Integrand size = 21, antiderivative size = 466 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^3 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 c^3 (b+a c)^2}-\frac {\left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c x^5}+\frac {(4 b+3 a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 c^2 (b+a c) x^3}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{15 c^3 (b+a c)^2 x}-\frac {\left (8 b^2+13 a b c+3 a^2 c^2\right ) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{15 c^{5/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {a (4 b+3 a c) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{15 c^{3/2} (b+a c)^2 \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
1/15*(3*a^2*c^2+13*a*b*c+8*b^2)*d^3*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^ 3/(a*c+b)^2-1/5*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c/x^5+1/15*(3* a*c+4*b)*d*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^2/(a*c+b)/x^3-1/1 5*(3*a^2*c^2+13*a*b*c+8*b^2)*d^2*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/ 2)/c^3/(a*c+b)^2/x-1/15*(3*a^2*c^2+13*a*b*c+8*b^2)*d^(5/2)*(1/(1+d*x^2/c)) ^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/2),(b/ (a*c+b))^(1/2))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^(5/2)/(a*c+b)^2/(c*(a* d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+1/15*a*(3*a*c+4*b)*d^(5/2)*(1/(1+d*x ^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)/c^(1/2)/(1+d*x^2/c)^(1/ 2),(b/(a*c+b))^(1/2))*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^(3/2)/(a*c+b)^2/ (c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 11.32 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (\sqrt {\frac {d}{c}} \left (c+d x^2\right ) \left (b^3 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )+3 a^3 c^2 \left (c^3+d^3 x^6\right )+a b^2 \left (9 c^3-8 c^2 d x^2+17 c d^2 x^4+8 d^3 x^6\right )+a^2 b c \left (9 c^3-4 c^2 d x^2+9 c d^2 x^4+13 d^3 x^6\right )\right )+i \left (8 b^3+21 a b^2 c+16 a^2 b c^2+3 a^3 c^3\right ) d^3 x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-i b \left (8 b^2+17 a b c+9 a^2 c^2\right ) d^3 x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{15 c^3 (b+a c)^2 \sqrt {\frac {d}{c}} x^5 \left (b+a \left (c+d x^2\right )\right )} \]
-1/15*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(Sqrt[d/c]*(c + d*x^2)*(b^3*( 3*c^2 - 4*c*d*x^2 + 8*d^2*x^4) + 3*a^3*c^2*(c^3 + d^3*x^6) + a*b^2*(9*c^3 - 8*c^2*d*x^2 + 17*c*d^2*x^4 + 8*d^3*x^6) + a^2*b*c*(9*c^3 - 4*c^2*d*x^2 + 9*c*d^2*x^4 + 13*d^3*x^6)) + I*(8*b^3 + 21*a*b^2*c + 16*a^2*b*c^2 + 3*a^3 *c^3)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*Elli pticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)] - I*b*(8*b^2 + 17*a*b*c + 9 *a^2*c^2)*d^3*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]* EllipticF[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a*c)]))/(c^3*(b + a*c)^2*Sqrt [d/c]*x^5*(b + a*(c + d*x^2)))
Time = 0.78 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.06, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2057, 2058, 377, 25, 27, 445, 27, 445, 25, 27, 406, 320, 388, 313}
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.
| \(\displaystyle \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {\sqrt {\frac {a c+a d x^2+b}{c+d x^2}}}{x^6}dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \int \frac {\sqrt {a d x^2+b+a c}}{x^6 \sqrt {d x^2+c}}dx}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 377 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\int -\frac {d \left (3 a d x^2+4 b+3 a c\right )}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {\int \frac {d \left (3 a d x^2+4 b+3 a c\right )}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \int \frac {3 a d x^2+4 b+3 a c}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {\int \frac {d \left (8 b^2+13 a c b+3 a^2 c^2+a (4 b+3 a c) d x^2\right )}{x^2 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \int \frac {8 b^2+13 a c b+3 a^2 c^2+a (4 b+3 a c) d x^2}{x^2 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (-\frac {\int -\frac {a d \left (\left (8 b^2+13 a c b+3 a^2 c^2\right ) d x^2+c (b+a c) (4 b+3 a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (\frac {\int \frac {a d \left (\left (8 b^2+13 a c b+3 a^2 c^2\right ) d x^2+c (b+a c) (4 b+3 a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (\frac {a d \int \frac {\left (8 b^2+13 a c b+3 a^2 c^2\right ) d x^2+c (b+a c) (4 b+3 a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (\frac {a d \left (d \left (3 a^2 c^2+13 a b c+8 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+c (a c+b) (3 a c+4 b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (\frac {a d \left (d \left (3 a^2 c^2+13 a b c+8 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} (3 a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (\frac {a d \left (d \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} (3 a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (-\frac {d \left (-\frac {d \left (\frac {a d \left (d \left (3 a^2 c^2+13 a b c+8 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} (3 a c+4 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {\left (3 a^2 c^2+13 a b c+8 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c (a c+b)}-\frac {(3 a c+4 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3 (a c+b)}\right )}{5 c}-\frac {\sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}\right )}{\sqrt {a c+a d x^2+b}}\) |
(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(-1/5*(Sqrt[c + d*x ^2]*Sqrt[b + a*c + a*d*x^2])/(c*x^5) - (d*(-1/3*((4*b + 3*a*c)*Sqrt[c + d* x^2]*Sqrt[b + a*c + a*d*x^2])/(c*(b + a*c)*x^3) - (d*(-(((8*b^2 + 13*a*b*c + 3*a^2*c^2)*Sqrt[c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/(c*(b + a*c)*x)) + (a*d*((8*b^2 + 13*a*b*c + 3*a^2*c^2)*d*((x*Sqrt[b + a*c + a*d*x^2])/(a*d*S qrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*EllipticE[ArcTan[(Sqrt[ d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d^(3/2)*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])) + (c^(3/2)*(4*b + 3*a*c)*Sqrt[b + a *c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(Sqrt[d ]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))]))) /(c*(b + a*c))))/(3*c*(b + a*c))))/(5*c)))/Sqrt[b + a*c + a*d*x^2]
3.4.30.3.1 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[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) , x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( m + 1))), x] - Simp[1/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) + 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b *c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m , 2, p, q, x]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[e Int[(a + b*x^2)^p*(c + d*x^2)^q, x], x] + Sim p[f Int[x^2*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, p, q}, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 7.87 (sec) , antiderivative size = 778, normalized size of antiderivative = 1.67
| method | result | size |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (3 a^{2} c^{2} d^{2} x^{4}+13 a c \,d^{2} b \,x^{4}-3 a^{2} c^{3} d \,x^{2}+8 b^{2} d^{2} x^{4}-7 a b \,c^{2} d \,x^{2}+3 a^{2} c^{4}-4 b^{2} c d \,x^{2}+6 a b \,c^{3}+3 b^{2} c^{2}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 c^{3} x^{5} \left (a c +b \right )^{2}}+\frac {a \,d^{3} \left (\frac {3 a^{2} c^{3} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {4 b^{2} c \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}+\frac {7 a b \,c^{2} \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}}-\frac {2 \left (3 a^{2} c^{2} d +13 a b c d +8 b^{2} d \right ) \left (a \,c^{2}+b c \right ) \sqrt {1+\frac {a d \,x^{2}}{a c +b}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )-E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {-1+\frac {2 a c d +b d}{d c a}}\right )\right )}{\sqrt {-\frac {a d}{a c +b}}\, \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \left (2 a c d +2 b d \right )}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}{15 \left (a c +b \right )^{2} c^{3} \left (a d \,x^{2}+a c +b \right )}\) | \(778\) |
| default | \(-\frac {\left (3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{2} d^{4} x^{8}+13 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b c \,d^{4} x^{8}-3 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{3} c^{3} d^{3} x^{5}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{3} d^{3} x^{6}+8 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} d^{4} x^{8}+6 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} b \,c^{2} d^{3} x^{5}-13 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a^{2} b \,c^{2} d^{3} x^{5}+22 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{2} d^{3} x^{6}+4 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,b^{2} c \,d^{3} x^{5}-8 \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) a \,b^{2} c \,d^{3} x^{5}+25 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c \,d^{3} x^{6}+5 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{3} d^{2} x^{4}+8 \sqrt {-\frac {a d}{a c +b}}\, b^{3} d^{3} x^{6}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{5} d \,x^{2}+9 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c^{2} d^{2} x^{4}+5 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{4} d \,x^{2}+4 \sqrt {-\frac {a d}{a c +b}}\, b^{3} c \,d^{2} x^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, a^{3} c^{6}+\sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c^{3} d \,x^{2}+9 \sqrt {-\frac {a d}{a c +b}}\, a^{2} b \,c^{5}-\sqrt {-\frac {a d}{a c +b}}\, b^{3} c^{2} d \,x^{2}+9 \sqrt {-\frac {a d}{a c +b}}\, a \,b^{2} c^{4}+3 \sqrt {-\frac {a d}{a c +b}}\, b^{3} c^{3}\right ) \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{15 \sqrt {a \,d^{2} x^{4}+2 a c d \,x^{2}+b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \left (a c +b \right )^{2} x^{5} c^{3} \sqrt {\left (a d \,x^{2}+a c +b \right ) \left (d \,x^{2}+c \right )}}\) | \(955\) |
-1/15*(d*x^2+c)*(3*a^2*c^2*d^2*x^4+13*a*b*c*d^2*x^4-3*a^2*c^3*d*x^2+8*b^2* d^2*x^4-7*a*b*c^2*d*x^2+3*a^2*c^4-4*b^2*c*d*x^2+6*a*b*c^3+3*b^2*c^2)/c^3/x ^5/(a*c+b)^2*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/15*a*d^3/(a*c+b)^2/c^3*(3 *a^2*c^3/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2 )/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b ))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))+4*b^2*c/(-a*d/(a*c+b))^(1/2)*(1+a *d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a *c^2+b*c)^(1/2)*EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^ (1/2))+7*a*b*c^2/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/c*d*x ^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)*EllipticF(x*(-a* d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-2*(3*a^2*c^2*d+13*a*b*c*d +8*b^2*d)*(a*c^2+b*c)/(-a*d/(a*c+b))^(1/2)*(1+a*d/(a*c+b)*x^2)^(1/2)*(1+1/ c*d*x^2)^(1/2)/(a*d^2*x^4+2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(2*a*c*d+2* b*d)*(EllipticF(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))-Ell ipticE(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+b*d)/d/c/a)^(1/2))))*((a*d*x^2+ a*c+b)/(d*x^2+c))^(1/2)*((a*d*x^2+a*c+b)*(d*x^2+c))^(1/2)/(a*d*x^2+a*c+b)
Time = 0.13 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=\frac {{\left (3 \, a^{3} c^{2} + 13 \, a^{2} b c + 8 \, a b^{2}\right )} \sqrt {-\frac {a d}{a c + b}} d^{4} x^{5} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (3 \, a^{3} c^{2} + 13 \, a^{2} b c + 8 \, a b^{2}\right )} d^{4} + {\left (3 \, a^{3} c^{3} + 10 \, a^{2} b c^{2} + 11 \, a b^{2} c + 4 \, b^{3}\right )} d^{3}\right )} \sqrt {-\frac {a d}{a c + b}} x^{5} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (3 \, a^{3} c^{3} + 16 \, a^{2} b c^{2} + 21 \, a b^{2} c + 8 \, b^{3}\right )} d^{3} x^{6} + 3 \, a^{3} c^{6} + 9 \, a^{2} b c^{5} + 9 \, a b^{2} c^{4} + 2 \, {\left (3 \, a^{2} b c^{3} + 5 \, a b^{2} c^{2} + 2 \, b^{3} c\right )} d^{2} x^{4} + 3 \, b^{3} c^{3} - {\left (a^{2} b c^{4} + 2 \, a b^{2} c^{3} + b^{3} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{15 \, {\left (a^{3} c^{6} + 3 \, a^{2} b c^{5} + 3 \, a b^{2} c^{4} + b^{3} c^{3}\right )} x^{5}} \]
1/15*((3*a^3*c^2 + 13*a^2*b*c + 8*a*b^2)*sqrt(-a*d/(a*c + b))*d^4*x^5*sqrt ((a*c^2 + b*c)/d^2)*elliptic_e(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/( a*c)) - ((3*a^3*c^2 + 13*a^2*b*c + 8*a*b^2)*d^4 + (3*a^3*c^3 + 10*a^2*b*c^ 2 + 11*a*b^2*c + 4*b^3)*d^3)*sqrt(-a*d/(a*c + b))*x^5*sqrt((a*c^2 + b*c)/d ^2)*elliptic_f(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - ((3*a^3* c^3 + 16*a^2*b*c^2 + 21*a*b^2*c + 8*b^3)*d^3*x^6 + 3*a^3*c^6 + 9*a^2*b*c^5 + 9*a*b^2*c^4 + 2*(3*a^2*b*c^3 + 5*a*b^2*c^2 + 2*b^3*c)*d^2*x^4 + 3*b^3*c ^3 - (a^2*b*c^4 + 2*a*b^2*c^3 + b^3*c^2)*d*x^2)*sqrt((a*d*x^2 + a*c + b)/( d*x^2 + c)))/((a^3*c^6 + 3*a^2*b*c^5 + 3*a*b^2*c^4 + b^3*c^3)*x^5)
\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=\int \frac {\sqrt {\frac {a c + a d x^{2} + b}{c + d x^{2}}}}{x^{6}}\, dx \]
\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=\int { \frac {\sqrt {a + \frac {b}{d x^{2} + c}}}{x^{6}} \,d x } \]
\[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=\int { \frac {\sqrt {a + \frac {b}{d x^{2} + c}}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {a+\frac {b}{c+d x^2}}}{x^6} \, dx=\int \frac {\sqrt {a+\frac {b}{d\,x^2+c}}}{x^6} \,d x \]