Integrand size = 22, antiderivative size = 72 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=\frac {137}{12} x \sqrt {2+3 x^2}+\frac {137}{36} x \left (2+3 x^2\right )^{3/2}+\frac {1}{45} (21-5 x) \left (2+3 x^2\right )^{5/2}+\frac {137 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{6 \sqrt {3}} \]
137/36*x*(3*x^2+2)^(3/2)+1/45*(21-5*x)*(3*x^2+2)^(5/2)+137/18*arcsinh(1/2* x*6^(1/2))*3^(1/2)+137/12*x*(3*x^2+2)^(1/2)
Time = 0.14 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.99 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{60} \sqrt {2+3 x^2} \left (-112-1115 x-336 x^2-605 x^3-252 x^4+60 x^5\right )-\frac {137 \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )}{6 \sqrt {3}} \]
-1/60*(Sqrt[2 + 3*x^2]*(-112 - 1115*x - 336*x^2 - 605*x^3 - 252*x^4 + 60*x ^5)) - (137*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/(6*Sqrt[3])
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {676, 211, 211, 222}
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.
\(\displaystyle \int (5-x) (2 x+3) \left (3 x^2+2\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 676 |
\(\displaystyle \frac {137}{9} \int \left (3 x^2+2\right )^{3/2}dx-\frac {1}{9} x \left (3 x^2+2\right )^{5/2}+\frac {7}{15} \left (3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {137}{9} \left (\frac {3}{2} \int \sqrt {3 x^2+2}dx+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )-\frac {1}{9} x \left (3 x^2+2\right )^{5/2}+\frac {7}{15} \left (3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 211 |
\(\displaystyle \frac {137}{9} \left (\frac {3}{2} \left (\int \frac {1}{\sqrt {3 x^2+2}}dx+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )-\frac {1}{9} x \left (3 x^2+2\right )^{5/2}+\frac {7}{15} \left (3 x^2+2\right )^{5/2}\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {137}{9} \left (\frac {3}{2} \left (\frac {\text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{\sqrt {3}}+\frac {1}{2} \sqrt {3 x^2+2} x\right )+\frac {1}{4} x \left (3 x^2+2\right )^{3/2}\right )-\frac {1}{9} x \left (3 x^2+2\right )^{5/2}+\frac {7}{15} \left (3 x^2+2\right )^{5/2}\) |
(7*(2 + 3*x^2)^(5/2))/15 - (x*(2 + 3*x^2)^(5/2))/9 + (137*((x*(2 + 3*x^2)^ (3/2))/4 + (3*((x*Sqrt[2 + 3*x^2])/2 + ArcSinh[Sqrt[3/2]*x]/Sqrt[3]))/2))/ 9
3.14.71.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(e*f + d*g)*((a + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + (Sim p[e*g*x*((a + c*x^2)^(p + 1)/(c*(2*p + 3))), x] - Simp[(a*e*g - c*d*f*(2*p + 3))/(c*(2*p + 3)) Int[(a + c*x^2)^p, x], x]) /; FreeQ[{a, c, d, e, f, g , p}, x] && !LeQ[p, -1]
Time = 0.27 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69
method | result | size |
risch | \(-\frac {\left (60 x^{5}-252 x^{4}-605 x^{3}-336 x^{2}-1115 x -112\right ) \sqrt {3 x^{2}+2}}{60}+\frac {137 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{18}\) | \(50\) |
default | \(\frac {137 x \left (3 x^{2}+2\right )^{\frac {3}{2}}}{36}+\frac {137 x \sqrt {3 x^{2}+2}}{12}+\frac {137 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{18}+\frac {7 \left (3 x^{2}+2\right )^{\frac {5}{2}}}{15}-\frac {x \left (3 x^{2}+2\right )^{\frac {5}{2}}}{9}\) | \(61\) |
trager | \(\left (-x^{5}+\frac {21}{5} x^{4}+\frac {121}{12} x^{3}+\frac {28}{5} x^{2}+\frac {223}{12} x +\frac {28}{15}\right ) \sqrt {3 x^{2}+2}-\frac {137 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{18}\) | \(67\) |
meijerg | \(\frac {15 \sqrt {3}\, \left (\frac {4 \sqrt {\pi }\, x \sqrt {3}\, \sqrt {2}\, \left (\frac {3 x^{2}}{8}+\frac {5}{8}\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{3}+\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )\right )}{2 \sqrt {\pi }}+\frac {7 \sqrt {2}\, \left (-\frac {8 \sqrt {\pi }}{15}+\frac {4 \sqrt {\pi }\, \left (\frac {9}{2} x^{4}+6 x^{2}+2\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{15}\right )}{2 \sqrt {\pi }}-\frac {2 \sqrt {3}\, \left (\frac {\sqrt {6}\, \sqrt {\pi }\, x \left (18 x^{4}+21 x^{2}+3\right ) \sqrt {\frac {3 x^{2}}{2}+1}}{36}-\frac {\sqrt {\pi }\, \operatorname {arcsinh}\left (\frac {x \sqrt {3}\, \sqrt {2}}{2}\right )}{6}\right )}{3 \sqrt {\pi }}\) | \(147\) |
-1/60*(60*x^5-252*x^4-605*x^3-336*x^2-1115*x-112)*(3*x^2+2)^(1/2)+137/18*a rcsinh(1/2*x*6^(1/2))*3^(1/2)
Time = 0.30 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.90 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{60} \, {\left (60 \, x^{5} - 252 \, x^{4} - 605 \, x^{3} - 336 \, x^{2} - 1115 \, x - 112\right )} \sqrt {3 \, x^{2} + 2} + \frac {137}{36} \, \sqrt {3} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) \]
-1/60*(60*x^5 - 252*x^4 - 605*x^3 - 336*x^2 - 1115*x - 112)*sqrt(3*x^2 + 2 ) + 137/36*sqrt(3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1)
Time = 0.62 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.53 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=- x^{5} \sqrt {3 x^{2} + 2} + \frac {21 x^{4} \sqrt {3 x^{2} + 2}}{5} + \frac {121 x^{3} \sqrt {3 x^{2} + 2}}{12} + \frac {28 x^{2} \sqrt {3 x^{2} + 2}}{5} + \frac {223 x \sqrt {3 x^{2} + 2}}{12} + \frac {28 \sqrt {3 x^{2} + 2}}{15} + \frac {137 \sqrt {3} \operatorname {asinh}{\left (\frac {\sqrt {6} x}{2} \right )}}{18} \]
-x**5*sqrt(3*x**2 + 2) + 21*x**4*sqrt(3*x**2 + 2)/5 + 121*x**3*sqrt(3*x**2 + 2)/12 + 28*x**2*sqrt(3*x**2 + 2)/5 + 223*x*sqrt(3*x**2 + 2)/12 + 28*sqr t(3*x**2 + 2)/15 + 137*sqrt(3)*asinh(sqrt(6)*x/2)/18
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.83 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{9} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} x + \frac {7}{15} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {137}{36} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {137}{12} \, \sqrt {3 \, x^{2} + 2} x + \frac {137}{18} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) \]
-1/9*(3*x^2 + 2)^(5/2)*x + 7/15*(3*x^2 + 2)^(5/2) + 137/36*(3*x^2 + 2)^(3/ 2)*x + 137/12*sqrt(3*x^2 + 2)*x + 137/18*sqrt(3)*arcsinh(1/2*sqrt(6)*x)
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=-\frac {1}{60} \, {\left ({\left ({\left ({\left (12 \, {\left (5 \, x - 21\right )} x - 605\right )} x - 336\right )} x - 1115\right )} x - 112\right )} \sqrt {3 \, x^{2} + 2} - \frac {137}{18} \, \sqrt {3} \log \left (-\sqrt {3} x + \sqrt {3 \, x^{2} + 2}\right ) \]
-1/60*((((12*(5*x - 21)*x - 605)*x - 336)*x - 1115)*x - 112)*sqrt(3*x^2 + 2) - 137/18*sqrt(3)*log(-sqrt(3)*x + sqrt(3*x^2 + 2))
Time = 10.58 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.69 \[ \int (5-x) (3+2 x) \left (2+3 x^2\right )^{3/2} \, dx=\frac {137\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {6}\,x}{2}\right )}{18}+\frac {\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}\,\left (-3\,x^5+\frac {63\,x^4}{5}+\frac {121\,x^3}{4}+\frac {84\,x^2}{5}+\frac {223\,x}{4}+\frac {28}{5}\right )}{3} \]