Integrand size = 25, antiderivative size = 131 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {1865 (5+6 x) \sqrt {2+5 x+3 x^2}}{663552}-\frac {1865 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{82944}+\frac {373 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{1728}+\frac {1}{168} (71-14 x) \left (2+5 x+3 x^2\right )^{7/2}-\frac {1865 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{1327104 \sqrt {3}} \]
-1865/82944*(5+6*x)*(3*x^2+5*x+2)^(3/2)+373/1728*(5+6*x)*(3*x^2+5*x+2)^(5/ 2)+1/168*(71-14*x)*(3*x^2+5*x+2)^(7/2)-1865/3981312*arctanh(1/6*(5+6*x)*3^ (1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+1865/663552*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.48 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-34777419-235223330 x-642995688 x^2-897818256 x^3-655212672 x^4-211154688 x^5-746496 x^6+10450944 x^7\right )-13055 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{13934592} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-34777419 - 235223330*x - 642995688*x^2 - 89781 8256*x^3 - 655212672*x^4 - 211154688*x^5 - 746496*x^6 + 10450944*x^7) - 13 055*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/13934592
Time = 0.27 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1225, 1087, 1087, 1087, 1092, 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.
| \(\displaystyle \int (5-x) (2 x+3) \left (3 x^2+5 x+2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {373}{48} \int \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {373}{48} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {373}{48} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {373}{48} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {373}{48} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {373}{48} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{168} (71-14 x) \left (3 x^2+5 x+2\right )^{7/2}\) |
((71 - 14*x)*(2 + 5*x + 3*x^2)^(7/2))/168 + (373*(((5 + 6*x)*(2 + 5*x + 3* x^2)^(5/2))/36 - (5*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(3/2))/24 + (-1/12*((5 + 6*x)*Sqrt[2 + 5*x + 3*x^2]) + ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])]/(24*Sqrt[3]))/16))/72))/48
3.25.35.3.1 Defintions of rubi rules used
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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(b + 2*c*x) *((a + b*x + c*x^2)^p/(2*c*(2*p + 1))), x] - Simp[p*((b^2 - 4*a*c)/(2*c*(2* p + 1))) Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(b*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x))*((a + b*x + c*x^2)^(p + 1)/(2*c^2*(p + 1)*(2*p + 3))), x] + Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c , d, e, f, g, p}, x] && !LeQ[p, -1]
Time = 0.33 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.61
| method | result | size |
| risch | \(-\frac {\left (10450944 x^{7}-746496 x^{6}-211154688 x^{5}-655212672 x^{4}-897818256 x^{3}-642995688 x^{2}-235223330 x -34777419\right ) \sqrt {3 x^{2}+5 x +2}}{4644864}-\frac {1865 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3981312}\) | \(80\) |
| trager | \(\left (-\frac {9}{4} x^{7}+\frac {9}{56} x^{6}+\frac {10183}{224} x^{5}+\frac {189587}{1344} x^{4}+\frac {692761}{3584} x^{3}+\frac {26791487}{193536} x^{2}+\frac {117611665}{2322432} x +\frac {11592473}{1548288}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {1865 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{3981312}\) | \(91\) |
| default | \(\frac {373 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1728}-\frac {1865 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{82944}+\frac {1865 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{663552}-\frac {1865 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{3981312}+\frac {71 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{168}-\frac {x \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{12}\) | \(117\) |
-1/4644864*(10450944*x^7-746496*x^6-211154688*x^5-655212672*x^4-897818256* x^3-642995688*x^2-235223330*x-34777419)*(3*x^2+5*x+2)^(1/2)-1865/3981312*l n(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.67 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{4644864} \, {\left (10450944 \, x^{7} - 746496 \, x^{6} - 211154688 \, x^{5} - 655212672 \, x^{4} - 897818256 \, x^{3} - 642995688 \, x^{2} - 235223330 \, x - 34777419\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1865}{7962624} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/4644864*(10450944*x^7 - 746496*x^6 - 211154688*x^5 - 655212672*x^4 - 89 7818256*x^3 - 642995688*x^2 - 235223330*x - 34777419)*sqrt(3*x^2 + 5*x + 2 ) + 1865/7962624*sqrt(3)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.76 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.74 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {9 x^{7}}{4} + \frac {9 x^{6}}{56} + \frac {10183 x^{5}}{224} + \frac {189587 x^{4}}{1344} + \frac {692761 x^{3}}{3584} + \frac {26791487 x^{2}}{193536} + \frac {117611665 x}{2322432} + \frac {11592473}{1548288}\right ) - \frac {1865 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{3981312} \]
sqrt(3*x**2 + 5*x + 2)*(-9*x**7/4 + 9*x**6/56 + 10183*x**5/224 + 189587*x* *4/1344 + 692761*x**3/3584 + 26791487*x**2/193536 + 117611665*x/2322432 + 11592473/1548288) - 1865*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2 ) + 5)/3981312
Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.11 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{12} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {71}{168} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {373}{288} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {1865}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {1865}{13824} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {9325}{82944} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {1865}{110592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {1865}{3981312} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {9325}{663552} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-1/12*(3*x^2 + 5*x + 2)^(7/2)*x + 71/168*(3*x^2 + 5*x + 2)^(7/2) + 373/288 *(3*x^2 + 5*x + 2)^(5/2)*x + 1865/1728*(3*x^2 + 5*x + 2)^(5/2) - 1865/1382 4*(3*x^2 + 5*x + 2)^(3/2)*x - 9325/82944*(3*x^2 + 5*x + 2)^(3/2) + 1865/11 0592*sqrt(3*x^2 + 5*x + 2)*x - 1865/3981312*sqrt(3)*log(2*sqrt(3)*sqrt(3*x ^2 + 5*x + 2) + 6*x + 5) + 9325/663552*sqrt(3*x^2 + 5*x + 2)
Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.64 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{4644864} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, x - 1\right )} x - 10183\right )} x - 189587\right )} x - 2078283\right )} x - 26791487\right )} x - 117611665\right )} x - 34777419\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1865}{3981312} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/4644864*(2*(12*(18*(8*(6*(36*(14*x - 1)*x - 10183)*x - 189587)*x - 2078 283)*x - 26791487)*x - 117611665)*x - 34777419)*sqrt(3*x^2 + 5*x + 2) + 18 65/3981312*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]