Integrand size = 25, antiderivative size = 251 \[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=\frac {1543648 \sqrt {x} (2+3 x)}{6567561 \sqrt {2+5 x+3 x^2}}-\frac {8 \sqrt {x} (397265+502911 x) \sqrt {2+5 x+3 x^2}}{2189187}+\frac {157160 \sqrt {x} \left (2+5 x+3 x^2\right )^{3/2}}{243243}-\frac {21620 x^{3/2} \left (2+5 x+3 x^2\right )^{3/2}}{34749}+\frac {656 x^{5/2} \left (2+5 x+3 x^2\right )^{3/2}}{1287}-\frac {10}{39} x^{7/2} \left (2+5 x+3 x^2\right )^{3/2}-\frac {1543648 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{6567561 \sqrt {2+5 x+3 x^2}}+\frac {349240 \sqrt {2} (1+x) \sqrt {\frac {2+3 x}{1+x}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{2189187 \sqrt {2+5 x+3 x^2}} \]
-21620/34749*x^(3/2)*(3*x^2+5*x+2)^(3/2)+656/1287*x^(5/2)*(3*x^2+5*x+2)^(3 /2)-10/39*x^(7/2)*(3*x^2+5*x+2)^(3/2)+157160/243243*(3*x^2+5*x+2)^(3/2)*x^ (1/2)+1543648/6567561*(2+3*x)*x^(1/2)/(3*x^2+5*x+2)^(1/2)-1543648/6567561* (1+x)^(3/2)*(1/(1+x))^(1/2)*EllipticE(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2 ^(1/2)*((2+3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)+349240/2189187*(1+x)^(3/2 )*(1/(1+x))^(1/2)*EllipticF(x^(1/2)/(1+x)^(1/2),1/2*I*2^(1/2))*2^(1/2)*((2 +3*x)/(1+x))^(1/2)/(3*x^2+5*x+2)^(1/2)-8/2189187*(397265+502911*x)*x^(1/2) *(3*x^2+5*x+2)^(1/2)
Result contains complex when optimal does not.
Time = 21.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.71 \[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=\frac {2 \left (1543648+2811400 x+670548 x^2-141444 x^3+58374 x^4+2892348 x^5+671895 x^6-10195794 x^7-7577955 x^8\right )+1543648 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right )|\frac {3}{2}\right )-495928 i \sqrt {2} \sqrt {1+\frac {1}{x}} \sqrt {3+\frac {2}{x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {2}{3}}}{\sqrt {x}}\right ),\frac {3}{2}\right )}{6567561 \sqrt {x} \sqrt {2+5 x+3 x^2}} \]
(2*(1543648 + 2811400*x + 670548*x^2 - 141444*x^3 + 58374*x^4 + 2892348*x^ 5 + 671895*x^6 - 10195794*x^7 - 7577955*x^8) + (1543648*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/ 2] - (495928*I)*Sqrt[2]*Sqrt[1 + x^(-1)]*Sqrt[3 + 2/x]*x^(3/2)*EllipticF[I *ArcSinh[Sqrt[2/3]/Sqrt[x]], 3/2])/(6567561*Sqrt[x]*Sqrt[2 + 5*x + 3*x^2])
Time = 0.48 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {1236, 1236, 27, 1236, 27, 1236, 27, 1231, 1240, 1503, 1413, 1456}
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.
| \(\displaystyle \int (2-5 x) x^{7/2} \sqrt {3 x^2+5 x+2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{39} \int x^{5/2} (164 x+35) \sqrt {3 x^2+5 x+2}dx-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{39} \left (\frac {2}{33} \int -\frac {5}{2} x^{3/2} (1081 x+328) \sqrt {3 x^2+5 x+2}dx+\frac {328}{33} \left (3 x^2+5 x+2\right )^{3/2} x^{5/2}\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \int x^{3/2} (1081 x+328) \sqrt {3 x^2+5 x+2}dx\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2}{27} \int -3 \sqrt {x} (3929 x+1081) \sqrt {3 x^2+5 x+2}dx+\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \int \sqrt {x} (3929 x+1081) \sqrt {3 x^2+5 x+2}dx\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {2}{21} \int -\frac {(55879 x+7858) \sqrt {3 x^2+5 x+2}}{2 \sqrt {x}}dx+\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{21} \int \frac {(55879 x+7858) \sqrt {3 x^2+5 x+2}}{\sqrt {x}}dx\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {1}{21} \left (\frac {2}{45} \int \frac {96478 x+43655}{\sqrt {x} \sqrt {3 x^2+5 x+2}}dx-\frac {2}{45} \sqrt {x} (502911 x+397265) \sqrt {3 x^2+5 x+2}\right )+\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1240 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{45} \int \frac {96478 x+43655}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}-\frac {2}{45} \sqrt {x} (502911 x+397265) \sqrt {3 x^2+5 x+2}\right )+\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1503 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{45} \left (43655 \int \frac {1}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+96478 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}\right )-\frac {2}{45} \sqrt {x} (502911 x+397265) \sqrt {3 x^2+5 x+2}\right )+\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1413 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{45} \left (96478 \int \frac {x}{\sqrt {3 x^2+5 x+2}}d\sqrt {x}+\frac {43655 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}\right )-\frac {2}{45} \sqrt {x} (502911 x+397265) \sqrt {3 x^2+5 x+2}\right )+\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
| \(\Big \downarrow \) 1456 |
| \(\displaystyle \frac {2}{39} \left (\frac {328}{33} x^{5/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {5}{33} \left (\frac {2162}{27} x^{3/2} \left (3 x^2+5 x+2\right )^{3/2}-\frac {2}{9} \left (\frac {1}{21} \left (\frac {4}{45} \left (\frac {43655 (x+1) \sqrt {\frac {3 x+2}{x+1}} \operatorname {EllipticF}\left (\arctan \left (\sqrt {x}\right ),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^2+5 x+2}}+96478 \left (\frac {\sqrt {x} (3 x+2)}{3 \sqrt {3 x^2+5 x+2}}-\frac {\sqrt {2} (x+1) \sqrt {\frac {3 x+2}{x+1}} E\left (\arctan \left (\sqrt {x}\right )|-\frac {1}{2}\right )}{3 \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2}{45} \sqrt {x} (502911 x+397265) \sqrt {3 x^2+5 x+2}\right )+\frac {7858}{21} \sqrt {x} \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )-\frac {10}{39} x^{7/2} \left (3 x^2+5 x+2\right )^{3/2}\) |
(-10*x^(7/2)*(2 + 5*x + 3*x^2)^(3/2))/39 + (2*((328*x^(5/2)*(2 + 5*x + 3*x ^2)^(3/2))/33 - (5*((2162*x^(3/2)*(2 + 5*x + 3*x^2)^(3/2))/27 - (2*((7858* Sqrt[x]*(2 + 5*x + 3*x^2)^(3/2))/21 + ((-2*Sqrt[x]*(397265 + 502911*x)*Sqr t[2 + 5*x + 3*x^2])/45 + (4*(96478*((Sqrt[x]*(2 + 3*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (Sqrt[2]*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticE[ArcTan[Sqrt[x ]], -1/2])/(3*Sqrt[2 + 5*x + 3*x^2])) + (43655*(1 + x)*Sqrt[(2 + 3*x)/(1 + x)]*EllipticF[ArcTan[Sqrt[x]], -1/2])/(Sqrt[2]*Sqrt[2 + 5*x + 3*x^2])))/4 5)/21))/9))/33))/39
3.11.34.3.1 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[((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*(d + e*x)^m*((a + b*x + 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 + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[2 Subst[Int[(f + g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b ^2 - 4*a*c, 2]}, Simp[(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q)*x^2)/(2*a + (b - q)*x^2)]/(2*a*Rt[(b - q)/(2*a), 2]*Sqrt[a + b*x^2 + c*x^4]))*EllipticF [ArcTan[Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[{a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[x*((b - q + 2*c*x^2)/(2*c*Sqrt[a + b*x^2 + c*x^4 ])), x] - Simp[Rt[(b - q)/(2*a), 2]*(2*a + (b - q)*x^2)*(Sqrt[(2*a + (b + q )*x^2)/(2*a + (b - q)*x^2)]/(2*c*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[ArcTan [Rt[(b - q)/(2*a), 2]*x], -2*(q/(b - q))], x] /; PosQ[(b - q)/a]] /; FreeQ[ {a, b, c}, x] && GtQ[b^2 - 4*a*c, 0]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[d Int[1/Sqrt[a + b*x^2 + c*x^4] , x], x] + Simp[e Int[x^2/Sqrt[a + b*x^2 + c*x^4], x], x] /; PosQ[(b + q) /a] || PosQ[(b - q)/a]] /; FreeQ[{a, b, c, d, e}, x] && GtQ[b^2 - 4*a*c, 0]
Time = 0.38 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.55
| method | result | size |
| default | \(-\frac {2 \left (22733865 x^{8}+30587382 x^{7}-2015685 x^{6}-8677044 x^{5}+633876 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-385912 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {6}\, \sqrt {-x}\, E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )-175122 x^{4}+424332 x^{3}+4934772 x^{2}+3143160 x \right )}{19702683 \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(137\) |
| risch | \(-\frac {2 \left (841995 x^{5}-270459 x^{4}-185220 x^{3}+167634 x^{2}-162396 x +174620\right ) \sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}{2189187}-\frac {\left (-\frac {349240 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}-\frac {771824 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right ) \sqrt {x \left (3 x^{2}+5 x +2\right )}}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(203\) |
| elliptic | \(\frac {\sqrt {x \left (3 x^{2}+5 x +2\right )}\, \left (-\frac {10 x^{5} \sqrt {3 x^{3}+5 x^{2}+2 x}}{13}+\frac {106 x^{4} \sqrt {3 x^{3}+5 x^{2}+2 x}}{429}+\frac {1960 x^{3} \sqrt {3 x^{3}+5 x^{2}+2 x}}{11583}-\frac {37252 x^{2} \sqrt {3 x^{3}+5 x^{2}+2 x}}{243243}+\frac {2776 x \sqrt {3 x^{3}+5 x^{2}+2 x}}{18711}-\frac {349240 \sqrt {3 x^{3}+5 x^{2}+2 x}}{2189187}+\frac {349240 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}+\frac {771824 \sqrt {6 x +4}\, \sqrt {3+3 x}\, \sqrt {-6 x}\, \left (\frac {E\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )}{3}-F\left (\frac {\sqrt {6 x +4}}{2}, i \sqrt {2}\right )\right )}{6567561 \sqrt {3 x^{3}+5 x^{2}+2 x}}\right )}{\sqrt {x}\, \sqrt {3 x^{2}+5 x +2}}\) | \(280\) |
-2/19702683/x^(1/2)/(3*x^2+5*x+2)^(1/2)*(22733865*x^8+30587382*x^7-2015685 *x^6-8677044*x^5+633876*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^(1/2)*(-x)^(1/2)*Ell ipticF(1/2*(6*x+4)^(1/2),I*2^(1/2))-385912*(6*x+4)^(1/2)*(3+3*x)^(1/2)*6^( 1/2)*(-x)^(1/2)*EllipticE(1/2*(6*x+4)^(1/2),I*2^(1/2))-175122*x^4+424332*x ^3+4934772*x^2+3143160*x)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.27 \[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=-\frac {2}{2189187} \, {\left (841995 \, x^{5} - 270459 \, x^{4} - 185220 \, x^{3} + 167634 \, x^{2} - 162396 \, x + 174620\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} \sqrt {x} - \frac {204560}{8444007} \, \sqrt {3} {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right ) - \frac {1543648}{6567561} \, \sqrt {3} {\rm weierstrassZeta}\left (\frac {28}{27}, \frac {80}{729}, {\rm weierstrassPInverse}\left (\frac {28}{27}, \frac {80}{729}, x + \frac {5}{9}\right )\right ) \]
-2/2189187*(841995*x^5 - 270459*x^4 - 185220*x^3 + 167634*x^2 - 162396*x + 174620)*sqrt(3*x^2 + 5*x + 2)*sqrt(x) - 204560/8444007*sqrt(3)*weierstras sPInverse(28/27, 80/729, x + 5/9) - 1543648/6567561*sqrt(3)*weierstrassZet a(28/27, 80/729, weierstrassPInverse(28/27, 80/729, x + 5/9))
\[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=- \int \left (- 2 x^{\frac {7}{2}} \sqrt {3 x^{2} + 5 x + 2}\right )\, dx - \int 5 x^{\frac {9}{2}} \sqrt {3 x^{2} + 5 x + 2}\, dx \]
-Integral(-2*x**(7/2)*sqrt(3*x**2 + 5*x + 2), x) - Integral(5*x**(9/2)*sqr t(3*x**2 + 5*x + 2), x)
\[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=\int { -\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} x^{\frac {7}{2}} \,d x } \]
\[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=\int { -\sqrt {3 \, x^{2} + 5 \, x + 2} {\left (5 \, x - 2\right )} x^{\frac {7}{2}} \,d x } \]
Timed out. \[ \int (2-5 x) x^{7/2} \sqrt {2+5 x+3 x^2} \, dx=-\int x^{7/2}\,\left (5\,x-2\right )\,\sqrt {3\,x^2+5\,x+2} \,d x \]