Integrand size = 32, antiderivative size = 435 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=-\frac {d^2 x \sqrt {e+f x^2}}{3 c (b c-a d) (d e-c f) \left (c+d x^2\right )^{3/2}}-\frac {d^{3/2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{3 c^{3/2} (b c-a d)^2 (d e-c f)^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}-\frac {d \sqrt {e} \sqrt {f} (a d (d e-3 c f)-2 b c (2 d e-3 c f)) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{3 c^2 (b c-a d)^2 (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {b^2 \sqrt {-c} \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} (b c-a d)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
-1/3*d*(a*d*(-3*c*f+d*e)-2*b*c*(-3*c*f+2*d*e))*(1/(1+f*x^2/e))^(1/2)*(1+f* x^2/e)^(1/2)*EllipticF(x*f^(1/2)/e^(1/2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/ 2))*e^(1/2)*f^(1/2)*(d*x^2+c)^(1/2)/c^2/(-a*d+b*c)^2/(-c*f+d*e)^2/(e*(d*x^ 2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-1/3*d^2*x*(f*x^2+e)^(1/2)/c/(-a*d+ b*c)/(-c*f+d*e)/(d*x^2+c)^(3/2)-1/3*d^(3/2)*(b*c*(-7*c*f+5*d*e)-2*a*d*(-2* c*f+d*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-c*f/d/e)^(1/2))*(f*x^2+e)^(1/2)/c^(3/2)/(-a*d+b*c )^2/(-c*f+d*e)^2/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)+b^2*Ellip ticPi(x*d^(1/2)/(-c)^(1/2),b*c/a/d,(c*f/d/e)^(1/2))*(-c)^(1/2)*(1+d*x^2/c) ^(1/2)*(1+f*x^2/e)^(1/2)/a/(-a*d+b*c)^2/d^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^ (1/2)
Result contains complex when optimal does not.
Time = 6.33 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\frac {a c d \left (\frac {d}{c}\right )^{3/2} x \left (e+f x^2\right ) \left (b c \left (-6 c d e+8 c^2 f-5 d^2 e x^2+7 c d f x^2\right )+a d \left (-5 c^2 f+2 d^2 e x^2+c d \left (3 e-4 f x^2\right )\right )\right )+i a d^2 e (2 a d (d e-2 c f)+b c (-5 d e+7 c f)) \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )+i a d (-d e+c f) (a d (2 d e-3 c f)+b c (-5 d e+6 c f)) \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )-3 i b^2 c^2 (d e-c f)^2 \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{3 a c^2 \sqrt {\frac {d}{c}} (b c-a d)^2 (d e-c f)^2 \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}} \]
(a*c*d*(d/c)^(3/2)*x*(e + f*x^2)*(b*c*(-6*c*d*e + 8*c^2*f - 5*d^2*e*x^2 + 7*c*d*f*x^2) + a*d*(-5*c^2*f + 2*d^2*e*x^2 + c*d*(3*e - 4*f*x^2))) + I*a*d ^2*e*(2*a*d*(d*e - 2*c*f) + b*c*(-5*d*e + 7*c*f))*(c + d*x^2)*Sqrt[1 + (d* x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)] + I*a*d*(-(d*e) + c*f)*(a*d*(2*d*e - 3*c*f) + b*c*(-5*d*e + 6*c*f))*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/ c]*x], (c*f)/(d*e)] - (3*I)*b^2*c^2*(d*e - c*f)^2*(c + d*x^2)*Sqrt[1 + (d* x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b*c)/(a*d), I*ArcSinh[Sqrt[d/c]*x] , (c*f)/(d*e)])/(3*a*c^2*Sqrt[d/c]*(b*c - a*d)^2*(d*e - c*f)^2*(c + d*x^2) ^(3/2)*Sqrt[e + f*x^2])
Time = 0.69 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {421, 402, 25, 400, 313, 320, 413, 413, 412}
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.
| \(\displaystyle \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 421 |
| \(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{(b c-a d)^2}-\frac {d \int \frac {b d x^2+2 b c-a d}{\left (d x^2+c\right )^{5/2} \sqrt {f x^2+e}}dx}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{(b c-a d)^2}-\frac {d \left (\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}-\frac {\int -\frac {d (b c-a d) f x^2+b c (5 d e-6 c f)-a d (2 d e-3 c f)}{\left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}}dx}{3 c (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{(b c-a d)^2}-\frac {d \left (\frac {\int \frac {d (b c-a d) f x^2+b c (5 d e-6 c f)-a d (2 d e-3 c f)}{\left (d x^2+c\right )^{3/2} \sqrt {f x^2+e}}dx}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{(b c-a d)^2}-\frac {d \left (\frac {\frac {f \left (a d (d e-3 c f)-b \left (4 c d e-6 c^2 f\right )\right ) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}+\frac {d (b c (5 d e-7 c f)-2 a d (d e-2 c f)) \int \frac {\sqrt {f x^2+e}}{\left (d x^2+c\right )^{3/2}}dx}{d e-c f}}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{(b c-a d)^2}-\frac {d \left (\frac {\frac {f \left (a d (d e-3 c f)-b \left (4 c d e-6 c^2 f\right )\right ) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}+\frac {\sqrt {d} \sqrt {e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b^2 \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{(b c-a d)^2}-\frac {d \left (\frac {\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} \left (a d (d e-3 c f)-b \left (4 c d e-6 c^2 f\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b^2 \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\left (b x^2+a\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {f x^2+e}}dx}{\sqrt {c+d x^2} (b c-a d)^2}-\frac {d \left (\frac {\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} \left (a d (d e-3 c f)-b \left (4 c d e-6 c^2 f\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {b^2 \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \int \frac {1}{\left (b x^2+a\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1}}dx}{\sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)^2}-\frac {d \left (\frac {\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} \left (a d (d e-3 c f)-b \left (4 c d e-6 c^2 f\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {b^2 \sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \operatorname {EllipticPi}\left (\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2} (b c-a d)^2}-\frac {d \left (\frac {\frac {\sqrt {e} \sqrt {f} \sqrt {c+d x^2} \left (a d (d e-3 c f)-b \left (4 c d e-6 c^2 f\right )\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {\sqrt {d} \sqrt {e+f x^2} (b c (5 d e-7 c f)-2 a d (d e-2 c f)) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{\sqrt {c} \sqrt {c+d x^2} (d e-c f) \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}}{3 c (d e-c f)}+\frac {d x \sqrt {e+f x^2} (b c-a d)}{3 c \left (c+d x^2\right )^{3/2} (d e-c f)}\right )}{(b c-a d)^2}\) |
-((d*((d*(b*c - a*d)*x*Sqrt[e + f*x^2])/(3*c*(d*e - c*f)*(c + d*x^2)^(3/2) ) + ((Sqrt[d]*(b*c*(5*d*e - 7*c*f) - 2*a*d*(d*e - 2*c*f))*Sqrt[e + f*x^2]* EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (c*f)/(d*e)])/(Sqrt[c]*(d*e - c *f)*Sqrt[c + d*x^2]*Sqrt[(c*(e + f*x^2))/(e*(c + d*x^2))]) + (Sqrt[e]*Sqrt [f]*(a*d*(d*e - 3*c*f) - b*(4*c*d*e - 6*c^2*f))*Sqrt[c + d*x^2]*EllipticF[ ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(c*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]))/(3*c*(d*e - c*f))))/(b*c - a*d )^2) + (b^2*Sqrt[-c]*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[(b *c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*(b*c - a *d)^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
3.1.82.3.1 Defintions of rubi rules used
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_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ (3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(Sqrt[a + b*x^2]* Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d) Int[Sqrt[a + b*x^ 2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & & PosQ[d/c]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[(((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_)^2)^(r_))/((a_) + (b_.)*( x_)^2), x_Symbol] :> Simp[b^2/(b*c - a*d)^2 Int[(c + d*x^2)^(q + 2)*((e + f*x^2)^r/(a + b*x^2)), x], x] - Simp[d/(b*c - a*d)^2 Int[(c + d*x^2)^q*( e + f*x^2)^r*(2*b*c - a*d + b*d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, r} , x] && LtQ[q, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(1324\) vs. \(2(460)=920\).
Time = 6.55 (sec) , antiderivative size = 1325, normalized size of antiderivative = 3.05
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1325\) |
| default | \(\text {Expression too large to display}\) | \(2062\) |
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(-1/3/c/(c*f-d *e)*x/(a*d-b*c)*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)/(x^2+c/d)^2-1/3*(d*f*x ^2+d*e)*d/c^2/(c*f-d*e)^2*x*(4*a*c*d*f-2*a*d^2*e-7*b*c^2*f+5*b*c*d*e)/(a*d -b*c)^2/((x^2+c/d)*(d*f*x^2+d*e))^(1/2)-1/3/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2) *(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^ (1/2),(-1+(c*f+d*e)/e/d)^(1/2))*d*f/c/(c*f-d*e)/(a*d-b*c)+4/3/(-d/c)^(1/2) *(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*E llipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))/(c*f-d*e)*d^2/c/(a*d-b*c )^2*a*f-2/3/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f* x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))/ (c*f-d*e)*d^3/c^2/(a*d-b*c)^2*a*e-7/3/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f* x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticF(x*(-d/c)^(1/2), (-1+(c*f+d*e)/e/d)^(1/2))/(c*f-d*e)*d/(a*d-b*c)^2*b*f+5/3/(-d/c)^(1/2)*(1+ d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*Ellip ticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))/(c*f-d*e)*d^2/c/(a*d-b*c)^2* b*e+4/3*d^3/(c*f-d*e)^2/c/(a*d-b*c)^2*e/(-d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+ f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticE(x*(-d/c)^(1/2 ),(-1+(c*f+d*e)/e/d)^(1/2))*a*f-2/3*d^4/(c*f-d*e)^2/c^2/(a*d-b*c)^2*e^2/(- d/c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c* e)^(1/2)*EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))*a-7/3*d^2/(...
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right ) \left (c + d x^{2}\right )^{\frac {5}{2}} \sqrt {e + f x^{2}}}\, dx \]
\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \]
\[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}} \sqrt {f x^{2} + e}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \left (c+d x^2\right )^{5/2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (b\,x^2+a\right )\,{\left (d\,x^2+c\right )}^{5/2}\,\sqrt {f\,x^2+e}} \,d x \]