Integrand size = 25, antiderivative size = 106 \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=-\frac {4797 \left (5+2 x^2\right ) \sqrt {3+5 x^2+x^4}}{1024}+\frac {123}{128} \left (5+2 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2}-\frac {1}{40} \left (27-10 x^2\right ) \left (3+5 x^2+x^4\right )^{5/2}+\frac {62361 \text {arctanh}\left (\frac {5+2 x^2}{2 \sqrt {3+5 x^2+x^4}}\right )}{2048} \]
123/128*(2*x^2+5)*(x^4+5*x^2+3)^(3/2)-1/40*(-10*x^2+27)*(x^4+5*x^2+3)^(5/2 )+62361/2048*arctanh(1/2*(2*x^2+5)/(x^4+5*x^2+3)^(1/2))-4797/1024*(2*x^2+5 )*(x^4+5*x^2+3)^(1/2)
Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.70 \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {\sqrt {3+5 x^2+x^4} \left (-77229+12390 x^2+5064 x^4+14960 x^6+9344 x^8+1280 x^{10}\right )}{5120}-\frac {62361 \log \left (-5-2 x^2+2 \sqrt {3+5 x^2+x^4}\right )}{2048} \]
(Sqrt[3 + 5*x^2 + x^4]*(-77229 + 12390*x^2 + 5064*x^4 + 14960*x^6 + 9344*x ^8 + 1280*x^10))/5120 - (62361*Log[-5 - 2*x^2 + 2*Sqrt[3 + 5*x^2 + x^4]])/ 2048
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.13, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1578, 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.
\(\displaystyle \int x^3 \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{2} \int x^2 \left (3 x^2+2\right ) \left (x^4+5 x^2+3\right )^{3/2}dx^2\) |
\(\Big \downarrow \) 1225 |
\(\displaystyle \frac {1}{2} \left (\frac {123}{8} \int \left (x^4+5 x^2+3\right )^{3/2}dx^2-\frac {1}{20} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{2} \left (\frac {123}{8} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \int \sqrt {x^4+5 x^2+3}dx^2\right )-\frac {1}{20} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}\right )\) |
\(\Big \downarrow \) 1087 |
\(\displaystyle \frac {1}{2} \left (\frac {123}{8} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \int \frac {1}{\sqrt {x^4+5 x^2+3}}dx^2\right )\right )-\frac {1}{20} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}\right )\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle \frac {1}{2} \left (\frac {123}{8} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{4} \int \frac {1}{4-x^4}d\frac {2 x^2+5}{\sqrt {x^4+5 x^2+3}}\right )\right )-\frac {1}{20} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {123}{8} \left (\frac {1}{8} \left (2 x^2+5\right ) \left (x^4+5 x^2+3\right )^{3/2}-\frac {39}{16} \left (\frac {1}{4} \left (2 x^2+5\right ) \sqrt {x^4+5 x^2+3}-\frac {13}{8} \text {arctanh}\left (\frac {2 x^2+5}{2 \sqrt {x^4+5 x^2+3}}\right )\right )\right )-\frac {1}{20} \left (27-10 x^2\right ) \left (x^4+5 x^2+3\right )^{5/2}\right )\) |
(-1/20*((27 - 10*x^2)*(3 + 5*x^2 + x^4)^(5/2)) + (123*(((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*Ar cTanh[(5 + 2*x^2)/(2*Sqrt[3 + 5*x^2 + x^4])])/8))/16))/8)/2
3.2.57.3.1 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 = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.59
method | result | size |
risch | \(\frac {\left (1280 x^{10}+9344 x^{8}+14960 x^{6}+5064 x^{4}+12390 x^{2}-77229\right ) \sqrt {x^{4}+5 x^{2}+3}}{5120}+\frac {62361 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2048}\) | \(63\) |
trager | \(\left (\frac {1}{4} x^{10}+\frac {73}{40} x^{8}+\frac {187}{64} x^{6}+\frac {633}{640} x^{4}+\frac {1239}{512} x^{2}-\frac {77229}{5120}\right ) \sqrt {x^{4}+5 x^{2}+3}+\frac {62361 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{2048}\) | \(66\) |
pseudoelliptic | \(\frac {62361 \ln \left (2 x^{2}+5+2 \sqrt {x^{4}+5 x^{2}+3}\right )}{2048}+\frac {\left (1280 x^{10}+9344 x^{8}+14960 x^{6}+5064 x^{4}+12390 x^{2}-77229\right ) \sqrt {x^{4}+5 x^{2}+3}}{5120}\) | \(67\) |
default | \(\frac {x^{10} \sqrt {x^{4}+5 x^{2}+3}}{4}+\frac {73 x^{8} \sqrt {x^{4}+5 x^{2}+3}}{40}+\frac {187 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{64}+\frac {633 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{640}+\frac {1239 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{512}-\frac {77229 \sqrt {x^{4}+5 x^{2}+3}}{5120}+\frac {62361 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2048}\) | \(121\) |
elliptic | \(\frac {x^{10} \sqrt {x^{4}+5 x^{2}+3}}{4}+\frac {73 x^{8} \sqrt {x^{4}+5 x^{2}+3}}{40}+\frac {187 x^{6} \sqrt {x^{4}+5 x^{2}+3}}{64}+\frac {633 x^{4} \sqrt {x^{4}+5 x^{2}+3}}{640}+\frac {1239 x^{2} \sqrt {x^{4}+5 x^{2}+3}}{512}-\frac {77229 \sqrt {x^{4}+5 x^{2}+3}}{5120}+\frac {62361 \ln \left (\frac {5}{2}+x^{2}+\sqrt {x^{4}+5 x^{2}+3}\right )}{2048}\) | \(121\) |
1/5120*(1280*x^10+9344*x^8+14960*x^6+5064*x^4+12390*x^2-77229)*(x^4+5*x^2+ 3)^(1/2)+62361/2048*ln(5/2+x^2+(x^4+5*x^2+3)^(1/2))
Time = 0.27 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{5120} \, {\left (1280 \, x^{10} + 9344 \, x^{8} + 14960 \, x^{6} + 5064 \, x^{4} + 12390 \, x^{2} - 77229\right )} \sqrt {x^{4} + 5 \, x^{2} + 3} - \frac {62361}{2048} \, \log \left (-2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} - 5\right ) \]
1/5120*(1280*x^10 + 9344*x^8 + 14960*x^6 + 5064*x^4 + 12390*x^2 - 77229)*s qrt(x^4 + 5*x^2 + 3) - 62361/2048*log(-2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) - 5 )
Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (94) = 188\).
Time = 1.16 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.79 \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=3 \left (\frac {x^{4}}{3} + \frac {5 x^{2}}{12} - \frac {17}{8}\right ) \sqrt {x^{4} + 5 x^{2} + 3} + \frac {19 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{6}}{4} + \frac {5 x^{4}}{24} - \frac {89 x^{2}}{96} + \frac {365}{64}\right )}{2} + \frac {17 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{8}}{5} + \frac {x^{6}}{8} - \frac {127 x^{4}}{240} + \frac {527 x^{2}}{192} - \frac {11143}{640}\right )}{2} + \frac {3 \sqrt {x^{4} + 5 x^{2} + 3} \left (\frac {x^{10}}{6} + \frac {x^{8}}{12} - \frac {11 x^{6}}{32} + \frac {107 x^{4}}{64} - \frac {2279 x^{2}}{256} + \frac {29049}{512}\right )}{2} + \frac {62361 \log {\left (2 x^{2} + 2 \sqrt {x^{4} + 5 x^{2} + 3} + 5 \right )}}{2048} \]
3*(x**4/3 + 5*x**2/12 - 17/8)*sqrt(x**4 + 5*x**2 + 3) + 19*sqrt(x**4 + 5*x **2 + 3)*(x**6/4 + 5*x**4/24 - 89*x**2/96 + 365/64)/2 + 17*sqrt(x**4 + 5*x **2 + 3)*(x**8/5 + x**6/8 - 127*x**4/240 + 527*x**2/192 - 11143/640)/2 + 3 *sqrt(x**4 + 5*x**2 + 3)*(x**10/6 + x**8/12 - 11*x**6/32 + 107*x**4/64 - 2 279*x**2/256 + 29049/512)/2 + 62361*log(2*x**2 + 2*sqrt(x**4 + 5*x**2 + 3) + 5)/2048
Time = 0.19 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.11 \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{4} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} x^{2} + \frac {123}{64} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} x^{2} - \frac {27}{40} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {5}{2}} - \frac {4797}{512} \, \sqrt {x^{4} + 5 \, x^{2} + 3} x^{2} + \frac {615}{128} \, {\left (x^{4} + 5 \, x^{2} + 3\right )}^{\frac {3}{2}} - \frac {23985}{1024} \, \sqrt {x^{4} + 5 \, x^{2} + 3} + \frac {62361}{2048} \, \log \left (2 \, x^{2} + 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
1/4*(x^4 + 5*x^2 + 3)^(5/2)*x^2 + 123/64*(x^4 + 5*x^2 + 3)^(3/2)*x^2 - 27/ 40*(x^4 + 5*x^2 + 3)^(5/2) - 4797/512*sqrt(x^4 + 5*x^2 + 3)*x^2 + 615/128* (x^4 + 5*x^2 + 3)^(3/2) - 23985/1024*sqrt(x^4 + 5*x^2 + 3) + 62361/2048*lo g(2*x^2 + 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Leaf count of result is larger than twice the leaf count of optimal. 179 vs. \(2 (88) = 176\).
Time = 0.32 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.69 \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\frac {1}{1024} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, x^{2} + 1\right )} x^{2} - 33\right )} x^{2} + 321\right )} x^{2} - 6837\right )} x^{2} + 87147\right )} + \frac {17}{3840} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, x^{2} + 5\right )} x^{2} - 127\right )} x^{2} + 2635\right )} x^{2} - 33429\right )} + \frac {19}{384} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, {\left (6 \, x^{2} + 5\right )} x^{2} - 89\right )} x^{2} + 1095\right )} + \frac {1}{8} \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (2 \, {\left (4 \, x^{2} + 5\right )} x^{2} - 51\right )} - \frac {62361}{2048} \, \log \left (2 \, x^{2} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} + 5\right ) \]
1/1024*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 + 87147) + 17/3840*sqrt(x^4 + 5*x^2 + 3)*(2*(4*(6*(8*x^2 + 5)*x^2 - 127)*x^2 + 2635)*x^2 - 33429) + 19/384*sqrt(x^4 + 5*x^2 + 3)*(2* (4*(6*x^2 + 5)*x^2 - 89)*x^2 + 1095) + 1/8*sqrt(x^4 + 5*x^2 + 3)*(2*(4*x^2 + 5)*x^2 - 51) - 62361/2048*log(2*x^2 - 2*sqrt(x^4 + 5*x^2 + 3) + 5)
Timed out. \[ \int x^3 \left (2+3 x^2\right ) \left (3+5 x^2+x^4\right )^{3/2} \, dx=\int x^3\,\left (3\,x^2+2\right )\,{\left (x^4+5\,x^2+3\right )}^{3/2} \,d x \]