Integrand size = 23, antiderivative size = 99 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {429}{256} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {11}{32} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{10} \left (3+5 x^2+x^4\right )^{5/2}-\frac {5577}{512} \text {arctanh}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right ) \]
-11/32*(2*x^2+5)*(x^4+5*x^2+3)^(3/2)+3/10*(x^4+5*x^2+3)^(5/2)-5577/512*arc tanh(1/2*(2*x^2+5)/(x^4+5*x^2+3)^(1/2))+429/256*(2*x^2+5)*(x^4+5*x^2+3)^(1 /2)
Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.70 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {\sqrt {3+5 x^2+x^4} \left (7581+2170 x^2+5304 x^4+2960 x^6+384 x^8\right )}{1280}+\frac {5577}{512} \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right ) \]
(Sqrt[3 + 5*x^2 + x^4]*(7581 + 2170*x^2 + 5304*x^4 + 2960*x^6 + 384*x^8))/ 1280 + (5577*Log[-5 - 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/512
Time = 0.25 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {1576, 1160, 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.
\(\displaystyle \int x \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1576 |
\(\displaystyle \frac {1}{2} \int \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}dx^2\) |
\(\Big \downarrow \) 1160 |
\(\displaystyle \frac {1}{2} \left (\frac {3}{5} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{2} \int \left (x^4+5 x^2+3\right )^{3/2}dx^2\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{2} \left (\frac {3}{5} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{2} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \int \sqrt {x^4+5 x^2+3}dx^2\right )\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{2} \left (\frac {3}{5} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{2} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx^2\right )\right )\right )\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {1}{2} \left (\frac {3}{5} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{2} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{4} \int \frac {1}{4-x^4}d\frac {2 x^2+5}{\sqrt {x^4+5 x^2+3}}\right )\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {3}{5} \left (x^4+5 x^2+3\right )^{5/2}-\frac {11}{2} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right )\right )\right )\) |
((3*(3 + 5*x^2 + x^4)^(5/2))/5 - (11*(((5 + 2*x^2)*(3 + 5*x^2 + x^4)^(3/2) )/8 - (39*(((5 + 2*x^2)*Sqrt[3 + 5*x^2 + x^4])/4 - (13*ArcTanh[(5 + 2*x^2) /(2*Sqrt[3 + 5*x^2 + x^4])])/8))/16))/2)/2
3.2.58.3.1 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]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Time = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {\left (384 x^{8}+2960 x^{6}+5304 x^{4}+2170 x^{2}+7581\right ) \sqrt {x^{4}+5 x^{2}+3}}{1280}-\frac {5577 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{512}\) | \(58\) |
trager | \(\left (\frac {3}{10} x^{8}+\frac {37}{16} x^{6}+\frac {663}{160} x^{4}+\frac {217}{128} x^{2}+\frac {7581}{1280}\right ) \sqrt {x^{4}+5 x^{2}+3}-\frac {5577 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{512}\) | \(61\) |
pseudoelliptic | \(-\frac {5577 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{512}+\frac {\left (384 x^{8}+2960 x^{6}+5304 x^{4}+2170 x^{2}+7581\right ) \sqrt {x^{4}+5 x^{2}+3}}{1280}\) | \(62\) |
default | \(-\frac {5577 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{512}+\frac {37 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{16}+\frac {663 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{160}+\frac {217 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{128}+\frac {7581 \sqrt {x^{4}+5 x^{2}+3}}{1280}+\frac {3 x^{8} \sqrt {x^{4}+5 x^{2}+3}}{10}\) | \(104\) |
elliptic | \(-\frac {5577 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{512}+\frac {37 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{16}+\frac {663 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{160}+\frac {217 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{128}+\frac {7581 \sqrt {x^{4}+5 x^{2}+3}}{1280}+\frac {3 x^{8} \sqrt {x^{4}+5 x^{2}+3}}{10}\) | \(104\) |
1/1280*(384*x^8+2960*x^6+5304*x^4+2170*x^2+7581)*(x^4+5*x^2+3)^(1/2)-5577/ 512*ln(5/2+x^2+(x^4+5*x^2+3)^(1/2))
Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.62 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{1280} \, {\left (384 \, x^{8} + 2960 \, x^{6} + 5304 \, x^{4} + 2170 \, x^{2} + 7581\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {5577}{512} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
1/1280*(384*x^8 + 2960*x^6 + 5304*x^4 + 2170*x^2 + 7581)*sqrt(x^4 + 5*x^2 + 3) + 5577/512*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5)
Time = 1.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.67 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=3 \left (\frac {x^{2}}{2} + \frac {5}{4}\right ) \sqrt {x^{4} + 5 x^{2} + 3} + \frac {19 \left (\frac {x^{4}}{3} + \frac {5 x^{2}}{12} - \frac {17}{8}\right ) \sqrt {x^{4} + 5 x^{2} + 3}}{2} + \frac {17 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{6}}{4} + \frac {5 x^{4}}{24} - \frac {89 x^{2}}{96} + \frac {365}{64}\right )}{2} + \frac {3 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{8}}{5} + \frac {x^{6}}{8} - \frac {127 x^{4}}{240} + \frac {527 x^{2}}{192} - \frac {11143}{640}\right )}{2} - \frac {5577 \log {\left (2 x^{2} + 2 \sqrt {x^{4} + 5 x^{2} + 3} + 5 \right )}}{512} \]
3*(x**2/2 + 5/4)*sqrt(x**4 + 5*x**2 + 3) + 19*(x**4/3 + 5*x**2/12 - 17/8)* sqrt(x**4 + 5*x**2 + 3)/2 + 17*sqrt(x**4 + 5*x**2 + 3)*(x**6/4 + 5*x**4/24 - 89*x**2/96 + 365/64)/2 + 3*sqrt(x**4 + 5*x**2 + 3)*(x**8/5 + x**6/8 - 1 27*x**4/240 + 527*x**2/192 - 11143/640)/2 - 5577*log(2*x**2 + 2*sqrt(x**4 + 5*x**2 + 3) + 5)/512
Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.02 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=-\frac {11}{16} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} + \frac {3}{10} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} + \frac {429}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {55}{32} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} + \frac {2145}{256} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {5577}{512} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
-11/16*(x^4 + 5*x^2 + 3)^(3/2)*x^2 + 3/10*(x^4 + 5*x^2 + 3)^(5/2) + 429/12 8*sqrt(x^4 + 5*x^2 + 3)*x^2 - 55/32*(x^4 + 5*x^2 + 3)^(3/2) + 2145/256*sqr t(x^4 + 5*x^2 + 3) - 5577/512*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Time = 0.29 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.53 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{1280} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + 5\right )} x^{2} - 127\right )} x^{2} + 2635\right )} x^{2} - 33429\right )} + \frac {17}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {19}{48} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} + 5\right )} x^{2} - 51\right )} + \frac {3}{4} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, x^{2} + 5\right )} + \frac {5577}{512} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
1/1280*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*(8*x^2 + 5)*x^2 - 127)*x^2 + 2635)*x ^2 - 33429) + 17/384*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*x^2 + 5)*x^2 - 89)*x^2 + 1095) + 19/48*sqrt(x^4 + 5*x^2 + 3)*(2*(4*x^2 + 5)*x^2 - 51) + 3/4*sqrt (x^4 + 5*x^2 + 3)*(2*x^2 + 5) + 5577/512*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Time = 7.79 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.28 \[ \int x \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {\left (x^2+\frac {5}{2}\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{4}-\frac {15\,x^2\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{16}-\frac {5577\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{512}+\frac {585\,\left (2\,x^2+5\right )\,\sqrt {x^4+5\,x^2+3}}{256}-\frac {39\,\left (\frac {x^2}{2}+\frac {5}{4}\right )\,\sqrt {x^4+5\,x^2+3}}{16}-\frac {75\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{32}+\frac {3\,{\left (x^4+5\,x^2+3\right )}^{5/2}}{10} \]
((x^2 + 5/2)*(5*x^2 + x^4 + 3)^(3/2))/4 - (15*x^2*(5*x^2 + x^4 + 3)^(3/2)) /16 - (5577*log((5*x^2 + x^4 + 3)^(1/2) + x^2 + 5/2))/512 + (585*(2*x^2 + 5)*(5*x^2 + x^4 + 3)^(1/2))/256 - (39*(x^2/2 + 5/4)*(5*x^2 + x^4 + 3)^(1/2 ))/16 - (75*(5*x^2 + x^4 + 3)^(3/2))/32 + (3*(5*x^2 + x^4 + 3)^(5/2))/10