Integrand size = 30, antiderivative size = 272 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {(b c-a d) x}{c (d e-c f) \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {\sqrt {f} (2 b c e-a d e-a c f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{c \sqrt {e} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}+\frac {\sqrt {e} (b d e+b c f-2 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{c \sqrt {f} (d e-c f)^2 \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
-(-a*d+b*c)*x/c/(-c*f+d*e)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)+(-2*a*d*f+b*c*f +b*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)*(d*x^2+c)^(1/2)/c/(-c*f+d*e )^2/f^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)-(-a*c*f-a*d*e+ 2*b*c*e)*(1/(1+f*x^2/e))^(1/2)*(1+f*x^2/e)^(1/2)*EllipticE(x*f^(1/2)/e^(1/ 2)/(1+f*x^2/e)^(1/2),(1-d*e/c/f)^(1/2))*f^(1/2)*(d*x^2+c)^(1/2)/c/(-c*f+d* e)^2/e^(1/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)
Result contains complex when optimal does not.
Time = 10.50 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {d}{c}} \left (\sqrt {\frac {d}{c}} x \left (a \left (c^2 f^2+c d f^2 x^2+d^2 e \left (e+f x^2\right )\right )-b c e \left (c f+d \left (e+2 f x^2\right )\right )\right )-i d e (2 b c e-a (d e+c f)) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i (b c-a d) e (-d e+c f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{d e (d e-c f)^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
(Sqrt[d/c]*(Sqrt[d/c]*x*(a*(c^2*f^2 + c*d*f^2*x^2 + d^2*e*(e + f*x^2)) - b *c*e*(c*f + d*(e + 2*f*x^2))) - I*d*e*(2*b*c*e - a*(d*e + c*f))*Sqrt[1 + ( d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticE[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e )] - I*(b*c - a*d)*e*(-(d*e) + c*f)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e ]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(d*e*(d*e - c*f)^2*Sqrt [c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.39 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {402, 25, 400, 313, 320}
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.
| \(\displaystyle \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle -\frac {\int -\frac {c (b e-a f)-(b c-a d) f x^2}{\sqrt {d x^2+c} \left (f x^2+e\right )^{3/2}}dx}{c (d e-c f)}-\frac {x (b c-a d)}{c \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {c (b e-a f)-(b c-a d) f x^2}{\sqrt {d x^2+c} \left (f x^2+e\right )^{3/2}}dx}{c (d e-c f)}-\frac {x (b c-a d)}{c \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)}\) |
| \(\Big \downarrow \) 400 |
| \(\displaystyle \frac {\frac {c (-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}-\frac {f (-a c f-a d e+2 b c e) \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d e-c f}}{c (d e-c f)}-\frac {x (b c-a d)}{c \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {c (-2 a d f+b c f+b d e) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{d e-c f}-\frac {\sqrt {f} \sqrt {c+d x^2} (-a c f-a d e+2 b c e) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{c (d e-c f)}-\frac {x (b c-a d)}{c \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\sqrt {e} \sqrt {c+d x^2} (-2 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {\sqrt {f} \sqrt {c+d x^2} (-a c f-a d e+2 b c e) E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{\sqrt {e} \sqrt {e+f x^2} (d e-c f) \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}}{c (d e-c f)}-\frac {x (b c-a d)}{c \sqrt {c+d x^2} \sqrt {e+f x^2} (d e-c f)}\) |
-(((b*c - a*d)*x)/(c*(d*e - c*f)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) + (-((S qrt[f]*(2*b*c*e - a*d*e - a*c*f)*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f] *x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[e]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/ (c*(e + f*x^2))]*Sqrt[e + f*x^2])) + (Sqrt[e]*(b*d*e + b*c*f - 2*a*d*f)*Sq rt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sq rt[f]*(d*e - c*f)*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]))/ (c*(d*e - c*f))
3.1.46.3.1 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]
Time = 4.98 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.89
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {2 d f \left (-\frac {\left (a c f +a d e -2 b c e \right ) x^{3}}{2 c e \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}-\frac {\left (c^{2} a \,f^{2}+a \,d^{2} e^{2}-b \,c^{2} e f -b c d \,e^{2}\right ) x}{2 c e \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) d f}\right )}{\sqrt {\left (x^{4}+\frac {\left (c f +d e \right ) x^{2}}{d f}+\frac {c e}{d f}\right ) d f}}+\frac {\left (\frac {a}{c e}-\frac {c^{2} a \,f^{2}+a \,d^{2} e^{2}-b \,c^{2} e f -b c d \,e^{2}}{c e \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {d \left (a c f +a d e -2 b c e \right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{c \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(515\) |
| default | \(\frac {\left (\sqrt {-\frac {d}{c}}\, a c d \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, a \,d^{2} e f \,x^{3}-2 \sqrt {-\frac {d}{c}}\, b c d e f \,x^{3}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e f +\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2}+\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b \,c^{2} e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2}-\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a c d e f -\sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a \,d^{2} e^{2}+2 \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) b c d \,e^{2}+\sqrt {-\frac {d}{c}}\, a \,c^{2} f^{2} x +\sqrt {-\frac {d}{c}}\, a \,d^{2} e^{2} x -\sqrt {-\frac {d}{c}}\, b \,c^{2} e f x -\sqrt {-\frac {d}{c}}\, b c d \,e^{2} x \right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{c \sqrt {-\frac {d}{c}}\, e \left (c f -d e \right )^{2} \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(581\) |
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(-2*d*f*(-1/2* (a*c*f+a*d*e-2*b*c*e)/c/e/(c^2*f^2-2*c*d*e*f+d^2*e^2)*x^3-1/2*(a*c^2*f^2+a *d^2*e^2-b*c^2*e*f-b*c*d*e^2)/c/e/(c^2*f^2-2*c*d*e*f+d^2*e^2)/d/f*x)/((x^4 +(c*f+d*e)/d/f*x^2+c*e/d/f)*d*f)^(1/2)+(a/c/e-(a*c^2*f^2+a*d^2*e^2-b*c^2*e *f-b*c*d*e^2)/c/e/(c^2*f^2-2*c*d*e*f+d^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)*EllipticF(x*(-d/ c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))+d*(a*c*f+a*d*e-2*b*c*e)/c/(c^2*f^2-2*c* d*e*f+d^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)*(EllipticF(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/ 2))-EllipticE(x*(-d/c)^(1/2),(-1+(c*f+d*e)/e/d)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 618 vs. \(2 (266) = 532\).
Time = 0.11 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.27 \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=-\frac {{\left (a c^{2} d^{2} e f + {\left (a c d^{3} f^{2} - {\left (2 \, b c d^{3} - a d^{4}\right )} e f\right )} x^{4} - {\left (2 \, b c^{2} d^{2} - a c d^{3}\right )} e^{2} + {\left (a c^{2} d^{2} f^{2} - {\left (2 \, b c d^{3} - a d^{4}\right )} e^{2} - 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} e f\right )} x^{2}\right )} \sqrt {c e} \sqrt {-\frac {d}{c}} E(\arcsin \left (x \sqrt {-\frac {d}{c}}\right )\,|\,\frac {c f}{d e}) + {\left ({\left ({\left (b c^{2} d^{2} + 2 \, b c d^{3} - a d^{4}\right )} e f + {\left (b c^{3} d - 2 \, a c^{2} d^{2} - a c d^{3}\right )} f^{2}\right )} x^{4} + {\left (b c^{3} d + 2 \, b c^{2} d^{2} - a c d^{3}\right )} e^{2} + {\left (b c^{4} - 2 \, a c^{3} d - a c^{2} d^{2}\right )} e f + {\left ({\left (b c^{2} d^{2} + 2 \, b c d^{3} - a d^{4}\right )} e^{2} + 2 \, {\left (b c^{3} d - {\left (a - b\right )} c^{2} d^{2} - a c d^{3}\right )} e f + {\left (b c^{4} - 2 \, a c^{3} d - a c^{2} d^{2}\right )} f^{2}\right )} x^{2}\right )} \sqrt {c e} \sqrt {-\frac {d}{c}} F(\arcsin \left (x \sqrt {-\frac {d}{c}}\right )\,|\,\frac {c f}{d e}) - {\left ({\left (a c^{2} d^{2} f^{2} - {\left (2 \, b c^{2} d^{2} - a c d^{3}\right )} e f\right )} x^{3} - {\left (b c^{3} d e f - a c^{3} d f^{2} + {\left (b c^{2} d^{2} - a c d^{3}\right )} e^{2}\right )} x\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{c^{3} d^{3} e^{4} - 2 \, c^{4} d^{2} e^{3} f + c^{5} d e^{2} f^{2} + {\left (c^{2} d^{4} e^{3} f - 2 \, c^{3} d^{3} e^{2} f^{2} + c^{4} d^{2} e f^{3}\right )} x^{4} + {\left (c^{2} d^{4} e^{4} - c^{3} d^{3} e^{3} f - c^{4} d^{2} e^{2} f^{2} + c^{5} d e f^{3}\right )} x^{2}} \]
-((a*c^2*d^2*e*f + (a*c*d^3*f^2 - (2*b*c*d^3 - a*d^4)*e*f)*x^4 - (2*b*c^2* d^2 - a*c*d^3)*e^2 + (a*c^2*d^2*f^2 - (2*b*c*d^3 - a*d^4)*e^2 - 2*(b*c^2*d ^2 - a*c*d^3)*e*f)*x^2)*sqrt(c*e)*sqrt(-d/c)*elliptic_e(arcsin(x*sqrt(-d/c )), c*f/(d*e)) + (((b*c^2*d^2 + 2*b*c*d^3 - a*d^4)*e*f + (b*c^3*d - 2*a*c^ 2*d^2 - a*c*d^3)*f^2)*x^4 + (b*c^3*d + 2*b*c^2*d^2 - a*c*d^3)*e^2 + (b*c^4 - 2*a*c^3*d - a*c^2*d^2)*e*f + ((b*c^2*d^2 + 2*b*c*d^3 - a*d^4)*e^2 + 2*( b*c^3*d - (a - b)*c^2*d^2 - a*c*d^3)*e*f + (b*c^4 - 2*a*c^3*d - a*c^2*d^2) *f^2)*x^2)*sqrt(c*e)*sqrt(-d/c)*elliptic_f(arcsin(x*sqrt(-d/c)), c*f/(d*e) ) - ((a*c^2*d^2*f^2 - (2*b*c^2*d^2 - a*c*d^3)*e*f)*x^3 - (b*c^3*d*e*f - a* c^3*d*f^2 + (b*c^2*d^2 - a*c*d^3)*e^2)*x)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e)) /(c^3*d^3*e^4 - 2*c^4*d^2*e^3*f + c^5*d*e^2*f^2 + (c^2*d^4*e^3*f - 2*c^3*d ^3*e^2*f^2 + c^4*d^2*e*f^3)*x^4 + (c^2*d^4*e^4 - c^3*d^3*e^3*f - c^4*d^2*e ^2*f^2 + c^5*d*e*f^3)*x^2)
\[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {a + b x^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int { \frac {b x^{2} + a}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {a+b x^2}{\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^{3/2}} \, dx=\int \frac {b\,x^2+a}{{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^{3/2}} \,d x \]