Integrand size = 25, antiderivative size = 66 \[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {-7-8 x^2}{39 \sqrt {3+5 x^2+x^4}}-\frac {\text {arctanh}\left (\frac {6+5 x^2}{2 \sqrt {3} \sqrt {3+5 x^2+x^4}}\right )}{3 \sqrt {3}} \]
-1/9*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4+5*x^2+3)^(1/2))*3^(1/2)+1/39*(-8*x ^2-7)/(x^4+5*x^2+3)^(1/2)
Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.94 \[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\frac {-7-8 x^2}{39 \sqrt {3+5 x^2+x^4}}+\frac {2 \text {arctanh}\left (\frac {x^2-\sqrt {3+5 x^2+x^4}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
(-7 - 8*x^2)/(39*Sqrt[3 + 5*x^2 + x^4]) + (2*ArcTanh[(x^2 - Sqrt[3 + 5*x^2 + x^4])/Sqrt[3]])/(3*Sqrt[3])
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1578, 1235, 27, 1154, 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.
\(\displaystyle \int \frac {3 x^2+2}{x \left (x^4+5 x^2+3\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1578 |
\(\displaystyle \frac {1}{2} \int \frac {3 x^2+2}{x^2 \left (x^4+5 x^2+3\right )^{3/2}}dx^2\) |
\(\Big \downarrow \) 1235 |
\(\displaystyle \frac {1}{2} \left (-\frac {2}{39} \int -\frac {13}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {2 \left (8 x^2+7\right )}{39 \sqrt {x^4+5 x^2+3}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \int \frac {1}{x^2 \sqrt {x^4+5 x^2+3}}dx^2-\frac {2 \left (8 x^2+7\right )}{39 \sqrt {x^4+5 x^2+3}}\right )\) |
\(\Big \downarrow \) 1154 |
\(\displaystyle \frac {1}{2} \left (-\frac {4}{3} \int \frac {1}{12-x^4}d\frac {5 x^2+6}{\sqrt {x^4+5 x^2+3}}-\frac {2 \left (8 x^2+7\right )}{39 \sqrt {x^4+5 x^2+3}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 \text {arctanh}\left (\frac {5 x^2+6}{2 \sqrt {3} \sqrt {x^4+5 x^2+3}}\right )}{3 \sqrt {3}}-\frac {2 \left (8 x^2+7\right )}{39 \sqrt {x^4+5 x^2+3}}\right )\) |
((-2*(7 + 8*x^2))/(39*Sqrt[3 + 5*x^2 + x^4]) - (2*ArcTanh[(6 + 5*x^2)/(2*S qrt[3]*Sqrt[3 + 5*x^2 + x^4])])/(3*Sqrt[3]))/2
3.2.97.3.1 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[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
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 = 0.34 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80
method | result | size |
risch | \(-\frac {8 x^{2}+7}{39 \sqrt {x^{4}+5 x^{2}+3}}-\frac {\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{9}\) | \(53\) |
pseudoelliptic | \(\frac {-13 \,\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}\, \sqrt {x^{4}+5 x^{2}+3}-24 x^{2}-21}{117 \sqrt {x^{4}+5 x^{2}+3}}\) | \(64\) |
default | \(-\frac {4 \left (2 x^{2}+5\right )}{39 \sqrt {x^{4}+5 x^{2}+3}}+\frac {1}{3 \sqrt {x^{4}+5 x^{2}+3}}-\frac {\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{9}\) | \(67\) |
elliptic | \(-\frac {4 \left (2 x^{2}+5\right )}{39 \sqrt {x^{4}+5 x^{2}+3}}+\frac {1}{3 \sqrt {x^{4}+5 x^{2}+3}}-\frac {\operatorname {arctanh}\left (\frac {\left (5 x^{2}+6\right ) \sqrt {3}}{6 \sqrt {x^{4}+5 x^{2}+3}}\right ) \sqrt {3}}{9}\) | \(67\) |
trager | \(-\frac {8 x^{2}+7}{39 \sqrt {x^{4}+5 x^{2}+3}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+6 \sqrt {x^{4}+5 x^{2}+3}+6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )}{x^{2}}\right )}{9}\) | \(71\) |
-1/39*(8*x^2+7)/(x^4+5*x^2+3)^(1/2)-1/9*arctanh(1/6*(5*x^2+6)*3^(1/2)/(x^4 +5*x^2+3)^(1/2))*3^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (52) = 104\).
Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.62 \[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {24 \, x^{4} - 13 \, \sqrt {3} {\left (x^{4} + 5 \, x^{2} + 3\right )} \log \left (\frac {25 \, x^{2} - 2 \, \sqrt {3} {\left (5 \, x^{2} + 6\right )} - 2 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (5 \, \sqrt {3} - 6\right )} + 30}{x^{2}}\right ) + 120 \, x^{2} + 3 \, \sqrt {x^{4} + 5 \, x^{2} + 3} {\left (8 \, x^{2} + 7\right )} + 72}{117 \, {\left (x^{4} + 5 \, x^{2} + 3\right )}} \]
-1/117*(24*x^4 - 13*sqrt(3)*(x^4 + 5*x^2 + 3)*log((25*x^2 - 2*sqrt(3)*(5*x ^2 + 6) - 2*sqrt(x^4 + 5*x^2 + 3)*(5*sqrt(3) - 6) + 30)/x^2) + 120*x^2 + 3 *sqrt(x^4 + 5*x^2 + 3)*(8*x^2 + 7) + 72)/(x^4 + 5*x^2 + 3)
\[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3 x^{2} + 2}{x \left (x^{4} + 5 x^{2} + 3\right )^{\frac {3}{2}}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.98 \[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {8 \, x^{2}}{39 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} - \frac {1}{9} \, \sqrt {3} \log \left (\frac {2 \, \sqrt {3} \sqrt {x^{4} + 5 \, x^{2} + 3}}{x^{2}} + \frac {6}{x^{2}} + 5\right ) - \frac {7}{39 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} \]
-8/39*x^2/sqrt(x^4 + 5*x^2 + 3) - 1/9*sqrt(3)*log(2*sqrt(3)*sqrt(x^4 + 5*x ^2 + 3)/x^2 + 6/x^2 + 5) - 7/39/sqrt(x^4 + 5*x^2 + 3)
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=-\frac {1}{9} \, \sqrt {3} \log \left (-x^{2} + \sqrt {3} + \sqrt {x^{4} + 5 \, x^{2} + 3}\right ) + \frac {1}{9} \, \sqrt {3} \log \left (-x^{2} - \sqrt {3} + \sqrt {x^{4} + 5 \, x^{2} + 3}\right ) - \frac {8 \, x^{2} + 7}{39 \, \sqrt {x^{4} + 5 \, x^{2} + 3}} \]
-1/9*sqrt(3)*log(-x^2 + sqrt(3) + sqrt(x^4 + 5*x^2 + 3)) + 1/9*sqrt(3)*log (-x^2 - sqrt(3) + sqrt(x^4 + 5*x^2 + 3)) - 1/39*(8*x^2 + 7)/sqrt(x^4 + 5*x ^2 + 3)
Timed out. \[ \int \frac {2+3 x^2}{x \left (3+5 x^2+x^4\right )^{3/2}} \, dx=\int \frac {3\,x^2+2}{x\,{\left (x^4+5\,x^2+3\right )}^{3/2}} \,d x \]