Integrand size = 29, antiderivative size = 280 \[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{a d x}-\frac {c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-(e*x^2+d)^(1/2)/a/d/x-c*arctan(x*(2*c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/( e*x^2+d)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(1+b/(-4*a*c+b^2)^(1/2))/a/(2 *c*d-e*(b-(-4*a*c+b^2)^(1/2)))^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-c*arctan (x*(2*c*d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*x^2+d)^(1/2)/(b+(-4*a*c+b^2)^ (1/2))^(1/2))*(1-b/(-4*a*c+b^2)^(1/2))/a/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(2*c *d-e*(b+(-4*a*c+b^2)^(1/2)))^(1/2)
Time = 10.80 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {\frac {\sqrt {d+e x^2}}{d x}+\frac {c \left (1+\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}{a} \]
-((Sqrt[d + e*x^2]/(d*x) + (c*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x ^2])])/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])* e]) + (c*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4* a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(Sqrt[b + Sqrt [b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]))/a)
Time = 0.72 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1626, 2009}
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.
\(\displaystyle \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 1626 |
\(\displaystyle \int \left (\frac {-b-c x^2}{a \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}+\frac {1}{a x^2 \sqrt {d+e x^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c \left (\frac {b}{\sqrt {b^2-4 a c}}+1\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {c \left (1-\frac {b}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {d+e x^2}}{a d x}\) |
-(Sqrt[d + e*x^2]/(a*d*x)) - (c*(1 + b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c *d - (b - Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[d + e *x^2])])/(a*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c ])*e]) - (c*(1 - b/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*x)/(Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[d + e*x^2])])/(a*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e])
3.4.91.3.1 Defintions of rubi rules used
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q, (f*x)^m/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[b^2 - 4*a *c, 0] && !IntegerQ[q] && IntegerQ[m]
Time = 0.62 (sec) , antiderivative size = 266, normalized size of antiderivative = 0.95
method | result | size |
default | \(-\frac {\sqrt {e \,x^{2}+d}}{a d x}+\frac {\sqrt {2}\, \left (\frac {\left (-2 a c d +b^{2} d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (2 a c d -b^{2} d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 a \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\) | \(266\) |
risch | \(-\frac {\sqrt {e \,x^{2}+d}}{a d x}+\frac {\sqrt {2}\, \left (\frac {\left (-2 a c d +b^{2} d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b \right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (2 a c d -b^{2} d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 a \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\) | \(266\) |
pseudoelliptic | \(-\frac {\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {2}\, d x \left (\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, b}{2}+d \left (a c -\frac {b^{2}}{2}\right )\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (-\frac {\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, b}{2}+d \left (a c -\frac {b^{2}}{2}\right )\right ) \sqrt {2}\, d x \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {e \,x^{2}+d}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, a x d}\) | \(366\) |
-(e*x^2+d)^(1/2)/a/d/x+1/2*2^(1/2)/a/(-d^2*(4*a*c-b^2))^(1/2)*((-2*a*c*d+b ^2*d+(-d^2*(4*a*c-b^2))^(1/2)*b)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^(1/2))*a) ^(1/2)*arctan(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((-2*a*e+b*d+(-d^2*(4*a*c-b^2))^ (1/2))*a)^(1/2))-(2*a*c*d-b^2*d+(-d^2*(4*a*c-b^2))^(1/2)*b)/((2*a*e-b*d+(- d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)*arctanh(a/x*(e*x^2+d)^(1/2)*2^(1/2)/((2*a *e-b*d+(-d^2*(4*a*c-b^2))^(1/2))*a)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 6431 vs. \(2 (236) = 472\).
Time = 12.66 (sec) , antiderivative size = 6431, normalized size of antiderivative = 22.97 \[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{2} \sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt {e x^{2} + d} x^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^2 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^2\,\sqrt {e\,x^2+d}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]