Integrand size = 25, antiderivative size = 127 \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {28379 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{2048}-\frac {2183}{768} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}+\frac {3}{14} x^4 \left (3+5 x^2+x^4\right )^{5/2}+\frac {\left (3313-1070 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}}{1680}-\frac {368927 \text {arctanh}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )}{4096} \]
-2183/768*(2*x^2+5)*(x^4+5*x^2+3)^(3/2)+3/14*x^4*(x^4+5*x^2+3)^(5/2)+1/168 0*(-1070*x^2+3313)*(x^4+5*x^2+3)^(5/2)-368927/4096*arctanh(1/2*(2*x^2+5)/( x^4+5*x^2+3)^(1/2))+28379/2048*(2*x^2+5)*(x^4+5*x^2+3)^(1/2)
Time = 0.18 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.62 \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {\sqrt {3+5 x^2+x^4} \left (9546951-1499570 x^2+283304 x^4+154800 x^6+482944 x^8+323840 x^{10}+46080 x^{12}\right )}{215040}+\frac {368927 \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right )}{4096} \]
(Sqrt[3 + 5*x^2 + x^4]*(9546951 - 1499570*x^2 + 283304*x^4 + 154800*x^6 + 482944*x^8 + 323840*x^10 + 46080*x^12))/215040 + (368927*Log[-5 - 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/4096
Time = 0.29 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {1578, 1236, 27, 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.
| \(\displaystyle \int x^5 \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {1}{2} \int x^4 \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}dx^2\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \int -\frac {1}{2} x^2 \left (107 x^2+36\right ) \left (x^4+5 x^2+3\right )^{3/2}dx^2+\frac {3}{7} \left (x^4+5 x^2+3\right )^{5/2} x^4\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{7} x^4 \left (x^4+5 x^2+3\right )^{5/2}-\frac {1}{14} \int x^2 \left (107 x^2+36\right ) \left (x^4+5 x^2+3\right )^{3/2}dx^2\right )\) |
| \(\Big \downarrow \) 1225 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{60} \left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}-\frac {15281}{24} \int \left (x^4+5 x^2+3\right )^{3/2}dx^2\right )+\frac {3}{7} \left (x^4+5 x^2+3\right )^{5/2} x^4\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{60} \left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}-\frac {15281}{24} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \int \sqrt {x^4+5 x^2+3}dx^2\right )\right )+\frac {3}{7} \left (x^4+5 x^2+3\right )^{5/2} x^4\right )\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{60} \left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}-\frac {15281}{24} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx^2\right )\right )\right )+\frac {3}{7} \left (x^4+5 x^2+3\right )^{5/2} x^4\right )\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{60} \left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}-\frac {15281}{24} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{4} \int \frac {1}{4-x^4}d\frac {2 x^2+5}{\sqrt {x^4+5 x^2+3}}\right )\right )\right )+\frac {3}{7} \left (x^4+5 x^2+3\right )^{5/2} x^4\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{14} \left (\frac {1}{60} \left (3313-1070 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}-\frac {15281}{24} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right )\right )\right )+\frac {3}{7} \left (x^4+5 x^2+3\right )^{5/2} x^4\right )\) |
((3*x^4*(3 + 5*x^2 + x^4)^(5/2))/7 + (((3313 - 1070*x^2)*(3 + 5*x^2 + x^4) ^(5/2))/60 - (15281*(((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))/24)/14)/2
3.2.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
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 = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(\frac {\left (46080 x^{12}+323840 x^{10}+482944 x^{8}+154800 x^{6}+283304 x^{4}-1499570 x^{2}+9546951\right ) \sqrt {x^{4}+5 x^{2}+3}}{215040}-\frac {368927 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4096}\) | \(68\) |
| trager | \(\left (\frac {3}{14} x^{12}+\frac {253}{168} x^{10}+\frac {539}{240} x^{8}+\frac {645}{896} x^{6}+\frac {5059}{3840} x^{4}-\frac {149957}{21504} x^{2}+\frac {3182317}{71680}\right ) \sqrt {x^{4}+5 x^{2}+3}-\frac {368927 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{4096}\) | \(71\) |
| pseudoelliptic | \(-\frac {368927 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{4096}+\frac {\left (46080 x^{12}+323840 x^{10}+482944 x^{8}+154800 x^{6}+283304 x^{4}-1499570 x^{2}+9546951\right ) \sqrt {x^{4}+5 x^{2}+3}}{215040}\) | \(72\) |
| default | \(\frac {3 x^{12} \sqrt {x^{4}+5 x^{2}+3}}{14}+\frac {253 x^{10} \sqrt {x^{4}+5 x^{2}+3}}{168}+\frac {539 x^{8} \sqrt {x^{4}+5 x^{2}+3}}{240}+\frac {645 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{896}+\frac {5059 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{3840}-\frac {149957 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{21504}+\frac {3182317 \sqrt {x^{4}+5 x^{2}+3}}{71680}-\frac {368927 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4096}\) | \(138\) |
| elliptic | \(\frac {3 x^{12} \sqrt {x^{4}+5 x^{2}+3}}{14}+\frac {253 x^{10} \sqrt {x^{4}+5 x^{2}+3}}{168}+\frac {539 x^{8} \sqrt {x^{4}+5 x^{2}+3}}{240}+\frac {645 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{896}+\frac {5059 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{3840}-\frac {149957 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{21504}+\frac {3182317 \sqrt {x^{4}+5 x^{2}+3}}{71680}-\frac {368927 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{4096}\) | \(138\) |
1/215040*(46080*x^12+323840*x^10+482944*x^8+154800*x^6+283304*x^4-1499570* x^2+9546951)*(x^4+5*x^2+3)^(1/2)-368927/4096*ln(5/2+x^2+(x^4+5*x^2+3)^(1/2 ))
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{215040} \, {\left (46080 \, x^{12} + 323840 \, x^{10} + 482944 \, x^{8} + 154800 \, x^{6} + 283304 \, x^{4} - 1499570 \, x^{2} + 9546951\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {368927}{4096} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
1/215040*(46080*x^12 + 323840*x^10 + 482944*x^8 + 154800*x^6 + 283304*x^4 - 1499570*x^2 + 9546951)*sqrt(x^4 + 5*x^2 + 3) + 368927/4096*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5)
Time = 1.21 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.72 \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=3 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{6}}{4} + \frac {5 x^{4}}{24} - \frac {89 x^{2}}{96} + \frac {365}{64}\right ) + \frac {19 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{8}}{5} + \frac {x^{6}}{8} - \frac {127 x^{4}}{240} + \frac {527 x^{2}}{192} - \frac {11143}{640}\right )}{2} + \frac {17 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{10}}{6} + \frac {x^{8}}{12} - \frac {11 x^{6}}{32} + \frac {107 x^{4}}{64} - \frac {2279 x^{2}}{256} + \frac {29049}{512}\right )}{2} + \frac {3 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{12}}{7} + \frac {5 x^{10}}{84} - \frac {29 x^{8}}{120} + \frac {509 x^{6}}{448} - \frac {3623 x^{4}}{640} + \frac {108481 x^{2}}{3584} - \frac {6918747}{35840}\right )}{2} - \frac {368927 \log {\left (2 x^{2} + 2 \sqrt {x^{4} + 5 x^{2} + 3} + 5 \right )}}{4096} \]
3*sqrt(x**4 + 5*x**2 + 3)*(x**6/4 + 5*x**4/24 - 89*x**2/96 + 365/64) + 19* sqrt(x**4 + 5*x**2 + 3)*(x**8/5 + x**6/8 - 127*x**4/240 + 527*x**2/192 - 1 1143/640)/2 + 17*sqrt(x**4 + 5*x**2 + 3)*(x**10/6 + x**8/12 - 11*x**6/32 + 107*x**4/64 - 2279*x**2/256 + 29049/512)/2 + 3*sqrt(x**4 + 5*x**2 + 3)*(x **12/7 + 5*x**10/84 - 29*x**8/120 + 509*x**6/448 - 3623*x**4/640 + 108481* x**2/3584 - 6918747/35840)/2 - 368927*log(2*x**2 + 2*sqrt(x**4 + 5*x**2 + 3) + 5)/4096
Time = 0.19 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.06 \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {3}{14} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} x^{4} - \frac {107}{168} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} x^{2} - \frac {2183}{384} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} + \frac {3313}{1680} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} + \frac {28379}{1024} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} - \frac {10915}{768} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} + \frac {141895}{2048} \, \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {368927}{4096} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
3/14*(x^4 + 5*x^2 + 3)^(5/2)*x^4 - 107/168*(x^4 + 5*x^2 + 3)^(5/2)*x^2 - 2 183/384*(x^4 + 5*x^2 + 3)^(3/2)*x^2 + 3313/1680*(x^4 + 5*x^2 + 3)^(5/2) + 28379/1024*sqrt(x^4 + 5*x^2 + 3)*x^2 - 10915/768*(x^4 + 5*x^2 + 3)^(3/2) + 141895/2048*sqrt(x^4 + 5*x^2 + 3) - 368927/4096*log(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Time = 0.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.63 \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{71680} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, {\left (12 \, x^{2} + 5\right )} x^{2} - 203\right )} x^{2} + 7635\right )} x^{2} - 76083\right )} x^{2} + 1627215\right )} x^{2} - 20756241\right )} + \frac {17}{3072} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, x^{2} + 1\right )} x^{2} - 33\right )} x^{2} + 321\right )} x^{2} - 6837\right )} x^{2} + 87147\right )} + \frac {19}{3840} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + 5\right )} x^{2} - 127\right )} x^{2} + 2635\right )} x^{2} - 33429\right )} + \frac {1}{64} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {368927}{4096} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
1/71680*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(2*(8*(10*(12*x^2 + 5)*x^2 - 203)*x^2 + 7635)*x^2 - 76083)*x^2 + 1627215)*x^2 - 20756241) + 17/3072*sqrt(x^4 + 5 *x^2 + 3)*(2*(4*(2*(8*(2*x^2 + 1)*x^2 - 33)*x^2 + 321)*x^2 - 6837)*x^2 + 8 7147) + 19/3840*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*(8*x^2 + 5)*x^2 - 127)*x^2 + 2635)*x^2 - 33429) + 1/64*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*x^2 + 5)*x^2 - 89)*x^2 + 1095) + 368927/4096*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Timed out. \[ \int x^5 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^5\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]