Integrand size = 21, antiderivative size = 494 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\frac {b \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{c x^5}+\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^3 x \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^4 (b+a c)}-\frac {(6 b+a c) \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^2 x^5}+\frac {(8 b+a c) d \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^3 x^3}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^2 \left (c+d x^2\right ) \sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{5 c^4 (b+a c) x}-\frac {\left (16 b^2+16 a b c+a^2 c^2\right ) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{5 c^{7/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {a (8 b+a c) d^{5/2} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{5 c^{5/2} (b+a c) \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \]
b*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c/x^5+1/5*(a^2*c^2+16*a*b*c+16*b^2)*d^ 3*x*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^4/(a*c+b)-1/5*(a*c+6*b)*(d*x^2+c)* ((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^2/x^5+1/5*(a*c+8*b)*d*(d*x^2+c)*((a*d* x^2+a*c+b)/(d*x^2+c))^(1/2)/c^3/x^3-1/5*(a^2*c^2+16*a*b*c+16*b^2)*d^2*(d*x ^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/c^4/(a*c+b)/x-1/5*(a^2*c^2+16*a*b* c+16*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^(7/2)/(a*c+b)/(c*(a*d*x^2+a*c+b)/(a*c+b)/(d*x^2+c))^(1/2)+1/5*a *(a*c+8*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^(5/2)/(a*c+b)/(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 = 11.23 (sec) , antiderivative size = 472, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (b^3 c^3+3 a b^2 c^4+3 a^2 b c^5+a^3 c^6-2 b^3 c^2 d x^2-3 a b^2 c^3 d x^2+a^3 c^5 d x^2+8 b^3 c d^2 x^4+13 a b^2 c^2 d^2 x^4+5 a^2 b c^3 d^2 x^4+16 b^3 d^3 x^6+40 a b^2 c d^3 x^6+24 a^2 b c^2 d^3 x^6+a^3 c^3 d^3 x^6+16 a b^2 d^4 x^8+16 a^2 b c d^4 x^8+a^3 c^2 d^4 x^8+i c \left (16 b^3+32 a b^2 c+17 a^2 b c^2+a^3 c^3\right ) d^2 \sqrt {\frac {d}{c}} x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )-8 i b c \left (2 b^2+3 a b c+a^2 c^2\right ) d^2 \sqrt {\frac {d}{c}} x^5 \sqrt {\frac {b+a c+a d x^2}{b+a c}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {a c}{b+a c}\right )\right )}{5 c^4 (b+a c) x^5 \left (b+a \left (c+d x^2\right )\right )} \]
-1/5*(Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(b^3*c^3 + 3*a*b^2*c^4 + 3*a^2 *b*c^5 + a^3*c^6 - 2*b^3*c^2*d*x^2 - 3*a*b^2*c^3*d*x^2 + a^3*c^5*d*x^2 + 8 *b^3*c*d^2*x^4 + 13*a*b^2*c^2*d^2*x^4 + 5*a^2*b*c^3*d^2*x^4 + 16*b^3*d^3*x ^6 + 40*a*b^2*c*d^3*x^6 + 24*a^2*b*c^2*d^3*x^6 + a^3*c^3*d^3*x^6 + 16*a*b^ 2*d^4*x^8 + 16*a^2*b*c*d^4*x^8 + a^3*c^2*d^4*x^8 + I*c*(16*b^3 + 32*a*b^2* c + 17*a^2*b*c^2 + a^3*c^3)*d^2*Sqrt[d/c]*x^5*Sqrt[(b + a*c + a*d*x^2)/(b + a*c)]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (a*c)/(b + a *c)] - (8*I)*b*c*(2*b^2 + 3*a*b*c + a^2*c^2)*d^2*Sqrt[d/c]*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^4*(b + a*c)*x^5*(b + a*(c + d*x^2)))
Time = 0.93 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.05, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {2057, 2058, 370, 25, 27, 445, 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.
| \(\displaystyle \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 2057 |
| \(\displaystyle \int \frac {\left (\frac {a c+a d x^2+b}{c+d x^2}\right )^{3/2}}{x^6}dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \int \frac {\left (a d x^2+b+a c\right )^{3/2}}{x^6 \left (d x^2+c\right )^{3/2}}dx}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}-\frac {\int -\frac {d \left (a (5 b+a c) d x^2+(b+a c) (6 b+a c)\right )}{x^6 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c d}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\int \frac {d \left (a (5 b+a c) d x^2+(b+a c) (6 b+a c)\right )}{x^6 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c d}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {\int \frac {a (5 b+a c) d x^2+(b+a c) (6 b+a c)}{x^6 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {\int \frac {3 (b+a c) d \left (a (6 b+a c) d x^2+(b+a c) (8 b+a c)\right )}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 c (a c+b)}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \int \frac {a (6 b+a c) d x^2+(b+a c) (8 b+a c)}{x^4 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {\int \frac {(b+a c) d \left (16 b^2+16 a c b+a^2 c^2+a (8 b+a c) d x^2\right )}{x^2 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 c (a c+b)}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \int \frac {16 b^2+16 a c b+a^2 c^2+a (8 b+a c) d x^2}{x^2 \sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (-\frac {\int -\frac {a d \left (\left (16 b^2+16 a c b+a^2 c^2\right ) d x^2+c (b+a c) (8 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (\frac {\int \frac {a d \left (\left (16 b^2+16 a c b+a^2 c^2\right ) d x^2+c (b+a c) (8 b+a c)\right )}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (\frac {a d \int \frac {\left (16 b^2+16 a c b+a^2 c^2\right ) d x^2+c (b+a c) (8 b+a c)}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (\frac {a d \left (d \left (a^2 c^2+16 a b c+16 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+c (a c+b) (a c+8 b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (\frac {a d \left (d \left (a^2 c^2+16 a b c+16 b^2\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {c^{3/2} (a c+8 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (\frac {a d \left (d \left (a^2 c^2+16 a b c+16 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {c^{3/2} (a c+8 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (\frac {-\frac {3 d \left (-\frac {d \left (\frac {a d \left (d \left (a^2 c^2+16 a b c+16 b^2\right ) \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} (a c+8 b) \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{c (a c+b)}-\frac {\left (a^2 c^2+16 a b c+16 b^2\right ) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{c x (a c+b)}\right )}{3 c}-\frac {(a c+8 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{3 c x^3}\right )}{5 c}-\frac {(a c+6 b) \sqrt {c+d x^2} \sqrt {a c+a d x^2+b}}{5 c x^5}}{c}+\frac {b \sqrt {a c+a d x^2+b}}{c x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a c+a d x^2+b}}\) |
(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*((b*Sqrt[b + a*c + a*d*x^2])/(c*x^5*Sqrt[c + d*x^2]) + (-1/5*((6*b + a*c)*Sqrt[c + d*x^2]*Sqr t[b + a*c + a*d*x^2])/(c*x^5) - (3*d*(-1/3*((8*b + a*c)*Sqrt[c + d*x^2]*Sq rt[b + a*c + a*d*x^2])/(c*x^3) - (d*(-(((16*b^2 + 16*a*b*c + a^2*c^2)*Sqrt [c + d*x^2]*Sqrt[b + a*c + a*d*x^2])/(c*(b + a*c)*x)) + (a*d*((16*b^2 + 16 *a*b*c + a^2*c^2)*d*((x*Sqrt[b + a*c + a*d*x^2])/(a*d*Sqrt[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)*(8*b + a*c)*Sqrt[b + a*c + a*d*x^2]*Ellipti cF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(Sqrt[d]*Sqrt[c + d*x^2]*Sqr t[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])))/(c*(b + a*c))))/(3*c )))/(5*c))/c))/Sqrt[b + a*c + a*d*x^2]
3.4.43.3.1 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[(-(b*c - a*d))*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(a*b*e*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(e*x) ^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*Simp[c*(b*c*2*(p + 1) + (b*c - a *d)*(m + 1)) + d*(b*c*2*(p + 1) + (b*c - a*d)*(m + 2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(1169\) vs. \(2(526)=1052\).
Time = 11.58 (sec) , antiderivative size = 1170, normalized size of antiderivative = 2.37
| method | result | size |
| risch | \(\text {Expression too large to display}\) | \(1170\) |
| default | \(\text {Expression too large to display}\) | \(1666\) |
-1/5*(d*x^2+c)*(a^2*c^2*d^2*x^4+11*a*b*c*d^2*x^4-a^2*c^3*d*x^2+11*b^2*d^2* x^4-4*a*b*c^2*d*x^2+a^2*c^4-3*b^2*c*d*x^2+2*a*b*c^3+b^2*c^2)/c^4/x^5/(a*c+ b)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)+1/5/c^4*d^3/(a*c+b)*(c^3*a^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*a^2*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))+3*a*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))-5*b^2*c*(a*c+b)*((a*d^2*x^2+a*c*d+b*d) /c/b*x/d/((x^2+c/d)*(a*d^2*x^2+a*c*d+b*d))^(1/2)+(1/c-(a*c*d+b*d)/c/b/d)/( -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*a*d/b/c*(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))-EllipticE(x*(-a*d/(a*c+b))^(1/2),(-1+(2*a*c*d+ b*d)/d/c/a)^(1/2))))-2*(a^3*c^2*d+11*a^2*b*c*d+11*a*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*...
Time = 0.11 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.80 \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\frac {{\left (a^{3} c^{2} + 16 \, a^{2} b c + 16 \, a b^{2}\right )} \sqrt {-\frac {a d}{a c + b}} d^{4} x^{5} \sqrt {\frac {a c^{2} + b c}{d^{2}}} E(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{3} c^{2} + 16 \, a^{2} b c + 16 \, a b^{2}\right )} d^{4} + {\left (a^{3} c^{3} + 10 \, a^{2} b c^{2} + 17 \, a b^{2} c + 8 \, b^{3}\right )} d^{3}\right )} \sqrt {-\frac {a d}{a c + b}} x^{5} \sqrt {\frac {a c^{2} + b c}{d^{2}}} F(\arcsin \left (\sqrt {-\frac {a d}{a c + b}} x\right )\,|\,\frac {a c + b}{a c}) - {\left ({\left (a^{3} c^{3} + 17 \, a^{2} b c^{2} + 32 \, a b^{2} c + 16 \, b^{3}\right )} d^{3} x^{6} + a^{3} c^{6} + 3 \, a^{2} b c^{5} + 3 \, a b^{2} c^{4} + {\left (7 \, a^{2} b c^{3} + 15 \, a b^{2} c^{2} + 8 \, b^{3} c\right )} d^{2} x^{4} + b^{3} c^{3} - 2 \, {\left (a^{2} b c^{4} + 2 \, a b^{2} c^{3} + b^{3} c^{2}\right )} d x^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{5 \, {\left (a^{2} c^{6} + 2 \, a b c^{5} + b^{2} c^{4}\right )} x^{5}} \]
1/5*((a^3*c^2 + 16*a^2*b*c + 16*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)) - ((a^3*c^2 + 16*a^2*b*c + 16*a*b^2)*d^4 + (a^3*c^3 + 10*a^2*b*c^2 + 1 7*a*b^2*c + 8*b^3)*d^3)*sqrt(-a*d/(a*c + b))*x^5*sqrt((a*c^2 + b*c)/d^2)*e lliptic_f(arcsin(sqrt(-a*d/(a*c + b))*x), (a*c + b)/(a*c)) - ((a^3*c^3 + 1 7*a^2*b*c^2 + 32*a*b^2*c + 16*b^3)*d^3*x^6 + a^3*c^6 + 3*a^2*b*c^5 + 3*a*b ^2*c^4 + (7*a^2*b*c^3 + 15*a*b^2*c^2 + 8*b^3*c)*d^2*x^4 + b^3*c^3 - 2*(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^2*c^6 + 2*a*b*c^5 + b^2*c^4)*x^5)
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int \frac {\left (\frac {a c + a d x^{2} + b}{c + d x^{2}}\right )^{\frac {3}{2}}}{x^{6}}\, dx \]
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (a + \frac {b}{d x^{2} + c}\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}}{x^6} \,d x \]