Integrand size = 25, antiderivative size = 81 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {259}{128} \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}-\frac {1}{48} \left (59-18 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {3367}{256} \text {arctanh}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right ) \]
-1/48*(-18*x^2+59)*(x^4+5*x^2+3)^(3/2)-3367/256*arctanh(1/2*(2*x^2+5)/(x^4 +5*x^2+3)^(1/2))+259/128*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.79 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {1}{384} \sqrt {3+5 x^2+x^4} \left (2469-374 x^2+248 x^4+144 x^6\right )+\frac {3367}{256} \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right ) \]
(Sqrt[3 + 5*x^2 + x^4]*(2469 - 374*x^2 + 248*x^4 + 144*x^6))/384 + (3367*L og[-5 - 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/256
Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1578, 1225, 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.
\(\displaystyle \int x^3 \left (3 x^2+2\right ) \sqrt {x^4+5 x^2+3} \, dx\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{2} \int x^2 \left (3 x^2+2\right ) \sqrt {x^4+5 x^2+3}dx^2\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {1}{2} \left (\frac {259}{16} \int \sqrt {x^4+5 x^2+3}dx^2-\frac {1}{24} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{2} \left (\frac {259}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx^2\right )-\frac {1}{24} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}\right )\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {1}{2} \left (\frac {259}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{4} \int \frac {1}{4-x^4}d\frac {2 x^2+5}{\sqrt {x^4+5 x^2+3}}\right )-\frac {1}{24} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {259}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right )-\frac {1}{24} \left (59-18 x^2\right ) \left (x^4+5 x^2+3\right )^{3/2}\right )\) |
(-1/24*((59 - 18*x^2)*(3 + 5*x^2 + x^4)^(3/2)) + (259*(((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
3.2.43.3.1 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]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(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] && Int egerQ[(m - 1)/2]
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65
method | result | size |
risch | \(\frac {\left (144 x^{6}+248 x^{4}-374 x^{2}+2469\right ) \sqrt {x^{4}+5 x^{2}+3}}{384}-\frac {3367 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(53\) |
trager | \(\left (\frac {3}{8} x^{6}+\frac {31}{48} x^{4}-\frac {187}{192} x^{2}+\frac {823}{128}\right ) \sqrt {x^{4}+5 x^{2}+3}-\frac {3367 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(56\) |
pseudoelliptic | \(-\frac {3367 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{256}+\frac {\left (288 x^{6}+496 x^{4}-748 x^{2}+4938\right ) \sqrt {x^{4}+5 x^{2}+3}}{768}\) | \(57\) |
default | \(\frac {3 x^{2} \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{8}-\frac {59 \left (x^{4}+5 x^{2}+3\right )^{\frac {3}{2}}}{48}+\frac {259 \left (2 x^{2}+5\right ) \sqrt {x^{4}+5 x^{2}+3}}{128}-\frac {3367 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}\) | \(74\) |
elliptic | \(\frac {823 \sqrt {x^{4}+5 x^{2}+3}}{128}-\frac {3367 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{256}+\frac {3 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{8}+\frac {31 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{48}-\frac {187 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{192}\) | \(87\) |
1/384*(144*x^6+248*x^4-374*x^2+2469)*(x^4+5*x^2+3)^(1/2)-3367/256*ln(5/2+x ^2+(x^4+5*x^2+3)^(1/2))
Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.69 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {1}{384} \, {\left (144 \, x^{6} + 248 \, x^{4} - 374 \, x^{2} + 2469\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {3367}{256} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
1/384*(144*x^6 + 248*x^4 - 374*x^2 + 2469)*sqrt(x^4 + 5*x^2 + 3) + 3367/25 6*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5)
Time = 0.73 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\left (\frac {x^{4}}{3} + \frac {5 x^{2}}{12} - \frac {17}{8}\right ) \sqrt {x^{4} + 5 x^{2} + 3} + \frac {3 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{6}}{4} + \frac {5 x^{4}}{24} - \frac {89 x^{2}}{96} + \frac {365}{64}\right )}{2} - \frac {3367 \log {\left (2 x^{2} + 2 \sqrt {x^{4} + 5 x^{2} + 3} + 5 \right )}}{256} \]
(x**4/3 + 5*x**2/12 - 17/8)*sqrt(x**4 + 5*x**2 + 3) + 3*sqrt(x**4 + 5*x**2 + 3)*(x**6/4 + 5*x**4/24 - 89*x**2/96 + 365/64)/2 - 3367*log(2*x**2 + 2*s qrt(x**4 + 5*x**2 + 3) + 5)/256
Time = 0.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {3}{8} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} + \frac {259}{64} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {59}{48} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} + \frac {1295}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {3367}{256} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
3/8*(x^4 + 5*x^2 + 3)^(3/2)*x^2 + 259/64*sqrt(x^4 + 5*x^2 + 3)*x^2 - 59/48 *(x^4 + 5*x^2 + 3)^(3/2) + 1295/128*sqrt(x^4 + 5*x^2 + 3) - 3367/256*log(2 *x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.09 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {1}{128} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {1}{24} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} + 5\right )} x^{2} - 51\right )} + \frac {3367}{256} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
1/128*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*x^2 + 5)*x^2 - 89)*x^2 + 1095) + 1/24 *sqrt(x^4 + 5*x^2 + 3)*(2*(4*x^2 + 5)*x^2 - 51) + 3367/256*log(2*x^2 - 2*s qrt(x^4 + 5*x^2 + 3) + 5)
Time = 7.68 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.05 \[ \int x^3 \left (2+3 x^2\right ) \sqrt {3+5 x^2+x^4} \, dx=\frac {3\,x^2\,{\left (x^4+5\,x^2+3\right )}^{3/2}}{8}-\frac {3367\,\ln \left (\sqrt {x^4+5\,x^2+3}+x^2+\frac {5}{2}\right )}{256}-\frac {9\,\left (\frac {x^2}{2}+\frac {5}{4}\right )\,\sqrt {x^4+5\,x^2+3}}{8}-\frac {59\,\sqrt {x^4+5\,x^2+3}\,\left (8\,x^4+10\,x^2-51\right )}{384} \]