Integrand size = 20, antiderivative size = 126 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {175 (5+6 x) \sqrt {2+5 x+3 x^2}}{82944}-\frac {175 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{10368}+\frac {35}{216} (5+6 x) \left (2+5 x+3 x^2\right )^{5/2}-\frac {1}{21} \left (2+5 x+3 x^2\right )^{7/2}-\frac {175 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{165888 \sqrt {3}} \]
-175/10368*(5+6*x)*(3*x^2+5*x+2)^(3/2)+35/216*(5+6*x)*(3*x^2+5*x+2)^(5/2)- 1/21*(3*x^2+5*x+2)^(7/2)-175/497664*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x +2)^(1/2))*3^(1/2)+175/82944*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.38 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-1568541-9651790 x-23110872 x^2-26388720 x^3-13454208 x^4-1347840 x^5+746496 x^6\right )-1225 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{1741824} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-1568541 - 9651790*x - 23110872*x^2 - 26388720* x^3 - 13454208*x^4 - 1347840*x^5 + 746496*x^6) - 1225*Sqrt[3]*ArcTanh[Sqrt [2/3 + (5*x)/3 + x^2]/(1 + x)])/1741824
Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1160, 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.
\(\displaystyle \int (5-x) \left (3 x^2+5 x+2\right )^{5/2} \, dx\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {35}{6} \int \left (3 x^2+5 x+2\right )^{5/2}dx-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {35}{6} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \int \left (3 x^2+5 x+2\right )^{3/2}dx\right )-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {35}{6} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )\right )-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {35}{6} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {35}{6} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {35}{6} \left (\frac {1}{36} (6 x+5) \left (3 x^2+5 x+2\right )^{5/2}-\frac {5}{72} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )\right )-\frac {1}{21} \left (3 x^2+5 x+2\right )^{7/2}\) |
-1/21*(2 + 5*x + 3*x^2)^(7/2) + (35*(((5 + 6*x)*(2 + 5*x + 3*x^2)^(5/2))/3 6 - (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))/6
3.25.36.3.1 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_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Time = 0.33 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60
method | result | size |
risch | \(-\frac {\left (746496 x^{6}-1347840 x^{5}-13454208 x^{4}-26388720 x^{3}-23110872 x^{2}-9651790 x -1568541\right ) \sqrt {3 x^{2}+5 x +2}}{580608}-\frac {175 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{497664}\) | \(75\) |
trager | \(\left (-\frac {9}{7} x^{6}+\frac {65}{28} x^{5}+\frac {3893}{168} x^{4}+\frac {61085}{1344} x^{3}+\frac {962953}{24192} x^{2}+\frac {4825895}{290304} x +\frac {522847}{193536}\right ) \sqrt {3 x^{2}+5 x +2}+\frac {175 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{497664}\) | \(86\) |
default | \(\frac {35 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{216}-\frac {175 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{10368}+\frac {175 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{82944}-\frac {175 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{497664}-\frac {\left (3 x^{2}+5 x +2\right )^{\frac {7}{2}}}{21}\) | \(102\) |
-1/580608*(746496*x^6-1347840*x^5-13454208*x^4-26388720*x^3-23110872*x^2-9 651790*x-1568541)*(3*x^2+5*x+2)^(1/2)-175/497664*ln(1/3*(5/2+3*x)*3^(1/2)+ (3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.66 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{580608} \, {\left (746496 \, x^{6} - 1347840 \, x^{5} - 13454208 \, x^{4} - 26388720 \, x^{3} - 23110872 \, x^{2} - 9651790 \, x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{995328} \, \sqrt {3} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/580608*(746496*x^6 - 1347840*x^5 - 13454208*x^4 - 26388720*x^3 - 231108 72*x^2 - 9651790*x - 1568541)*sqrt(3*x^2 + 5*x + 2) + 175/995328*sqrt(3)*l og(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.59 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- \frac {9 x^{6}}{7} + \frac {65 x^{5}}{28} + \frac {3893 x^{4}}{168} + \frac {61085 x^{3}}{1344} + \frac {962953 x^{2}}{24192} + \frac {4825895 x}{290304} + \frac {522847}{193536}\right ) - \frac {175 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{497664} \]
sqrt(3*x**2 + 5*x + 2)*(-9*x**6/7 + 65*x**5/28 + 3893*x**4/168 + 61085*x** 3/1344 + 962953*x**2/24192 + 4825895*x/290304 + 522847/193536) - 175*sqrt( 3)*log(6*x + 2*sqrt(3)*sqrt(3*x**2 + 5*x + 2) + 5)/497664
Time = 0.27 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.03 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{21} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {35}{36} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {175}{216} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {175}{1728} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {875}{10368} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {175}{13824} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {175}{497664} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {875}{82944} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-1/21*(3*x^2 + 5*x + 2)^(7/2) + 35/36*(3*x^2 + 5*x + 2)^(5/2)*x + 175/216* (3*x^2 + 5*x + 2)^(5/2) - 175/1728*(3*x^2 + 5*x + 2)^(3/2)*x - 875/10368*( 3*x^2 + 5*x + 2)^(3/2) + 175/13824*sqrt(3*x^2 + 5*x + 2)*x - 175/497664*sq rt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 875/82944*sqrt(3*x^ 2 + 5*x + 2)
Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.63 \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=-\frac {1}{580608} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (6 \, {\left (36 \, x - 65\right )} x - 3893\right )} x - 61085\right )} x - 962953\right )} x - 4825895\right )} x - 1568541\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {175}{497664} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/580608*(2*(12*(18*(8*(6*(36*x - 65)*x - 3893)*x - 61085)*x - 962953)*x - 4825895)*x - 1568541)*sqrt(3*x^2 + 5*x + 2) + 175/497664*sqrt(3)*log(abs (-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) \left (2+5 x+3 x^2\right )^{5/2} \, dx=\int -\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2} \,d x \]