Integrand size = 26, antiderivative size = 420 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\frac {\left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{15 a^2 c^2 x}-\frac {\left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{5 c x^5}-\frac {(b c-4 a d) \left (c+d x^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}}}{15 a c^2 x^3}+\frac {\sqrt {d} \left (2 b^2 c^2+3 a b c d-8 a^2 d^2\right ) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 a^2 c^{5/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {d} (b c-4 a d) \sqrt {\frac {b e}{d}-\frac {(b c-a d) e}{d \left (c+d x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 a^2 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \] Output:
1/15*(-8*a^2*d^2+3*a*b*c*d+2*b^2*c^2)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/ 2)/a^2/c^2/x-1/5*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/c/x^5-1/ 15*(-4*a*d+b*c)*(d*x^2+c)*(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)/a/c^2/x^3 +1/15*d^(1/2)*(-8*a^2*d^2+3*a*b*c*d+2*b^2*c^2)*(b*e/d-(-a*d+b*c)*e/d/(d*x^ 2+c))^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2 ))/a^2/c^(5/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)-1/15*b*d^(1/2)*(-4*a*d+b*c) *(b*e/d-(-a*d+b*c)*e/d/(d*x^2+c))^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c ^(1/2)),(1-b*c/a/d)^(1/2))/a^2/c^(3/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 2.89 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=-\frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 b^2 c^2 x^4+a b c x^2 \left (c-3 d x^2\right )+a^2 \left (3 c^2-4 c d x^2+8 d^2 x^4\right )\right )+i b c \left (-2 b^2 c^2-3 a b c d+8 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-2 i b c \left (-b^2 c^2-a b c d+2 a^2 d^2\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 a^2 \sqrt {\frac {b}{a}} c^3 x^5 \left (a+b x^2\right )} \] Input:
Integrate[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^6,x]
Output:
-1/15*(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2 )*(-2*b^2*c^2*x^4 + a*b*c*x^2*(c - 3*d*x^2) + a^2*(3*c^2 - 4*c*d*x^2 + 8*d ^2*x^4)) + I*b*c*(-2*b^2*c^2 - 3*a*b*c*d + 8*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)] - ( 2*I)*b*c*(-(b^2*c^2) - a*b*c*d + 2*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)]))/(a^2*Sqrt[b /a]*c^3*x^5*(a + b*x^2))
Time = 1.12 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.03, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2058, 377, 445, 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.
\(\displaystyle \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \int \frac {\sqrt {b x^2+a}}{x^6 \sqrt {d x^2+c}}dx}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 377 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {\int \frac {-3 b d x^2+b c-4 a d}{x^4 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 445 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\int \frac {2 b^2 c^2+3 a b d c-8 a^2 d^2+b d (b c-4 a d) x^2}{x^2 \sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 445 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {-\frac {\int -\frac {b d \left (\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {\int \frac {b d \left (\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \int \frac {\left (2 b^2 c^2+3 a b d c-8 a^2 d^2\right ) x^2+a c (b c-4 a d)}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 406 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c (b c-4 a d) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}} \left (\frac {-\frac {\frac {b d \left (\left (-8 a^2 d^2+3 a b c d+2 b^2 c^2\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )+\frac {c^{3/2} \sqrt {a+b x^2} (b c-4 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} \left (\frac {2 b^2 c}{a}-\frac {8 a d^2}{c}+3 b d\right )}{x}}{3 a c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-4 a d)}{3 a c x^3}}{5 c}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{5 c x^5}\right )}{\sqrt {a+b x^2}}\) |
Input:
Int[Sqrt[(e*(a + b*x^2))/(c + d*x^2)]/x^6,x]
Output:
(Sqrt[(e*(a + b*x^2))/(c + d*x^2)]*Sqrt[c + d*x^2]*(-1/5*(Sqrt[a + b*x^2]* Sqrt[c + d*x^2])/(c*x^5) + (-1/3*((b*c - 4*a*d)*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(a*c*x^3) - (-((((2*b^2*c)/a + 3*b*d - (8*a*d^2)/c)*Sqrt[a + b*x^2] *Sqrt[c + d*x^2])/x) + (b*d*((2*b^2*c^2 + 3*a*b*c*d - 8*a^2*d^2)*((x*Sqrt[ a + b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTa n[(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)*(b*c - 4*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*a*c))/(5*c))) /Sqrt[a + b*x^2]
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_.)*((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 = 8.54 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {\left (d \,x^{2}+c \right ) \left (8 a^{2} d^{2} x^{4}-3 a b c d \,x^{4}-2 b^{2} c^{2} x^{4}-4 a^{2} c d \,x^{2}+a b \,c^{2} x^{2}+3 a^{2} c^{2}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}}{15 c^{3} x^{5} a^{2}}+\frac {d b \left (-\frac {2 \left (8 a^{2} d^{2}-3 a b c d -2 b^{2} c^{2}\right ) a c e \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}\, \left (a d e +b c e +e \left (a d -b c \right )\right )}-\frac {a b \,c^{2} \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}+\frac {4 a^{2} c d \sqrt {1+\frac {x^{2} b}{a}}\, \sqrt {1+\frac {x^{2} d}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d e +b c e}{c b e}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d e \,x^{4}+a d e \,x^{2}+b c e \,x^{2}+a c e}}\right ) \sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right ) e}}{15 a^{2} c^{3} \left (b \,x^{2}+a \right )}\) | \(539\) |
default | \(-\frac {\sqrt {\frac {e \left (b \,x^{2}+a \right )}{d \,x^{2}+c}}\, \left (d \,x^{2}+c \right ) \left (8 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{3} x^{8}-3 \sqrt {-\frac {b}{a}}\, a \,b^{2} c \,d^{2} x^{8}-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} d \,x^{8}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{5}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{5}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{5}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b c \,d^{2} x^{5}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{2} c^{2} d \,x^{5}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{3} c^{3} x^{5}+8 \sqrt {-\frac {b}{a}}\, a^{3} d^{3} x^{6}+\sqrt {-\frac {b}{a}}\, a^{2} b c \,d^{2} x^{6}-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} d \,x^{6}-2 \sqrt {-\frac {b}{a}}\, b^{3} c^{3} x^{6}+4 \sqrt {-\frac {b}{a}}\, a^{3} c \,d^{2} x^{4}-3 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{2} d \,x^{4}-\sqrt {-\frac {b}{a}}\, a \,b^{2} c^{3} x^{4}-\sqrt {-\frac {b}{a}}\, a^{3} c^{2} d \,x^{2}+4 \sqrt {-\frac {b}{a}}\, a^{2} b \,c^{3} x^{2}+3 \sqrt {-\frac {b}{a}}\, a^{3} c^{3}\right )}{15 \sqrt {\left (d \,x^{2}+c \right ) \left (b \,x^{2}+a \right )}\, c^{3} x^{5} a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {d b \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}}\) | \(708\) |
Input:
int((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/15*(d*x^2+c)*(8*a^2*d^2*x^4-3*a*b*c*d*x^4-2*b^2*c^2*x^4-4*a^2*c*d*x^2+a *b*c^2*x^2+3*a^2*c^2)/c^3/x^5/a^2*(e*(b*x^2+a)/(d*x^2+c))^(1/2)+1/15/a^2*d *b/c^3*(-2*(8*a^2*d^2-3*a*b*c*d-2*b^2*c^2)*a*c*e/(-b/a)^(1/2)*(1+1/a*x^2*b )^(1/2)*(1+1/c*x^2*d)^(1/2)/(b*d*e*x^4+a*d*e*x^2+b*c*e*x^2+a*c*e)^(1/2)/(a *d*e+b*c*e+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)))-a*b*c^2/ (-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*(1+1/c*x^2*d)^(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))+4*a^2*c*d/(-b/a)^(1/2)*(1+1/a*x^2*b)^(1/2)*(1+1/c*x^2*d)^(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)))*(e*(b*x^2+a)/(d*x^2+c))^(1/2)*((d*x^2+c)*(b*x^2+a)* e)^(1/2)/(b*x^2+a)
Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=-\frac {{\left (2 \, b^{3} c^{2} d + 3 \, a b^{2} c d^{2} - 8 \, a^{2} b d^{3}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{5} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, b^{3} c^{2} d + {\left (a^{2} b + 3 \, a b^{2}\right )} c d^{2} - 4 \, {\left (a^{3} + 2 \, a^{2} b\right )} d^{3}\right )} \sqrt {\frac {a c e}{d^{2}}} x^{5} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - 8 \, a^{3} d^{3}\right )} x^{6} - 3 \, a^{3} c^{3} + 2 \, {\left (a b^{2} c^{3} + a^{2} b c^{2} d - 2 \, a^{3} c d^{2}\right )} x^{4} - {\left (a^{2} b c^{3} - a^{3} c^{2} d\right )} x^{2}\right )} \sqrt {\frac {b e x^{2} + a e}{d x^{2} + c}}}{15 \, a^{3} c^{3} x^{5}} \] Input:
integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x, algorithm="fricas")
Output:
-1/15*((2*b^3*c^2*d + 3*a*b^2*c*d^2 - 8*a^2*b*d^3)*sqrt(a*c*e/d^2)*x^5*sqr t(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*b^3*c^2*d + (a^2* b + 3*a*b^2)*c*d^2 - 4*(a^3 + 2*a^2*b)*d^3)*sqrt(a*c*e/d^2)*x^5*sqrt(-b/a) *elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((2*a*b^2*c^2*d + 3*a^2*b*c *d^2 - 8*a^3*d^3)*x^6 - 3*a^3*c^3 + 2*(a*b^2*c^3 + a^2*b*c^2*d - 2*a^3*c*d ^2)*x^4 - (a^2*b*c^3 - a^3*c^2*d)*x^2)*sqrt((b*e*x^2 + a*e)/(d*x^2 + c)))/ (a^3*c^3*x^5)
Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\text {Timed out} \] Input:
integrate((e*(b*x**2+a)/(d*x**2+c))**(1/2)/x**6,x)
Output:
Timed out
\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\int { \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{6}} \,d x } \] Input:
integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x, algorithm="maxima")
Output:
integrate(sqrt((b*x^2 + a)*e/(d*x^2 + c))/x^6, x)
\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\int { \frac {\sqrt {\frac {{\left (b x^{2} + a\right )} e}{d x^{2} + c}}}{x^{6}} \,d x } \] Input:
integrate((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x, algorithm="giac")
Output:
integrate(sqrt((b*x^2 + a)*e/(d*x^2 + c))/x^6, x)
Timed out. \[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\int \frac {\sqrt {\frac {e\,\left (b\,x^2+a\right )}{d\,x^2+c}}}{x^6} \,d x \] Input:
int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^6,x)
Output:
int(((e*(a + b*x^2))/(c + d*x^2))^(1/2)/x^6, x)
\[ \int \frac {\sqrt {\frac {e \left (a+b x^2\right )}{c+d x^2}}}{x^6} \, dx=\sqrt {e}\, \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{d \,x^{8}+c \,x^{6}}d x \right ) \] Input:
int((e*(b*x^2+a)/(d*x^2+c))^(1/2)/x^6,x)
Output:
sqrt(e)*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(c*x**6 + d*x**8),x)