Integrand size = 26, antiderivative size = 480 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=-\frac {(b c-a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{c d x^5}+\frac {d \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e x \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{5 a c^4}+\frac {(5 b c-6 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^2 d x^5}-\frac {(7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 c^3 x^3}-\frac {\left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (c+d x^2\right )}{5 a c^4 x}-\frac {\sqrt {d} \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{5 a c^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (7 b c-8 a d) e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{5 a c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]
-(-a*d+b*c)*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c/d/x^5+1/5*d*(16*a^2*d^2-16*a *b*c*d+b^2*c^2)*e*x*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^4+1/5*(-6*a*d+5*b*c) *e*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^2/d/x^5-1/5*(-8*a*d+7*b*c)*e* (d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/c^3/x^3-1/5*(16*a^2*d^2-16*a*b*c*d +b^2*c^2)*e*(d*x^2+c)*(e*(b*x^2+a)/(d*x^2+c))^(1/2)/a/c^4/x-1/5*(16*a^2*d^ 2-16*a*b*c*d+b^2*c^2)*e*(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),(1-b*c/a/d)^(1/2))*d^(1/2)*(e*(b*x^2+a )/(d*x^2+c))^(1/2)/a/c^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)-1/5*b*(-8*a*d +7*b*c)*e*(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),(1-b*c/a/d)^(1/2))*d^(1/2)*(e*(b*x^2+a)/(d*x^2+c))^( 1/2)/a/c^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 6.23 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.67 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=-\frac {\sqrt {\frac {b}{a}} e \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (b^2 c^2 x^4 \left (c+d x^2\right )+a b c x^2 \left (2 c^2-9 c d x^2-16 d^2 x^4\right )+a^2 \left (c^3-2 c^2 d x^2+8 c d^2 x^4+16 d^3 x^6\right )\right )+i b c \left (b^2 c^2-16 a b c d+16 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c \left (b^2 c^2-9 a b c d+8 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{5 b c^4 x^5 \left (a+b x^2\right )} \]
-1/5*(Sqrt[b/a]*e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2) *(b^2*c^2*x^4*(c + d*x^2) + a*b*c*x^2*(2*c^2 - 9*c*d*x^2 - 16*d^2*x^4) + a ^2*(c^3 - 2*c^2*d*x^2 + 8*c*d^2*x^4 + 16*d^3*x^6)) + I*b*c*(b^2*c^2 - 16*a *b*c*d + 16*a^2*d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE [I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*b*c*(b^2*c^2 - 9*a*b*c*d + 8*a^2 *d^2)*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt [b/a]*x], (a*d)/(b*c)]))/(b*c^4*x^5*(a + b*x^2))
Time = 0.76 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2058, 370, 445, 27, 445, 25, 27, 445, 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.
| \(\displaystyle \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {\left (b x^2+a\right )^{3/2}}{x^6 \left (d x^2+c\right )^{3/2}}dx}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 370 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {\int \frac {b (4 b c-5 a d) x^2+a (5 b c-6 a d)}{x^6 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {\int \frac {3 a d \left (b (5 b c-6 a d) x^2+a (7 b c-8 a d)\right )}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \int \frac {b (5 b c-6 a d) x^2+a (7 b c-8 a d)}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (-\frac {\int -\frac {a \left (b^2 c^2-16 a b d c+16 a^2 d^2-b d (7 b c-8 a d) x^2\right )}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {\int \frac {a \left (b^2 c^2-16 a b d c+16 a^2 d^2-b d (7 b c-8 a d) x^2\right )}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {\int \frac {b^2 c^2-16 a b d c+16 a^2 d^2-b d (7 b c-8 a d) x^2}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {-\frac {\int \frac {b d \left (a c (7 b c-8 a d)-\left (b^2 c^2-16 a b d c+16 a^2 d^2\right ) x^2\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {b^2 c}{a}+\frac {16 a d^2}{c}-16 b d\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {-\frac {b d \int \frac {a c (7 b c-8 a d)-\left (b^2 c^2-16 a b d c+16 a^2 d^2\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {b^2 c}{a}+\frac {16 a d^2}{c}-16 b d\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {-\frac {b d \left (a c (7 b c-8 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {b^2 c}{a}+\frac {16 a d^2}{c}-16 b d\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {-\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {b^2 c}{a}+\frac {16 a d^2}{c}-16 b d\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {-\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {b^2 c}{a}+\frac {16 a d^2}{c}-16 b d\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {e \sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (-\frac {-\frac {3 d \left (\frac {-\frac {b d \left (\frac {c^{3/2} \sqrt {a+b x^2} (7 b c-8 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (16 a^2 d^2-16 a b c d+b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {b^2 c}{a}+\frac {16 a d^2}{c}-16 b d\right )}{x}}{3 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (7 b c-8 a d)}{3 c x^3}\right )}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (5 b c-6 a d)}{5 c x^5}}{c d}-\frac {\sqrt {a+b x^2} (b c-a d)}{c d x^5 \sqrt {c+d x^2}}\right )}{\sqrt {a+b x^2}}\) |
(e*Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-(((b*c - a*d)*Sqrt[ a + b*x^2])/(c*d*x^5*Sqrt[c + d*x^2])) - (-1/5*((5*b*c - 6*a*d)*Sqrt[a + b *x^2]*Sqrt[c + d*x^2])/(c*x^5) - (3*d*(-1/3*((7*b*c - 8*a*d)*Sqrt[a + b*x^ 2]*Sqrt[c + d*x^2])/(c*x^3) + (-((((b^2*c)/a - 16*b*d + (16*a*d^2)/c)*Sqrt [a + b*x^2]*Sqrt[c + d*x^2])/x) - (b*d*(-((b^2*c^2 - 16*a*b*c*d + 16*a^2*d ^2)*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*El lipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c* (a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))) + (c^(3/2)*(7*b*c - 8*a*d )*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)]) /(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/(a*c))/ (3*c)))/(5*c))/(c*d)))/Sqrt[a + b*x^2]
3.3.88.3.1 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_.)*((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 = 10.05 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.89
| method | result | size |
| risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (11 a^{2} d^{2} x^{4}-11 b d a c \,x^{4}+b^{2} c^{2} x^{4}-3 a^{2} c d \,x^{2}+2 a b \,c^{2} x^{2}+a^{2} c^{2}\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{5 c^{4} x^{5} a}+\frac {d \left (-\frac {2 b^{2} a \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {3 a^{2} b c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}-\frac {2 \left (11 a^{2} b \,d^{2}-11 a \,b^{2} c d +b^{3} c^{2}\right ) a c e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}-5 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a c \left (\frac {\left (b d e \,x^{2}+e d a \right ) x}{c \left (a d -b c \right ) e \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d e \,x^{2}+e d a \right )}}+\frac {\left (\frac {1}{c}-\frac {d a}{c \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {2 b d a e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {e d a +e b c}{c b e}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (e d a +e b c +e \left (a d -b c \right )\right )}\right )\right ) e \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) e \left (b \,x^{2}+a \right )}}{5 c^{4} a \left (b \,x^{2}+a \right )}\) | \(908\) |
| default | \(\text {Expression too large to display}\) | \(1197\) |
-1/5*(d*x^2+c)*(11*a^2*d^2*x^4-11*a*b*c*d*x^4+b^2*c^2*x^4-3*a^2*c*d*x^2+2* a*b*c^2*x^2+a^2*c^2)/c^4/x^5/a*e*(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/5/c^4/a*d *(-2*b^2*a*c^2/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/(b*d*e*x ^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b* c*e)/c/b/e)^(1/2))+3*a^2*b*c*d/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2 )^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/ 2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-2*(11*a^2*b*d^2-11*a*b^2*c*d+b^3*c^2)*a *c*e/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/(b*d*e*x^4+a*d*e*x ^2+b*c*e*x^2+a*c*e)^(1/2)/(e*d*a+e*b*c+e*(a*d-b*c))*(EllipticF(x*(-b/a)^(1 /2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d*e+b* c*e)/c/b/e)^(1/2)))-5*(a^2*d^2-2*a*b*c*d+b^2*c^2)*a*c*((b*d*e*x^2+a*d*e)/c /(a*d-b*c)*x/e/((x^2+c/d)*(b*d*e*x^2+a*d*e))^(1/2)+(1/c-d*a/c/(a*d-b*c))/( -b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c *e*x^2+a*c*e)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2 ))+2*b*d/(a*d-b*c)*a*e/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+1/c*d*x^2)^(1/2)/ (b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)/(e*d*a+e*b*c+e*(a*d-b*c))*(Ell ipticF(x*(-b/a)^(1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2))-EllipticE(x*(-b/a)^( 1/2),(-1+(a*d*e+b*c*e)/c/b/e)^(1/2)))))*e/(b*x^2+a)*(e*(b*x^2+a)/(d*x^2+c) )^(1/2)*((d*x^2+c)*e*(b*x^2+a))^(1/2)
Time = 0.12 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.61 \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\frac {{\left (b^{3} c^{2} d - 16 \, a b^{2} c d^{2} + 16 \, a^{2} b d^{3}\right )} \sqrt {\frac {a c e}{d^{2}}} e x^{5} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{3} c^{2} d - {\left (7 \, a^{2} b + 16 \, a b^{2}\right )} c d^{2} + 8 \, {\left (a^{3} + 2 \, a^{2} b\right )} d^{3}\right )} \sqrt {\frac {a c e}{d^{2}}} e x^{5} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (a b^{2} c^{2} d - 16 \, a^{2} b c d^{2} + 16 \, a^{3} d^{3}\right )} e x^{6} + a^{3} c^{3} e + {\left (a b^{2} c^{3} - 9 \, a^{2} b c^{2} d + 8 \, a^{3} c d^{2}\right )} e x^{4} + 2 \, {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} e x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{5 \, a^{2} c^{4} x^{5}} \]
1/5*((b^3*c^2*d - 16*a*b^2*c*d^2 + 16*a^2*b*d^3)*sqrt(a*c*e/d^2)*e*x^5*sqr t(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (b^3*c^2*d - (7*a^2* b + 16*a*b^2)*c*d^2 + 8*(a^3 + 2*a^2*b)*d^3)*sqrt(a*c*e/d^2)*e*x^5*sqrt(-b /a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((a*b^2*c^2*d - 16*a^2*b *c*d^2 + 16*a^3*d^3)*e*x^6 + a^3*c^3*e + (a*b^2*c^3 - 9*a^2*b*c^2*d + 8*a^ 3*c*d^2)*e*x^4 + 2*(a^2*b*c^3 - a^3*c^2*d)*e*x^2)*sqrt((b*e*x^2 + a*e)/(d* x^2 + c)))/(a^2*c^4*x^5)
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\text {Timed out} \]
\[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{6}} \,d x } \]
\[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {\left (\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}\right )^{\frac {3}{2}}}{x^{6}} \,d x } \]
Timed out. \[ \int \frac {\left (\frac {e \left (a+b x^2\right )}{c+d x^2}\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}\right )}^{3/2}}{x^6} \,d x \]