Integrand size = 25, antiderivative size = 154 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {9793 (5+6 x) \sqrt {2+5 x+3 x^2}}{47775744}+\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{5971968}-\frac {9793 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{622080}+\frac {1399 (5+6 x) \left (2+5 x+3 x^2\right )^{7/2}}{8640}+\frac {1}{810} (265-54 x) \left (2+5 x+3 x^2\right )^{9/2}+\frac {9793 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{95551488 \sqrt {3}} \]
9793/5971968*(5+6*x)*(3*x^2+5*x+2)^(3/2)-9793/622080*(5+6*x)*(3*x^2+5*x+2) ^(5/2)+1399/8640*(5+6*x)*(3*x^2+5*x+2)^(7/2)+1/810*(265-54*x)*(3*x^2+5*x+2 )^(9/2)+9793/286654464*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^ (1/2)-9793/47775744*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.75 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-2726071095-25257845290 x-100612822920 x^2-224097754320 x^3-302902600320 x^4-250227954432 x^5-117850567680 x^6-23529056256 x^7+2269347840 x^8+1289945088 x^9\right )+48965 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{716636160} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-2726071095 - 25257845290*x - 100612822920*x^2 - 224097754320*x^3 - 302902600320*x^4 - 250227954432*x^5 - 117850567680*x^ 6 - 23529056256*x^7 + 2269347840*x^8 + 1289945088*x^9) + 48965*Sqrt[3]*Arc Tanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/716636160
Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1225, 1087, 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.
| \(\displaystyle \int (5-x) (2 x+3) \left (3 x^2+5 x+2\right )^{7/2} \, dx\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1399}{180} \int \left (3 x^2+5 x+2\right )^{7/2}dx+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1399}{180} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \int \left (3 x^2+5 x+2\right )^{5/2}dx\right )+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1399}{180} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )\right )+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1399}{180} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )\right )+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1399}{180} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1399}{180} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1399}{180} \left (\frac {1}{48} (6 x+5) \left (3 x^2+5 x+2\right )^{7/2}-\frac {7}{96} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )\right )+\frac {1}{810} (265-54 x) \left (3 x^2+5 x+2\right )^{9/2}\) |
((265 - 54*x)*(2 + 5*x + 3*x^2)^(9/2))/810 + (1399*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(7/2))/48 - (7*(((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))/96))/180
3.25.50.3.1 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 = 90, normalized size of antiderivative = 0.58
| method | result | size |
| risch | \(-\frac {\left (1289945088 x^{9}+2269347840 x^{8}-23529056256 x^{7}-117850567680 x^{6}-250227954432 x^{5}-302902600320 x^{4}-224097754320 x^{3}-100612822920 x^{2}-25257845290 x -2726071095\right ) \sqrt {3 x^{2}+5 x +2}}{238878720}+\frac {9793 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{286654464}\) | \(90\) |
| trager | \(\left (-\frac {27}{5} x^{9}-\frac {19}{2} x^{8}+\frac {47279}{480} x^{7}+\frac {94723}{192} x^{6}+\frac {36201961}{34560} x^{5}+\frac {52587257}{41472} x^{4}+\frac {311246881}{331776} x^{3}+\frac {838440191}{1990656} x^{2}+\frac {2525784529}{23887872} x +\frac {181738073}{15925248}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {9793 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{286654464}\) | \(101\) |
| default | \(\frac {1399 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{8640}-\frac {9793 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{622080}+\frac {9793 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{5971968}-\frac {9793 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{47775744}+\frac {9793 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{286654464}+\frac {53 \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{162}-\frac {x \left (3 x^{2}+5 x +2\right )^{\frac {9}{2}}}{15}\) | \(136\) |
-1/238878720*(1289945088*x^9+2269347840*x^8-23529056256*x^7-117850567680*x ^6-250227954432*x^5-302902600320*x^4-224097754320*x^3-100612822920*x^2-252 57845290*x-2726071095)*(3*x^2+5*x+2)^(1/2)+9793/286654464*ln(1/3*(5/2+3*x) *3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.30 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.64 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{238878720} \, {\left (1289945088 \, x^{9} + 2269347840 \, x^{8} - 23529056256 \, x^{7} - 117850567680 \, x^{6} - 250227954432 \, x^{5} - 302902600320 \, x^{4} - 224097754320 \, x^{3} - 100612822920 \, x^{2} - 25257845290 \, x - 2726071095\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {9793}{573308928} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/238878720*(1289945088*x^9 + 2269347840*x^8 - 23529056256*x^7 - 11785056 7680*x^6 - 250227954432*x^5 - 302902600320*x^4 - 224097754320*x^3 - 100612 822920*x^2 - 25257845290*x - 2726071095)*sqrt(3*x^2 + 5*x + 2) + 9793/5733 08928*sqrt(3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120 *x + 49)
Time = 0.97 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {27 x^{9}}{5} - \frac {19 x^{8}}{2} + \frac {47279 x^{7}}{480} + \frac {94723 x^{6}}{192} + \frac {36201961 x^{5}}{34560} + \frac {52587257 x^{4}}{41472} + \frac {311246881 x^{3}}{331776} + \frac {838440191 x^{2}}{1990656} + \frac {2525784529 x}{23887872} + \frac {181738073}{15925248}\right ) + \frac {9793 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{286654464} \]
sqrt(3*x**2 + 5*x + 2)*(-27*x**9/5 - 19*x**8/2 + 47279*x**7/480 + 94723*x* *6/192 + 36201961*x**5/34560 + 52587257*x**4/41472 + 311246881*x**3/331776 + 838440191*x**2/1990656 + 2525784529*x/23887872 + 181738073/15925248) + 9793*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/286654464
Time = 0.27 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.13 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{15} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} x + \frac {53}{162} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}} + \frac {1399}{1440} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {1399}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {9793}{103680} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x - \frac {9793}{124416} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {9793}{995328} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {48965}{5971968} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {9793}{7962624} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {9793}{286654464} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {48965}{47775744} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-1/15*(3*x^2 + 5*x + 2)^(9/2)*x + 53/162*(3*x^2 + 5*x + 2)^(9/2) + 1399/14 40*(3*x^2 + 5*x + 2)^(7/2)*x + 1399/1728*(3*x^2 + 5*x + 2)^(7/2) - 9793/10 3680*(3*x^2 + 5*x + 2)^(5/2)*x - 9793/124416*(3*x^2 + 5*x + 2)^(5/2) + 979 3/995328*(3*x^2 + 5*x + 2)^(3/2)*x + 48965/5971968*(3*x^2 + 5*x + 2)^(3/2) - 9793/7962624*sqrt(3*x^2 + 5*x + 2)*x + 9793/286654464*sqrt(3)*log(2*sqr t(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 48965/47775744*sqrt(3*x^2 + 5*x + 2)
Time = 0.34 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\frac {1}{238878720} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (2 \, {\left (48 \, {\left (54 \, x + 95\right )} x - 47279\right )} x - 473615\right )} x - 36201961\right )} x - 262936285\right )} x - 1556234405\right )} x - 4192200955\right )} x - 12628922645\right )} x - 2726071095\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {9793}{286654464} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/238878720*(2*(12*(6*(8*(6*(36*(2*(48*(54*x + 95)*x - 47279)*x - 473615) *x - 36201961)*x - 262936285)*x - 1556234405)*x - 4192200955)*x - 12628922 645)*x - 2726071095)*sqrt(3*x^2 + 5*x + 2) - 9793/286654464*sqrt(3)*log(ab s(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{7/2} \, dx=-\int \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2} \,d x \]