Integrand size = 25, antiderivative size = 108 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {839 (5+6 x) \sqrt {2+5 x+3 x^2}}{20736}+\frac {839 (5+6 x) \left (2+5 x+3 x^2\right )^{3/2}}{2592}+\frac {1}{270} (161-30 x) \left (2+5 x+3 x^2\right )^{5/2}+\frac {839 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{41472 \sqrt {3}} \]
839/2592*(5+6*x)*(3*x^2+5*x+2)^(3/2)+1/270*(161-30*x)*(3*x^2+5*x+2)^(5/2)+ 839/124416*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-839/20 736*(5+6*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.70 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=\frac {-3 \sqrt {2+5 x+3 x^2} \left (-561921-2406950 x-3567288 x^2-2032560 x^3-210816 x^4+103680 x^5\right )+4195 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{311040} \]
(-3*Sqrt[2 + 5*x + 3*x^2]*(-561921 - 2406950*x - 3567288*x^2 - 2032560*x^3 - 210816*x^4 + 103680*x^5) + 4195*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^ 2]/(1 + x)])/311040
Time = 0.23 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {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.
\(\displaystyle \int (5-x) (2 x+3) \left (3 x^2+5 x+2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {839}{108} \int \left (3 x^2+5 x+2\right )^{3/2}dx+\frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {839}{108} \left (\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}-\frac {1}{16} \int \sqrt {3 x^2+5 x+2}dx\right )+\frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {839}{108} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {839}{108} \left (\frac {1}{16} \left (\frac {1}{12} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {839}{108} \left (\frac {1}{16} \left (\frac {\text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{24 \sqrt {3}}-\frac {1}{12} (6 x+5) \sqrt {3 x^2+5 x+2}\right )+\frac {1}{24} (6 x+5) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{270} (161-30 x) \left (3 x^2+5 x+2\right )^{5/2}\) |
((161 - 30*x)*(2 + 5*x + 3*x^2)^(5/2))/270 + (839*(((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))/108
3.25.21.3.1 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 = 0.32 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.65
method | result | size |
risch | \(-\frac {\left (103680 x^{5}-210816 x^{4}-2032560 x^{3}-3567288 x^{2}-2406950 x -561921\right ) \sqrt {3 x^{2}+5 x +2}}{103680}+\frac {839 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{124416}\) | \(70\) |
trager | \(\left (-x^{5}+\frac {61}{30} x^{4}+\frac {941}{48} x^{3}+\frac {148637}{4320} x^{2}+\frac {240695}{10368} x +\frac {187307}{34560}\right ) \sqrt {3 x^{2}+5 x +2}-\frac {839 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{124416}\) | \(81\) |
default | \(\frac {839 \left (5+6 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}{2592}-\frac {839 \left (5+6 x \right ) \sqrt {3 x^{2}+5 x +2}}{20736}+\frac {839 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{124416}+\frac {161 \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{270}-\frac {x \left (3 x^{2}+5 x +2\right )^{\frac {5}{2}}}{9}\) | \(98\) |
-1/103680*(103680*x^5-210816*x^4-2032560*x^3-3567288*x^2-2406950*x-561921) *(3*x^2+5*x+2)^(1/2)+839/124416*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/ 2))*3^(1/2)
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{103680} \, {\left (103680 \, x^{5} - 210816 \, x^{4} - 2032560 \, x^{3} - 3567288 \, x^{2} - 2406950 \, x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {839}{248832} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) \]
-1/103680*(103680*x^5 - 210816*x^4 - 2032560*x^3 - 3567288*x^2 - 2406950*x - 561921)*sqrt(3*x^2 + 5*x + 2) + 839/248832*sqrt(3)*log(4*sqrt(3)*sqrt(3 *x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49)
Time = 0.67 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.74 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=\sqrt {3 x^{2} + 5 x + 2} \left (- x^{5} + \frac {61 x^{4}}{30} + \frac {941 x^{3}}{48} + \frac {148637 x^{2}}{4320} + \frac {240695 x}{10368} + \frac {187307}{34560}\right ) + \frac {839 \sqrt {3} \log {\left (6 x + 2 \sqrt {3} \sqrt {3 x^{2} + 5 x + 2} + 5 \right )}}{124416} \]
sqrt(3*x**2 + 5*x + 2)*(-x**5 + 61*x**4/30 + 941*x**3/48 + 148637*x**2/432 0 + 240695*x/10368 + 187307/34560) + 839*sqrt(3)*log(6*x + 2*sqrt(3)*sqrt( 3*x**2 + 5*x + 2) + 5)/124416
Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.07 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{9} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {161}{270} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} + \frac {839}{432} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + \frac {4195}{2592} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {839}{3456} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {839}{124416} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {4195}{20736} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-1/9*(3*x^2 + 5*x + 2)^(5/2)*x + 161/270*(3*x^2 + 5*x + 2)^(5/2) + 839/432 *(3*x^2 + 5*x + 2)^(3/2)*x + 4195/2592*(3*x^2 + 5*x + 2)^(3/2) - 839/3456* sqrt(3*x^2 + 5*x + 2)*x + 839/124416*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5* x + 2) + 6*x + 5) - 4195/20736*sqrt(3*x^2 + 5*x + 2)
Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.69 \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\frac {1}{103680} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (8 \, {\left (30 \, x - 61\right )} x - 4705\right )} x - 148637\right )} x - 1203475\right )} x - 561921\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {839}{124416} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) \]
-1/103680*(2*(12*(18*(8*(30*x - 61)*x - 4705)*x - 148637)*x - 1203475)*x - 561921)*sqrt(3*x^2 + 5*x + 2) - 839/124416*sqrt(3)*log(abs(-2*sqrt(3)*(sq rt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5))
Timed out. \[ \int (5-x) (3+2 x) \left (2+5 x+3 x^2\right )^{3/2} \, dx=-\int \left (2\,x+3\right )\,\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2} \,d x \]