Integrand size = 30, antiderivative size = 381 \[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\frac {\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 d^2 f \sqrt {e+f x^2}}+\frac {(b d e-2 b c f+5 a d f) x \sqrt {c+d x^2} \sqrt {e+f x^2}}{15 d f}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\sqrt {e} \left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{15 d^2 f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}}-\frac {e^{3/2} (b d e+b c f-10 a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{15 d f^{3/2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt {e+f x^2}} \]
1/15*(5*a*d*f*(c*f+d*e)-2*b*(c^2*f^2-c*d*e*f+d^2*e^2))*x*(d*x^2+c)^(1/2)/d ^2/f/(f*x^2+e)^(1/2)-1/15*e^(3/2)*(-10*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))*(d*x^2+c)^(1/2)/d/f^(3/2)/(e*(d*x^2+c)/c/(f*x^2+e))^(1/2)/( f*x^2+e)^(1/2)-1/15*(5*a*d*f*(c*f+d*e)-2*b*(c^2*f^2-c*d*e*f+d^2*e^2))*(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))*e^(1/2)*(d*x^2+c)^(1/2)/d^2/f^(3/2)/(e*(d*x^2+c )/c/(f*x^2+e))^(1/2)/(f*x^2+e)^(1/2)+1/5*b*x*(d*x^2+c)^(3/2)*(f*x^2+e)^(1/ 2)/d+1/15*(5*a*d*f-2*b*c*f+b*d*e)*x*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/d/f
Result contains complex when optimal does not.
Time = 1.60 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.70 \[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\frac {\sqrt {\frac {d}{c}} f x \left (c+d x^2\right ) \left (e+f x^2\right ) \left (b c f+5 a d f+b d \left (e+3 f x^2\right )\right )+i e \left (-5 a d f (d e+c f)+2 b \left (d^2 e^2-c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-i e (-d e+c f) (-2 b d e+b c f+5 a d f) \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )}{15 d \sqrt {\frac {d}{c}} f^2 \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
(Sqrt[d/c]*f*x*(c + d*x^2)*(e + f*x^2)*(b*c*f + 5*a*d*f + b*d*(e + 3*f*x^2 )) + I*e*(-5*a*d*f*(d*e + c*f) + 2*b*(d^2*e^2 - c*d*e*f + c^2*f^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*e*(-(d*e) + c*f)*(-2*b*d*e + b*c*f + 5*a*d*f)*Sqrt[1 + (d*x^2)/ c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)])/(15 *d*Sqrt[d/c]*f^2*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.49 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {403, 25, 403, 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.
| \(\displaystyle \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int -\frac {\sqrt {d x^2+c} \left ((b c-5 a d) e-(b d e-2 b c f+5 a d f) x^2\right )}{\sqrt {f x^2+e}}dx}{5 d}+\frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\int \frac {\sqrt {d x^2+c} \left ((b c-5 a d) e-(b d e-2 b c f+5 a d f) x^2\right )}{\sqrt {f x^2+e}}dx}{5 d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\frac {\int \frac {c e (b d e+b c f-10 a d f)-\left (5 a d f (d e+c f)-2 b \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-2 b c f+b d e)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\frac {c e (-10 a d f+b c f+b d e) \int \frac {1}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx-\left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-2 b c f+b d e)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\frac {\frac {e^{3/2} \sqrt {c+d x^2} (-10 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {f x^2+e}}dx}{3 f}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-2 b c f+b d e)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\frac {\frac {e^{3/2} \sqrt {c+d x^2} (-10 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {e \int \frac {\sqrt {d x^2+c}}{\left (f x^2+e\right )^{3/2}}dx}{d}\right )}{3 f}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-2 b c f+b d e)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b x \left (c+d x^2\right )^{3/2} \sqrt {e+f x^2}}{5 d}-\frac {\frac {\frac {e^{3/2} \sqrt {c+d x^2} (-10 a d f+b c f+b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{\sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\left (5 a d f (c f+d e)-2 b \left (c^2 f^2-c d e f+d^2 e^2\right )\right ) \left (\frac {x \sqrt {c+d x^2}}{d \sqrt {e+f x^2}}-\frac {\sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\right )}{3 f}-\frac {x \sqrt {c+d x^2} \sqrt {e+f x^2} (5 a d f-2 b c f+b d e)}{3 f}}{5 d}\) |
(b*x*(c + d*x^2)^(3/2)*Sqrt[e + f*x^2])/(5*d) - (-1/3*((b*d*e - 2*b*c*f + 5*a*d*f)*x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/f + (-((5*a*d*f*(d*e + c*f) - 2*b*(d^2*e^2 - c*d*e*f + c^2*f^2))*((x*Sqrt[c + d*x^2])/(d*Sqrt[e + f*x^2] ) - (Sqrt[e]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d *e)/(c*f)])/(d*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^ 2]))) + (e^(3/2)*(b*d*e + b*c*f - 10*a*d*f)*Sqrt[c + d*x^2]*EllipticF[ArcT an[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(Sqrt[f]*Sqrt[(e*(c + d*x^2))/( c*(e + f*x^2))]*Sqrt[e + f*x^2]))/(3*f))/(5*d)
3.1.24.3.1 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[(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[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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]
Time = 4.44 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.13
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {b \,x^{3} \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{5}+\frac {\left (a d f +b c f +b d e -\frac {b \left (4 c f +4 d e \right )}{5}\right ) x \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}{3 d f}+\frac {\left (a c e -\frac {\left (a d f +b c f +b d e -\frac {b \left (4 c f +4 d e \right )}{5}\right ) c e}{3 d f}\right ) \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (a c f +a d e +\frac {2 b c e}{5}-\frac {\left (a d f +b c f +b d e -\frac {b \left (4 c f +4 d e \right )}{5}\right ) \left (2 c f +2 d e \right )}{3 d f}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(431\) |
| risch | \(\frac {x \left (3 b d f \,x^{2}+5 a d f +b c f +b d e \right ) \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}{15 d f}+\frac {\left (-\frac {b \,c^{2} e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {b c d \,e^{2} \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {10 a c d e f \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\left (5 a c d \,f^{2}+5 a \,d^{2} e f -2 b \,c^{2} f^{2}+2 b c d e f -2 b \,d^{2} e^{2}\right ) e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}\right ) \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}{15 d f \sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(538\) |
| default | \(\frac {\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, \left (3 \sqrt {-\frac {d}{c}}\, b \,d^{2} f^{3} x^{7}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} f^{3} x^{5}+4 \sqrt {-\frac {d}{c}}\, b c d \,f^{3} x^{5}+4 \sqrt {-\frac {d}{c}}\, b \,d^{2} e \,f^{2} x^{5}+5 \sqrt {-\frac {d}{c}}\, a c d \,f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, a \,d^{2} e \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, b \,c^{2} f^{3} x^{3}+5 \sqrt {-\frac {d}{c}}\, b c d e \,f^{2} x^{3}+\sqrt {-\frac {d}{c}}\, b \,d^{2} e^{2} f \,x^{3}+5 \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^{2}-5 \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} 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^{2} e \,f^{2}-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 ) b c d \,e^{2} f +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 \,d^{2} e^{3}+5 \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^{2}+5 \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} f -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^{2} e \,f^{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} f -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 \,d^{2} e^{3}+5 \sqrt {-\frac {d}{c}}\, a c d e \,f^{2} x +\sqrt {-\frac {d}{c}}\, b \,c^{2} e \,f^{2} x +\sqrt {-\frac {d}{c}}\, b c d \,e^{2} f x \right )}{15 \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right ) d \,f^{2} \sqrt {-\frac {d}{c}}}\) | \(865\) |
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(1/5*b*x^3*(d* f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+1/3*(a*d*f+b*c*f+b*d*e-1/5*b*(4*c*f+4*d*e ))/d/f*x*(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)+(a*c*e-1/3*(a*d*f+b*c*f+b*d*e -1/5*b*(4*c*f+4*d*e))/d/f*c*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)*EllipticF(x*(-d/c)^(1/2),(-1+(c* f+d*e)/e/d)^(1/2))-(a*c*f+a*d*e+2/5*b*c*e-1/3*(a*d*f+b*c*f+b*d*e-1/5*b*(4* c*f+4*d*e))/d/f*(2*c*f+2*d*e))*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)/f*(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)) ))
Time = 0.11 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.80 \[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\frac {{\left (2 \, b d^{2} e^{3} - {\left (2 \, b c d + 5 \, a d^{2}\right )} e^{2} f + {\left (2 \, b c^{2} - 5 \, a c d\right )} e f^{2}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} E(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) - {\left (2 \, b d^{2} e^{3} - {\left (2 \, b c d + 5 \, a d^{2}\right )} e^{2} f + {\left (2 \, b c^{2} - {\left (5 \, a - b\right )} c d\right )} e f^{2} + {\left (b c^{2} - 10 \, a c d\right )} f^{3}\right )} \sqrt {d f} x \sqrt {-\frac {e}{f}} F(\arcsin \left (\frac {\sqrt {-\frac {e}{f}}}{x}\right )\,|\,\frac {c f}{d e}) + {\left (3 \, b d^{2} f^{3} x^{4} - 2 \, b d^{2} e^{2} f + {\left (2 \, b c d + 5 \, a d^{2}\right )} e f^{2} - {\left (2 \, b c^{2} - 5 \, a c d\right )} f^{3} + {\left (b d^{2} e f^{2} + {\left (b c d + 5 \, a d^{2}\right )} f^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}}{15 \, d^{2} f^{3} x} \]
1/15*((2*b*d^2*e^3 - (2*b*c*d + 5*a*d^2)*e^2*f + (2*b*c^2 - 5*a*c*d)*e*f^2 )*sqrt(d*f)*x*sqrt(-e/f)*elliptic_e(arcsin(sqrt(-e/f)/x), c*f/(d*e)) - (2* b*d^2*e^3 - (2*b*c*d + 5*a*d^2)*e^2*f + (2*b*c^2 - (5*a - b)*c*d)*e*f^2 + (b*c^2 - 10*a*c*d)*f^3)*sqrt(d*f)*x*sqrt(-e/f)*elliptic_f(arcsin(sqrt(-e/f )/x), c*f/(d*e)) + (3*b*d^2*f^3*x^4 - 2*b*d^2*e^2*f + (2*b*c*d + 5*a*d^2)* e*f^2 - (2*b*c^2 - 5*a*c*d)*f^3 + (b*d^2*e*f^2 + (b*c*d + 5*a*d^2)*f^3)*x^ 2)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e))/(d^2*f^3*x)
\[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\int \left (a + b x^{2}\right ) \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}\, dx \]
\[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\int { {\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} \,d x } \]
\[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\int { {\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e} \,d x } \]
Timed out. \[ \int \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2} \, dx=\int \left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e} \,d x \]