)^(7/2) + 211761*((x*(2 + 3*x^2)^(5/2))/6 + (5*( (x*(2 + 3*x^2)^(3/2))/4 + (3*((x*Sqrt[2 + 3*x^2])/2 + ArcSinh[Sqrt[3/2]*x] /Sqrt[3]))/2))/3))/27)/5))/33
3.14.81.3.1 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)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
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[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) ), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp [c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x ] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && Eq Q[f, 0])
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.49
| method | result | size |
| risch | \(-\frac {\left (699840 x^{10}+769824 x^{9}-12921120 x^{8}-50615928 x^{7}-93646260 x^{6}-129966606 x^{5}-150762600 x^{4}-127123425 x^{3}-92160240 x^{2}-64370295 x -19537120\right ) \sqrt {3 x^{2}+2}}{53460}+\frac {4991 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{18}\) | \(75\) |
| trager | \(\left (-\frac {144}{11} x^{10}-\frac {72}{5} x^{9}+\frac {7976}{33} x^{8}+\frac {4734}{5} x^{7}+\frac {173419}{99} x^{6}+\frac {24311}{10} x^{5}+\frac {279190}{99} x^{4}+\frac {28535}{12} x^{3}+\frac {1536004}{891} x^{2}+\frac {14449}{12} x +\frac {976856}{2673}\right ) \sqrt {3 x^{2}+2}-\frac {4991 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{18}\) | \(92\) |
| default | \(\frac {4991 x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{90}+\frac {4991 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{36}+\frac {4991 x \sqrt {3 x^{2}+2}}{12}+\frac {4991 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{18}+\frac {122107 \left (3 x^{2}+2\right )^{\frac {7}{2}}}{2673}-\frac {16 x^{4} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{33}+\frac {8840 x^{2} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{891}-\frac {8 x^{3} \left (3 x^{2}+2\right )^{\frac {7}{2}}}{15}+\frac {542 x \left (3 x^{2}+2\right )^{\frac {7}{2}}}{15}\) | \(115\) |
| meijerg | \(-\frac {2025 \sqrt {3}\, \left (-\frac {8 \sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (\frac {3}{8} x^{4}+\frac {13}{16} x^{2}+\frac {11}{16}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{3}\right )}{2 \sqrt {\pi }}-\frac {4995 \sqrt {2}\, \left (\frac {16 \sqrt {\pi }}{105}-\frac {8 \sqrt {\pi }\, \left (\frac {27}{4} x^{6}+\frac {27}{2} x^{4}+9 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{105}\right )}{2 \sqrt {\pi }}-\frac {1440 \sqrt {3}\, \left (-\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (162 x^{6}+306 x^{4}+177 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{720}+\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{24}\right )}{\sqrt {\pi }}-\frac {440 \sqrt {2}\, \left (-\frac {32 \sqrt {\pi }}{945}+\frac {4 \sqrt {\pi }\, \left (-\frac {567}{4} x^{8}-\frac {513}{2} x^{6}-135 x^{4}-6 x^{2}+8\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{945}\right )}{\sqrt {\pi }}+\frac {160 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (-648 x^{8}-1134 x^{6}-558 x^{4}-15 x^{2}+15\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{2400}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{80}\right )}{9 \sqrt {\pi }}+\frac {160 \sqrt {2}\, \left (\frac {128 \sqrt {\pi }}{10395}-\frac {8 \sqrt {\pi }\, \left (\frac {15309}{16} x^{10}+\frac {13041}{8} x^{8}+\frac {3051}{4} x^{6}+\frac {27}{2} x^{4}-12 x^{2}+16\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{10395}\right )}{9 \sqrt {\pi }}\) | \(332\) |
-1/53460*(699840*x^10+769824*x^9-12921120*x^8-50615928*x^7-93646260*x^6-12 9966606*x^5-150762600*x^4-127123425*x^3-92160240*x^2-64370295*x-19537120)* (3*x^2+2)^(1/2)+4991/18*arcsinh(1/2*x*6^(1/2))*3^(1/2)
Time = 0.33 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.58 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{53460} \, {\left (699840 \, x^{10} + 769824 \, x^{9} - 12921120 \, x^{8} - 50615928 \, x^{7} - 93646260 \, x^{6} - 129966606 \, x^{5} - 150762600 \, x^{4} - 127123425 \, x^{3} - 92160240 \, x^{2} - 64370295 \, x - 19537120\right )} \sqrt {3 \, x^{2} + 2} + \frac {4991}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
-1/53460*(699840*x^10 + 769824*x^9 - 12921120*x^8 - 50615928*x^7 - 9364626 0*x^6 - 129966606*x^5 - 150762600*x^4 - 127123425*x^3 - 92160240*x^2 - 643 70295*x - 19537120)*sqrt(3*x^2 + 2) + 4991/36*sqrt(3)*log(-sqrt(3)*sqrt(3* x^2 + 2)*x - 3*x^2 - 1)
Time = 8.23 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.29 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=- \frac {144 x^{10} \sqrt {3 x^{2} + 2}}{11} - \frac {72 x^{9} \sqrt {3 x^{2} + 2}}{5} + \frac {7976 x^{8} \sqrt {3 x^{2} + 2}}{33} + \frac {4734 x^{7} \sqrt {3 x^{2} + 2}}{5} + \frac {173419 x^{6} \sqrt {3 x^{2} + 2}}{99} + \frac {24311 x^{5} \sqrt {3 x^{2} + 2}}{10} + \frac {279190 x^{4} \sqrt {3 x^{2} + 2}}{99} + \frac {28535 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {1536004 x^{2} \sqrt {3 x^{2} + 2}}{891} + \frac {14449 x \sqrt {3 x^{2} + 2}}{12} + \frac {976856 \sqrt {3 x^{2} + 2}}{2673} + \frac {4991 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \]
-144*x**10*sqrt(3*x**2 + 2)/11 - 72*x**9*sqrt(3*x**2 + 2)/5 + 7976*x**8*sq rt(3*x**2 + 2)/33 + 4734*x**7*sqrt(3*x**2 + 2)/5 + 173419*x**6*sqrt(3*x**2 + 2)/99 + 24311*x**5*sqrt(3*x**2 + 2)/10 + 279190*x**4*sqrt(3*x**2 + 2)/9 9 + 28535*x**3*sqrt(3*x**2 + 2)/12 + 1536004*x**2*sqrt(3*x**2 + 2)/891 + 1 4449*x*sqrt(3*x**2 + 2)/12 + 976856*sqrt(3*x**2 + 2)/2673 + 4991*sqrt(3)*a sinh(sqrt(6)*x/2)/18
Time = 0.31 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.74 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {16}{33} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{4} - \frac {8}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{3} + \frac {8840}{891} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x^{2} + \frac {542}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} x + \frac {122107}{2673} \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}} + \frac {4991}{90} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {4991}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {4991}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {4991}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
-16/33*(3*x^2 + 2)^(7/2)*x^4 - 8/15*(3*x^2 + 2)^(7/2)*x^3 + 8840/891*(3*x^ 2 + 2)^(7/2)*x^2 + 542/15*(3*x^2 + 2)^(7/2)*x + 122107/2673*(3*x^2 + 2)^(7 /2) + 4991/90*(3*x^2 + 2)^(5/2)*x + 4991/36*(3*x^2 + 2)^(3/2)*x + 4991/12* sqrt(3*x^2 + 2)*x + 4991/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)
Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.53 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=-\frac {1}{53460} \, {\left (3 \, {\left ({\left (9 \, {\left (2 \, {\left ({\left (2 \, {\left (6 \, {\left (4 \, {\left (27 \, {\left (10 \, x + 11\right )} x - 4985\right )} x - 78111\right )} x - 867095\right )} x - 2406789\right )} x - 2791900\right )} x - 4708275\right )} x - 30720080\right )} x - 21456765\right )} x - 19537120\right )} \sqrt {3 \, x^{2} + 2} - \frac {4991}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
-1/53460*(3*((9*(2*((2*(6*(4*(27*(10*x + 11)*x - 4985)*x - 78111)*x - 8670 95)*x - 2406789)*x - 2791900)*x - 4708275)*x - 30720080)*x - 21456765)*x - 19537120)*sqrt(3*x^2 + 2) - 4991/18*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
Time = 10.60 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.49 \[ \int (5-x) (3+2 x)^4 \left (2+3 x^2\right )^{5/2} \, dx=\frac {4991\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-\frac {432\,x^{10}}{11}-\frac {216\,x^9}{5}+\frac {7976\,x^8}{11}+\frac {14202\,x^7}{5}+\frac {173419\,x^6}{33}+\frac {72933\,x^5}{10}+\frac {279190\,x^4}{33}+\frac {28535\,x^3}{4}+\frac {1536004\,x^2}{297}+\frac {14449\,x}{4}+\frac {976856}{891}\right )}{3} \]