Integrand size = 27, antiderivative size = 206 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {249299 (5+6 x) \sqrt {2+5 x+3 x^2}}{35831808}-\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{4478976}+\frac {249299 (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}}{466560}+\frac {3298 (3+2 x)^2 \left (2+5 x+3 x^2\right )^{7/2}}{4455}+\frac {41}{110} (3+2 x)^3 \left (2+5 x+3 x^2\right )^{7/2}-\frac {1}{33} (3+2 x)^4 \left (2+5 x+3 x^2\right )^{7/2}+\frac {(7405817+3365726 x) \left (2+5 x+3 x^2\right )^{7/2}}{1496880}-\frac {249299 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{71663616 \sqrt {3}} \]
-249299/4478976*(5+6*x)*(3*x^2+5*x+2)^(3/2)+249299/466560*(5+6*x)*(3*x^2+5 *x+2)^(5/2)+3298/4455*(3+2*x)^2*(3*x^2+5*x+2)^(7/2)+41/110*(3+2*x)^3*(3*x^ 2+5*x+2)^(7/2)-1/33*(3+2*x)^4*(3*x^2+5*x+2)^(7/2)+1/1496880*(7405817+33657 26*x)*(3*x^2+5*x+2)^(7/2)-249299/214990848*arctanh(1/6*(5+6*x)*3^(1/2)/(3* x^2+5*x+2)^(1/2))*3^(1/2)+249299/35831808*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.79 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.49 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-2261297826735-19521700361210 x-73069860056520 x^2-155155370878800 x^3-204855126595200 x^4-172473366866688 x^5-90095929758720 x^6-25759323039744 x^7-1932170526720 x^8+875872714752 x^9+180592312320 x^{10}\right )-95980115 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{41385738240} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-2261297826735 - 19521700361210*x - 73069860056 520*x^2 - 155155370878800*x^3 - 204855126595200*x^4 - 172473366866688*x^5 - 90095929758720*x^6 - 25759323039744*x^7 - 1932170526720*x^8 + 8758727147 52*x^9 + 180592312320*x^10) - 95980115*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/41385738240
Time = 0.41 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.15, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1236, 27, 1236, 27, 1236, 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.
| \(\displaystyle \int (5-x) (2 x+3)^4 \left (3 x^2+5 x+2\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{33} \int \frac {1}{2} (2 x+3)^3 (738 x+1127) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{66} \int (2 x+3)^3 (738 x+1127) \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{30} \int 3 (2 x+3)^2 (13192 x+17943) \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \int (2 x+3)^2 (13192 x+17943) \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \int (2 x+3) (480818 x+655267) \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \left (\frac {2742289}{8} \int \left (3 x^2+5 x+2\right )^{5/2}dx+\frac {1}{84} (3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \left (\frac {2742289}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )+\frac {1}{84} (3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \left (\frac {2742289}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )+\frac {1}{84} (3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \left (\frac {2742289}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \left (\frac {2742289}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{66} \left (\frac {1}{10} \left (\frac {1}{27} \left (\frac {2742289}{8} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )+\frac {1}{84} (3365726 x+7405817) \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {13192}{27} (2 x+3)^2 \left (3 x^2+5 x+2\right )^{7/2}\right )+\frac {123}{5} (2 x+3)^3 \left (3 x^2+5 x+2\right )^{7/2}\right )-\frac {1}{33} (2 x+3)^4 \left (3 x^2+5 x+2\right )^{7/2}\) |
-1/33*((3 + 2*x)^4*(2 + 5*x + 3*x^2)^(7/2)) + ((123*(3 + 2*x)^3*(2 + 5*x + 3*x^2)^(7/2))/5 + ((13192*(3 + 2*x)^2*(2 + 5*x + 3*x^2)^(7/2))/27 + (((74 05817 + 3365726*x)*(2 + 5*x + 3*x^2)^(7/2))/84 + (2742289*(((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))/8)/27)/10)/66
3.25.32.3.1 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 = 0.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(-\frac {\left (180592312320 x^{10}+875872714752 x^{9}-1932170526720 x^{8}-25759323039744 x^{7}-90095929758720 x^{6}-172473366866688 x^{5}-204855126595200 x^{4}-155155370878800 x^{3}-73069860056520 x^{2}-19521700361210 x -2261297826735\right ) \sqrt {3 x^{2}+5 x +2}}{13795246080}-\frac {249299 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{214990848}\) | \(95\) |
| trager | \(\left (-\frac {144}{11} x^{10}-\frac {3492}{55} x^{9}+\frac {4622}{33} x^{8}+\frac {7394353}{3960} x^{7}+\frac {72415067}{11088} x^{6}+\frac {24952744049}{1995840} x^{5}+\frac {35565126145}{2395008} x^{4}+\frac {215493570665}{19160064} x^{3}+\frac {608915500471}{114960384} x^{2}+\frac {1952170036121}{1379524608} x +\frac {150753188449}{919683072}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {249299 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{214990848}\) | \(106\) |
| default | \(\frac {249299 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{466560}-\frac {249299 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{4478976}+\frac {249299 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{35831808}-\frac {249299 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{214990848}+\frac {5753773 \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{299376}-\frac {16 x^{4} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{33}+\frac {4 x^{3} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{55}+\frac {8762 x^{2} \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{891}+\frac {2642401 x \left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{106920}\) | \(168\) |
-1/13795246080*(180592312320*x^10+875872714752*x^9-1932170526720*x^8-25759 323039744*x^7-90095929758720*x^6-172473366866688*x^5-204855126595200*x^4-1 55155370878800*x^3-73069860056520*x^2-19521700361210*x-2261297826735)*(3*x ^2+5*x+2)^(1/2)-249299/214990848*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 = 103, normalized size of antiderivative = 0.50 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{13795246080} \, {\left (180592312320 \, x^{10} + 875872714752 \, x^{9} - 1932170526720 \, x^{8} - 25759323039744 \, x^{7} - 90095929758720 \, x^{6} - 172473366866688 \, x^{5} - 204855126595200 \, x^{4} - 155155370878800 \, x^{3} - 73069860056520 \, x^{2} - 19521700361210 \, x - 2261297826735\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {249299}{429981696} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/13795246080*(180592312320*x^10 + 875872714752*x^9 - 1932170526720*x^8 - 25759323039744*x^7 - 90095929758720*x^6 - 172473366866688*x^5 - 204855126 595200*x^4 - 155155370878800*x^3 - 73069860056520*x^2 - 19521700361210*x - 2261297826735)*sqrt(3*x^2 + 5*x + 2) + 249299/429981696*sqrt(3)*log(-4*sq rt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.84 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.57 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {144 x^{10}}{11} - \frac {3492 x^{9}}{55} + \frac {4622 x^{8}}{33} + \frac {7394353 x^{7}}{3960} + \frac {72415067 x^{6}}{11088} + \frac {24952744049 x^{5}}{1995840} + \frac {35565126145 x^{4}}{2395008} + \frac {215493570665 x^{3}}{19160064} + \frac {608915500471 x^{2}}{114960384} + \frac {1952170036121 x}{1379524608} + \frac {150753188449}{919683072}\right ) - \frac {249299 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{214990848} \]
sqrt(3*x**2 + 5*x + 2)*(-144*x**10/11 - 3492*x**9/55 + 4622*x**8/33 + 7394 353*x**7/3960 + 72415067*x**6/11088 + 24952744049*x**5/1995840 + 355651261 45*x**4/2395008 + 215493570665*x**3/19160064 + 608915500471*x**2/114960384 + 1952170036121*x/1379524608 + 150753188449/919683072) - 249299*sqrt(3)*l og(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/214990848
Time = 0.28 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.95 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {16}{33} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{4} + \frac {4}{55} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{3} + \frac {8762}{891} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x^{2} + \frac {2642401}{106920} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} x + \frac {5753773}{299376} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {249299}{77760} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {249299}{93312} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {249299}{746496} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {1246495}{4478976} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {249299}{5971968} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {249299}{214990848} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {1246495}{35831808} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-16/33*(3*x^2 + 5*x + 2)^(7/2)*x^4 + 4/55*(3*x^2 + 5*x + 2)^(7/2)*x^3 + 87 62/891*(3*x^2 + 5*x + 2)^(7/2)*x^2 + 2642401/106920*(3*x^2 + 5*x + 2)^(7/2 )*x + 5753773/299376*(3*x^2 + 5*x + 2)^(7/2) + 249299/77760*(3*x^2 + 5*x + 2)^(5/2)*x + 249299/93312*(3*x^2 + 5*x + 2)^(5/2) - 249299/746496*(3*x^2 + 5*x + 2)^(3/2)*x - 1246495/4478976*(3*x^2 + 5*x + 2)^(3/2) + 249299/5971 968*sqrt(3*x^2 + 5*x + 2)*x - 249299/214990848*sqrt(3)*log(2*sqrt(3)*sqrt( 3*x^2 + 5*x + 2) + 6*x + 5) + 1246495/35831808*sqrt(3*x^2 + 5*x + 2)
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.48 \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{13795246080} \, {\left (2 \, {\left (12 \, {\left (6 \, {\left (8 \, {\left (6 \, {\left (36 \, {\left (14 \, {\left (48 \, {\left (54 \, {\left (20 \, x + 97\right )} x - 11555\right )} x - 7394353\right )} x - 362075335\right )} x - 24952744049\right )} x - 177825630725\right )} x - 1077467853325\right )} x - 3044577502355\right )} x - 9760850180605\right )} x - 2261297826735\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {249299}{214990848} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/13795246080*(2*(12*(6*(8*(6*(36*(14*(48*(54*(20*x + 97)*x - 11555)*x - 7394353)*x - 362075335)*x - 24952744049)*x - 177825630725)*x - 10774678533 25)*x - 3044577502355)*x - 9760850180605)*x - 2261297826735)*sqrt(3*x^2 + 5*x + 2) + 249299/214990848*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3 *x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x)^4 \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\int {\left (2\,x+3\right )}^4\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]