Integrand size = 29, antiderivative size = 443 \[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {\sqrt {d+e x^2}}{5 a d x^5}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^3 x}-\frac {c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \]
-1/5*(e*x^2+d)^(1/2)/a/d/x^5+1/3*b*(e*x^2+d)^(1/2)/a^2/d/x^3+4/15*e*(e*x^2 +d)^(1/2)/a/d^2/x^3-(-a*c+b^2)*(e*x^2+d)^(1/2)/a^3/d/x-2/3*b*e*(e*x^2+d)^( 1/2)/a^2/d^2/x-8/15*e^2*(e*x^2+d)^(1/2)/a/d^3/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))*(b^2 -a*c+b*(-3*a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^3/(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))*(b^2-a*c-b*(-3 *a*c+b^2)/(-4*a*c+b^2)^(1/2))/a^3/(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 = 11.23 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=-\frac {\frac {15 \left (b^2-a c\right ) \sqrt {d+e x^2}}{d x}-\frac {5 a b \left (d-2 e x^2\right ) \sqrt {d+e x^2}}{d^2 x^3}+\frac {a^2 \sqrt {d+e x^2} \left (3 d^2-4 d e x^2+8 e^2 x^4\right )}{d^3 x^5}+\frac {15 c \left (b^2-a c+\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} x}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}+\frac {15 c \left (b^2-a c-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} x}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{\sqrt {b+\sqrt {b^2-4 a c}} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}}{15 a^3} \]
-1/15*((15*(b^2 - a*c)*Sqrt[d + e*x^2])/(d*x) - (5*a*b*(d - 2*e*x^2)*Sqrt[ d + e*x^2])/(d^2*x^3) + (a^2*Sqrt[d + e*x^2]*(3*d^2 - 4*d*e*x^2 + 8*e^2*x^ 4))/(d^3*x^5) + (15*c*(b^2 - a*c + (b*(b^2 - 3*a*c))/Sqrt[b^2 - 4*a*c])*Ar cTan[(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 + Sqr t[b^2 - 4*a*c])*e]) + (15*c*(b^2 - a*c - (b*(b^2 - 3*a*c))/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^3
Time = 1.25 (sec) , antiderivative size = 443, 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.
| \(\displaystyle \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1626 |
| \(\displaystyle \int \left (\frac {b^2-a c}{a^3 x^2 \sqrt {d+e x^2}}+\frac {-c x^2 \left (b^2-a c\right )-b \left (b^2-2 a c\right )}{a^3 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )}-\frac {b}{a^2 x^4 \sqrt {d+e x^2}}+\frac {1}{a x^6 \sqrt {d+e x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {c \left (\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}{\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {b-\sqrt {b^2-4 a c}} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}-\frac {c \left (-\frac {b \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}-a c+b^2\right ) \arctan \left (\frac {x \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}{\sqrt {\sqrt {b^2-4 a c}+b} \sqrt {d+e x^2}}\right )}{a^3 \sqrt {\sqrt {b^2-4 a c}+b} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\left (b^2-a c\right ) \sqrt {d+e x^2}}{a^3 d x}-\frac {2 b e \sqrt {d+e x^2}}{3 a^2 d^2 x}+\frac {b \sqrt {d+e x^2}}{3 a^2 d x^3}-\frac {8 e^2 \sqrt {d+e x^2}}{15 a d^3 x}+\frac {4 e \sqrt {d+e x^2}}{15 a d^2 x^3}-\frac {\sqrt {d+e x^2}}{5 a d x^5}\) |
-1/5*Sqrt[d + e*x^2]/(a*d*x^5) + (b*Sqrt[d + e*x^2])/(3*a^2*d*x^3) + (4*e* Sqrt[d + e*x^2])/(15*a*d^2*x^3) - ((b^2 - a*c)*Sqrt[d + e*x^2])/(a^3*d*x) - (2*b*e*Sqrt[d + e*x^2])/(3*a^2*d^2*x) - (8*e^2*Sqrt[d + e*x^2])/(15*a*d^ 3*x) - (c*(b^2 - a*c + (b*(b^2 - 3*a*c))/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^3*Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4 *a*c])*e]) - (c*(b^2 - a*c - (b*(b^2 - 3*a*c))/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]]*Sq rt[d + e*x^2])])/(a^3*Sqrt[b + Sqrt[b^2 - 4*a*c]]*Sqrt[2*c*d - (b + Sqrt[b ^2 - 4*a*c])*e])
3.4.93.3.1 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 = 0.98 (sec) , antiderivative size = 410, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(-\frac {\sqrt {e \,x^{2}+d}\, \left (8 a^{2} e^{2} x^{4}+10 a b d e \,x^{4}-15 a c \,d^{2} x^{4}+15 b^{2} d^{2} x^{4}-4 a^{2} d e \,x^{2}-5 a b \,d^{2} x^{2}+3 a^{2} d^{2}\right )}{15 d^{3} a^{3} x^{5}}-\frac {\sqrt {2}\, \left (\frac {\left (-2 a^{2} c^{2} d +4 a \,b^{2} c d -d \,b^{4}+2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a b c -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b^{3}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (2 a^{2} c^{2} d -4 a \,b^{2} c d +d \,b^{4}+2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a b c -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b^{3}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 a^{3} \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\) | \(410\) |
| default | \(\frac {-\frac {\sqrt {e \,x^{2}+d}}{5 d \,x^{5}}-\frac {4 e \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{5 d}}{a}-\frac {b \left (-\frac {\sqrt {e \,x^{2}+d}}{3 d \,x^{3}}+\frac {2 e \sqrt {e \,x^{2}+d}}{3 d^{2} x}\right )}{a^{2}}-\frac {\left (-a c +b^{2}\right ) \sqrt {e \,x^{2}+d}}{a^{3} d x}-\frac {\sqrt {2}\, \left (\frac {\left (-2 a^{2} c^{2} d +4 a \,b^{2} c d -d \,b^{4}+2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a b c -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b^{3}\right ) \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (-2 a e +b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}-\frac {\left (2 a^{2} c^{2} d -4 a \,b^{2} c d +d \,b^{4}+2 \sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, a b c -\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\, b^{3}\right ) \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{\sqrt {\left (2 a e -b d +\sqrt {-d^{2} \left (4 a c -b^{2}\right )}\right ) a}}\right )}{2 a^{3} \sqrt {-d^{2} \left (4 a c -b^{2}\right )}}\) | \(454\) |
| pseudoelliptic | \(-\frac {-5 \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (a b c -\frac {1}{2} b^{3}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right )\right ) \sqrt {2}\, d^{3} x^{5} \operatorname {arctanh}\left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (-5 \left (\left (-a b c +\frac {1}{2} b^{3}\right ) \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}+d \left (-2 a \,b^{2} c +\frac {1}{2} b^{4}+a^{2} c^{2}\right )\right ) \sqrt {2}\, d^{3} x^{5} \arctan \left (\frac {a \sqrt {e \,x^{2}+d}\, \sqrt {2}}{x \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}}\right )+\sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \left (\left (\frac {8}{3} e^{2} x^{4}-\frac {4}{3} e d \,x^{2}+d^{2}\right ) a^{2}-\frac {5 d \left (\left (3 c \,x^{2}+b \right ) d -2 e \,x^{2} b \right ) x^{2} a}{3}+5 b^{2} d^{2} x^{4}\right ) \sqrt {e \,x^{2}+d}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{5 \sqrt {\left (2 a e -b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {\left (-2 a e +b d +\sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) a}\, \sqrt {-4 d^{2} \left (a c -\frac {b^{2}}{4}\right )}\, x^{5} a^{3} d^{3}}\) | \(475\) |
-1/15*(e*x^2+d)^(1/2)*(8*a^2*e^2*x^4+10*a*b*d*e*x^4-15*a*c*d^2*x^4+15*b^2* d^2*x^4-4*a^2*d*e*x^2-5*a*b*d^2*x^2+3*a^2*d^2)/d^3/a^3/x^5-1/2/a^3*2^(1/2) /(-d^2*(4*a*c-b^2))^(1/2)*((-2*a^2*c^2*d+4*a*b^2*c*d-d*b^4+2*(-d^2*(4*a*c- b^2))^(1/2)*a*b*c-(-d^2*(4*a*c-b^2))^(1/2)*b^3)/((-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^2*c^2*d-4*a*b^2*c*d+d*b^4+2*(-d^2*(4* a*c-b^2))^(1/2)*a*b*c-(-d^2*(4*a*c-b^2))^(1/2)*b^3)/((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. 9998 vs. \(2 (381) = 762\).
Time = 165.00 (sec) , antiderivative size = 9998, normalized size of antiderivative = 22.57 \[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
\[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^{6} \sqrt {d + e x^{2}} \left (a + b x^{2} + c x^{4}\right )}\, dx \]
\[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )} \sqrt {e x^{2} + d} x^{6}} \,d x } \]
Timed out. \[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{x^6 \sqrt {d+e x^2} \left (a+b x^2+c x^4\right )} \, dx=\int \frac {1}{x^6\,\sqrt {e\,x^2+d}\,\left (c\,x^4+b\,x^2+a\right )} \,d x \]