Integrand size = 27, antiderivative size = 133 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1129 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {1129 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}-\frac {1}{21} (3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}+\frac {(5827+2370 x) \left (2+5 x+3 x^2\right )^{5/2}}{1890}+\frac {1129 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}} \]
1129/2592*(5+6*x)*(3*x^2+5*x+2)^(3/2)-1/21*(3+2*x)^2*(3*x^2+5*x+2)^(5/2)+1 /1890*(5827+2370*x)*(3*x^2+5*x+2)^(5/2)+1129/124416*arctanh(1/6*(5+6*x)*3^ (1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-1129/20736*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.41 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.61 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-10669737-51971350 x-94861176 x^2-79049520 x^3-27084672 x^4-311040 x^5+1244160 x^6\right )+39515 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{2177280} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-10669737 - 51971350*x - 94861176*x^2 - 7904952 0*x^3 - 27084672*x^4 - 311040*x^5 + 1244160*x^6) + 39515*Sqrt[3]*ArcTanh[S qrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/2177280
Time = 0.28 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.11, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1236, 27, 1225, 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.
| \(\displaystyle \int (5-x) (2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{21} \int \frac {1}{2} (2 x+3) (474 x+721) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{42} \int (2 x+3) (474 x+721) \left (3 x^2+5 x+2\right )^{3/2}dx-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{42} \left (\frac {7903}{18} \int \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {1}{45} (2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{42} \left (\frac {7903}{18} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )+\frac {1}{45} (2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{42} \left (\frac {7903}{18} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{45} (2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{42} \left (\frac {7903}{18} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{45} (2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{42} \left (\frac {7903}{18} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{45} (2370 x+5827) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {1}{21} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}\) |
-1/21*((3 + 2*x)^2*(2 + 5*x + 3*x^2)^(5/2)) + (((5827 + 2370*x)*(2 + 5*x + 3*x^2)^(5/2))/45 + (7903*(((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))/18)/42
3.25.20.3.1 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[((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]
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])
Time = 0.32 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {\left (1244160 x^{6}-311040 x^{5}-27084672 x^{4}-79049520 x^{3}-94861176 x^{2}-51971350 x -10669737\right ) \sqrt {3 x^{2}+5 x +2}}{725760}+\frac {1129 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{124416}\) | \(75\) |
| trager | \(\left (-\frac {12}{7} x^{6}+\frac {3}{7} x^{5}+\frac {7837}{210} x^{4}+\frac {12199}{112} x^{3}+\frac {3952549}{30240} x^{2}+\frac {5197135}{72576} x +\frac {3556579}{241920}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {1129 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{124416}\) | \(86\) |
| default | \(\frac {1129 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {1129 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{20736}+\frac {1129 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{124416}+\frac {5017 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{1890}-\frac {4 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{21}+\frac {43 x \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{63}\) | \(115\) |
-1/725760*(1244160*x^6-311040*x^5-27084672*x^4-79049520*x^3-94861176*x^2-5 1971350*x-10669737)*(3*x^2+5*x+2)^(1/2)+1129/124416*ln(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 = 83, normalized size of antiderivative = 0.62 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{725760} \, {\left (1244160 \, x^{6} - 311040 \, x^{5} - 27084672 \, x^{4} - 79049520 \, x^{3} - 94861176 \, x^{2} - 51971350 \, x - 10669737\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {1129}{248832} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/725760*(1244160*x^6 - 311040*x^5 - 27084672*x^4 - 79049520*x^3 - 948611 76*x^2 - 51971350*x - 10669737)*sqrt(3*x^2 + 5*x + 2) + 1129/248832*sqrt(3 )*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.58 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.68 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {12 x^{6}}{7} + \frac {3 x^{5}}{7} + \frac {7837 x^{4}}{210} + \frac {12199 x^{3}}{112} + \frac {3952549 x^{2}}{30240} + \frac {5197135 x}{72576} + \frac {3556579}{241920}\right ) + \frac {1129 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{124416} \]
sqrt(3*x**2 + 5*x + 2)*(-12*x**6/7 + 3*x**5/7 + 7837*x**4/210 + 12199*x**3 /112 + 3952549*x**2/30240 + 5197135*x/72576 + 3556579/241920) + 1129*sqrt( 3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/124416
Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {4}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x^{2} + \frac {43}{63} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {5017}{1890} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {1129}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {5645}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {1129}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {1129}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {5645}{20736} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-4/21*(3*x^2 + 5*x + 2)^(5/2)*x^2 + 43/63*(3*x^2 + 5*x + 2)^(5/2)*x + 5017 /1890*(3*x^2 + 5*x + 2)^(5/2) + 1129/432*(3*x^2 + 5*x + 2)^(3/2)*x + 5645/ 2592*(3*x^2 + 5*x + 2)^(3/2) - 1129/3456*sqrt(3*x^2 + 5*x + 2)*x + 1129/12 4416*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 5645/20736*s qrt(3*x^2 + 5*x + 2)
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.59 \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{725760} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (90 \, {\left (4 \, x - 1\right )} x - 7837\right )} x - 182985\right )} x - 3952549\right )} x - 25985675\right )} x - 10669737\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {1129}{124416} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/725760*(2*(12*(18*(8*(90*(4*x - 1)*x - 7837)*x - 182985)*x - 3952549)*x - 25985675)*x - 10669737)*sqrt(3*x^2 + 5*x + 2) - 1129/124416*sqrt(3)*log (abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x)^2 \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\int {\left (2\,x+3\right )}^2\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]