Integrand size = 30, antiderivative size = 356 \[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\frac {\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) x \sqrt {3+f x^2}}{15 d f^2 \sqrt {2+d x^2}}-\frac {(6 b d-2 b f-5 a d f) x \sqrt {2+d x^2} \sqrt {3+f x^2}}{15 d f}+\frac {b x \sqrt {2+d x^2} \left (3+f x^2\right )^{3/2}}{5 f}-\frac {\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \sqrt {3+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|1-\frac {2 f}{3 d}\right )}{5 \sqrt {3} d^{3/2} f^2 \sqrt {2+d x^2} \sqrt {\frac {3+f x^2}{2+d x^2}}}-\frac {2 (3 b d+2 b f-10 a d f) \sqrt {3+f x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {2}}\right ),1-\frac {2 f}{3 d}\right )}{5 \sqrt {3} d^{3/2} f \sqrt {2+d x^2} \sqrt {\frac {3+f x^2}{2+d x^2}}} \] Output:
1/15*(5*a*d*f*(3*d+2*f)-2*b*(9*d^2-6*d*f+4*f^2))*x*(f*x^2+3)^(1/2)/d/f^2/( d*x^2+2)^(1/2)-1/15*(-5*a*d*f+6*b*d-2*b*f)*x*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/ 2)/d/f+1/5*b*x*(d*x^2+2)^(1/2)*(f*x^2+3)^(3/2)/f-1/15*(5*a*d*f*(3*d+2*f)-2 *b*(9*d^2-6*d*f+4*f^2))*(f*x^2+3)^(1/2)*EllipticE(d^(1/2)*x*2^(1/2)/(2*d*x ^2+4)^(1/2),1/3*(9-6*f/d)^(1/2))*3^(1/2)/d^(3/2)/f^2/(d*x^2+2)^(1/2)/((f*x ^2+3)/(d*x^2+2))^(1/2)-2/15*(-10*a*d*f+3*b*d+2*b*f)*(f*x^2+3)^(1/2)*Invers eJacobiAM(arctan(1/2*d^(1/2)*x*2^(1/2)),1/3*(9-6*f/d)^(1/2))*3^(1/2)/d^(3/ 2)/f/(d*x^2+2)^(1/2)/((f*x^2+3)/(d*x^2+2))^(1/2)
Result contains complex when optimal does not.
Time = 1.61 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.52 \[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\frac {\sqrt {d} f x \sqrt {2+d x^2} \sqrt {3+f x^2} \left (2 b f+5 a d f+3 b d \left (1+f x^2\right )\right )+i \sqrt {3} \left (-5 a d f (3 d+2 f)+2 b \left (9 d^2-6 d f+4 f^2\right )\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+i \sqrt {3} (3 d-2 f) (-6 b d+2 b f+5 a d f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right ),\frac {2 f}{3 d}\right )}{15 d^{3/2} f^2} \] Input:
Integrate[(a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2],x]
Output:
(Sqrt[d]*f*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]*(2*b*f + 5*a*d*f + 3*b*d*(1 + f*x^2)) + I*Sqrt[3]*(-5*a*d*f*(3*d + 2*f) + 2*b*(9*d^2 - 6*d*f + 4*f^2))* EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + I*Sqrt[3]*(3*d - 2*f)*(-6*b*d + 2*b*f + 5*a*d*f)*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)])/(15*d^(3/2)*f^2)
Time = 0.44 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.92, 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.
| \(\displaystyle \int \sqrt {d x^2+2} \sqrt {f x^2+3} \left (a+b x^2\right ) \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int -\frac {\sqrt {d x^2+2} \left (3 (2 b-5 a d)-(3 b d+5 a f d-4 b f) x^2\right )}{\sqrt {f x^2+3}}dx}{5 d}+\frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}-\frac {\int \frac {\sqrt {d x^2+2} \left (3 (2 b-5 a d)-(3 b d+5 a f d-4 b f) x^2\right )}{\sqrt {f x^2+3}}dx}{5 d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}-\frac {\frac {\int \frac {6 (3 b d-10 a f d+2 b f)-\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 f d+4 f^2\right )\right ) x^2}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx}{3 f}-\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}-\frac {\frac {6 (-10 a d f+3 b d+2 b f) \int \frac {1}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx-\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \int \frac {x^2}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx}{3 f}-\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}-\frac {\frac {\frac {3 \sqrt {2} \sqrt {d x^2+2} (-10 a d f+3 b d+2 b f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{\sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \int \frac {x^2}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx}{3 f}-\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}-\frac {\frac {\frac {3 \sqrt {2} \sqrt {d x^2+2} (-10 a d f+3 b d+2 b f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{\sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \left (\frac {x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}}-\frac {3 \int \frac {\sqrt {d x^2+2}}{\left (f x^2+3\right )^{3/2}}dx}{d}\right )}{3 f}-\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{3 f}}{5 d}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b x \left (d x^2+2\right )^{3/2} \sqrt {f x^2+3}}{5 d}-\frac {\frac {\frac {3 \sqrt {2} \sqrt {d x^2+2} (-10 a d f+3 b d+2 b f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{\sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}-\left (5 a d f (3 d+2 f)-2 b \left (9 d^2-6 d f+4 f^2\right )\right ) \left (\frac {x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}}-\frac {\sqrt {2} \sqrt {d x^2+2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}\right )}{3 f}-\frac {x \sqrt {d x^2+2} \sqrt {f x^2+3} (5 a d f+3 b d-4 b f)}{3 f}}{5 d}\) |
Input:
Int[(a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2],x]
Output:
(b*x*(2 + d*x^2)^(3/2)*Sqrt[3 + f*x^2])/(5*d) - (-1/3*((3*b*d - 4*b*f + 5* a*d*f)*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/f + (-((5*a*d*f*(3*d + 2*f) - 2* b*(9*d^2 - 6*d*f + 4*f^2))*((x*Sqrt[2 + d*x^2])/(d*Sqrt[3 + f*x^2]) - (Sqr t[2]*Sqrt[2 + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f )])/(d*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]))) + (3*Sqrt[ 2]*(3*b*d + 2*b*f - 10*a*d*f)*Sqrt[2 + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x) /Sqrt[3]], 1 - (3*d)/(2*f)])/(Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3 + f*x^2]))/(3*f))/(5*d)
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 = 4.93 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.10
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (x^{2} d +2\right )}\, \left (\frac {b \,x^{3} \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}{5}+\frac {\left (a d f +3 b d +2 f b -\frac {b \left (12 d +8 f \right )}{5}\right ) x \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}{3 d f}+\frac {\left (6 a -\frac {2 \left (a d f +3 b d +2 f b -\frac {b \left (12 d +8 f \right )}{5}\right )}{d f}\right ) \sqrt {3 f \,x^{2}+9}\, \sqrt {2 x^{2} d +4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{2 \sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}-\frac {\left (3 a d +2 a f +\frac {12 b}{5}-\frac {\left (a d f +3 b d +2 f b -\frac {b \left (12 d +8 f \right )}{5}\right ) \left (6 d +4 f \right )}{3 d f}\right ) \sqrt {3 f \,x^{2}+9}\, \sqrt {2 x^{2} d +4}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}\, d}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {x^{2} d +2}}\) | \(390\) |
| risch | \(\frac {x \left (3 b d f \,x^{2}+5 a d f +3 b d +2 f b \right ) \sqrt {f \,x^{2}+3}\, \sqrt {x^{2} d +2}}{15 d f}+\frac {\left (-\frac {\left (15 a \,d^{2} f +10 a d \,f^{2}-18 b \,d^{2}+12 b d f -8 b \,f^{2}\right ) \sqrt {3 f \,x^{2}+9}\, \sqrt {2 x^{2} d +4}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}\, d}-\frac {9 b d \sqrt {3 f \,x^{2}+9}\, \sqrt {2 x^{2} d +4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}-\frac {6 f b \sqrt {3 f \,x^{2}+9}\, \sqrt {2 x^{2} d +4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}+\frac {30 a d f \sqrt {3 f \,x^{2}+9}\, \sqrt {2 x^{2} d +4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}\right ) \sqrt {\left (f \,x^{2}+3\right ) \left (x^{2} d +2\right )}}{15 d f \sqrt {f \,x^{2}+3}\, \sqrt {x^{2} d +2}}\) | \(471\) |
| default | \(\frac {\sqrt {f \,x^{2}+3}\, \sqrt {x^{2} d +2}\, \left (3 b \,d^{3} f^{2} x^{7} \sqrt {-f}+5 a \,d^{3} f^{2} x^{5} \sqrt {-f}+12 b \,d^{3} f \,x^{5} \sqrt {-f}+8 b \,d^{2} f^{2} x^{5} \sqrt {-f}+15 a \,d^{3} f \,x^{3} \sqrt {-f}+10 a \,d^{2} f^{2} x^{3} \sqrt {-f}+15 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) a \,d^{2} f \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}-10 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) a d \,f^{2} \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}+15 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) a \,d^{2} f \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}+10 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) a d \,f^{2} \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}+9 b \,d^{3} x^{3} \sqrt {-f}+30 b \,d^{2} f \,x^{3} \sqrt {-f}+4 b d \,f^{2} x^{3} \sqrt {-f}+9 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) b \,d^{2} \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}-18 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) b d f \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}+8 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) b \,f^{2} \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}-18 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) b \,d^{2} \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}+12 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) b d f \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}-8 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {d}{f}}}{2}\right ) b \,f^{2} \sqrt {x^{2} d +2}\, \sqrt {f \,x^{2}+3}+30 a \,d^{2} f x \sqrt {-f}+18 b \,d^{2} x \sqrt {-f}+12 b d f x \sqrt {-f}\right )}{15 \left (d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6\right ) d^{2} f \sqrt {-f}}\) | \(775\) |
Input:
int((b*x^2+a)*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
((f*x^2+3)*(d*x^2+2))^(1/2)/(f*x^2+3)^(1/2)/(d*x^2+2)^(1/2)*(1/5*b*x^3*(d* f*x^4+3*d*x^2+2*f*x^2+6)^(1/2)+1/3*(a*d*f+3*b*d+2*f*b-1/5*b*(12*d+8*f))/d/ f*x*(d*f*x^4+3*d*x^2+2*f*x^2+6)^(1/2)+1/2*(6*a-2*(a*d*f+3*b*d+2*f*b-1/5*b* (12*d+8*f))/d/f)/(-3*f)^(1/2)*(3*f*x^2+9)^(1/2)*(2*d*x^2+4)^(1/2)/(d*f*x^4 +3*d*x^2+2*f*x^2+6)^(1/2)*EllipticF(1/3*x*(-3*f)^(1/2),1/2*(-4+2*(3*d+2*f) /f)^(1/2))-(3*a*d+2*a*f+12/5*b-1/3*(a*d*f+3*b*d+2*f*b-1/5*b*(12*d+8*f))/d/ f*(6*d+4*f))/(-3*f)^(1/2)*(3*f*x^2+9)^(1/2)*(2*d*x^2+4)^(1/2)/(d*f*x^4+3*d *x^2+2*f*x^2+6)^(1/2)/d*(EllipticF(1/3*x*(-3*f)^(1/2),1/2*(-4+2*(3*d+2*f)/ f)^(1/2))-EllipticE(1/3*x*(-3*f)^(1/2),1/2*(-4+2*(3*d+2*f)/f)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.77 \[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\frac {3 \, \sqrt {3} {\left (18 \, b d^{2} - 2 \, {\left (5 \, a d - 4 \, b\right )} f^{2} - 3 \, {\left (5 \, a d^{2} + 4 \, b d\right )} f\right )} \sqrt {d f} x \sqrt {-\frac {1}{f}} E(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{f}}}{x}\right )\,|\,\frac {2 \, f}{3 \, d}) + \sqrt {3} {\left (4 \, {\left (5 \, a d - b\right )} f^{3} - 54 \, b d^{2} + 6 \, {\left ({\left (5 \, a - b\right )} d - 4 \, b\right )} f^{2} + 9 \, {\left (5 \, a d^{2} + 4 \, b d\right )} f\right )} \sqrt {d f} x \sqrt {-\frac {1}{f}} F(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{f}}}{x}\right )\,|\,\frac {2 \, f}{3 \, d}) + {\left (3 \, b d^{2} f^{3} x^{4} - 18 \, b d^{2} f + 2 \, {\left (5 \, a d - 4 \, b\right )} f^{3} + 3 \, {\left (5 \, a d^{2} + 4 \, b d\right )} f^{2} + {\left (3 \, b d^{2} f^{2} + {\left (5 \, a d^{2} + 2 \, b d\right )} f^{3}\right )} x^{2}\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{15 \, d^{2} f^{3} x} \] Input:
integrate((b*x^2+a)*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2),x, algorithm="fricas")
Output:
1/15*(3*sqrt(3)*(18*b*d^2 - 2*(5*a*d - 4*b)*f^2 - 3*(5*a*d^2 + 4*b*d)*f)*s qrt(d*f)*x*sqrt(-1/f)*elliptic_e(arcsin(sqrt(3)*sqrt(-1/f)/x), 2/3*f/d) + sqrt(3)*(4*(5*a*d - b)*f^3 - 54*b*d^2 + 6*((5*a - b)*d - 4*b)*f^2 + 9*(5*a *d^2 + 4*b*d)*f)*sqrt(d*f)*x*sqrt(-1/f)*elliptic_f(arcsin(sqrt(3)*sqrt(-1/ f)/x), 2/3*f/d) + (3*b*d^2*f^3*x^4 - 18*b*d^2*f + 2*(5*a*d - 4*b)*f^3 + 3* (5*a*d^2 + 4*b*d)*f^2 + (3*b*d^2*f^2 + (5*a*d^2 + 2*b*d)*f^3)*x^2)*sqrt(d* x^2 + 2)*sqrt(f*x^2 + 3))/(d^2*f^3*x)
\[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\int \left (a + b x^{2}\right ) \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}\, dx \] Input:
integrate((b*x**2+a)*(d*x**2+2)**(1/2)*(f*x**2+3)**(1/2),x)
Output:
Integral((a + b*x**2)*sqrt(d*x**2 + 2)*sqrt(f*x**2 + 3), x)
\[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\int { {\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3} \,d x } \] Input:
integrate((b*x^2+a)*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2),x, algorithm="maxima")
Output:
integrate((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3), x)
\[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\int { {\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3} \,d x } \] Input:
integrate((b*x^2+a)*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3), x)
Timed out. \[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\int \left (b\,x^2+a\right )\,\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3} \,d x \] Input:
int((a + b*x^2)*(d*x^2 + 2)^(1/2)*(f*x^2 + 3)^(1/2),x)
Output:
int((a + b*x^2)*(d*x^2 + 2)^(1/2)*(f*x^2 + 3)^(1/2), x)
\[ \int \left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2} \, dx=\frac {5 \sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, a d f x +3 \sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, b d f \,x^{3}+3 \sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, b d x +2 \sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, b f x +15 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, x^{2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) a \,d^{2} f +10 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, x^{2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) a d \,f^{2}-18 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, x^{2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) b \,d^{2}+12 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, x^{2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) b d f -8 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, x^{2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) b \,f^{2}+60 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) a d f -18 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) b d -12 \left (\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}{d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}d x \right ) b f}{15 d f} \] Input:
int((b*x^2+a)*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2),x)
Output:
(5*sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2)*a*d*f*x + 3*sqrt(f*x**2 + 3)*sqrt(d*x **2 + 2)*b*d*f*x**3 + 3*sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2)*b*d*x + 2*sqrt(f *x**2 + 3)*sqrt(d*x**2 + 2)*b*f*x + 15*int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2)*x**2)/(d*f*x**4 + 3*d*x**2 + 2*f*x**2 + 6),x)*a*d**2*f + 10*int((sqrt( f*x**2 + 3)*sqrt(d*x**2 + 2)*x**2)/(d*f*x**4 + 3*d*x**2 + 2*f*x**2 + 6),x) *a*d*f**2 - 18*int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2)*x**2)/(d*f*x**4 + 3* d*x**2 + 2*f*x**2 + 6),x)*b*d**2 + 12*int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2)*x**2)/(d*f*x**4 + 3*d*x**2 + 2*f*x**2 + 6),x)*b*d*f - 8*int((sqrt(f*x** 2 + 3)*sqrt(d*x**2 + 2)*x**2)/(d*f*x**4 + 3*d*x**2 + 2*f*x**2 + 6),x)*b*f* *2 + 60*int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2))/(d*f*x**4 + 3*d*x**2 + 2*f *x**2 + 6),x)*a*d*f - 18*int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2))/(d*f*x**4 + 3*d*x**2 + 2*f*x**2 + 6),x)*b*d - 12*int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2))/(d*f*x**4 + 3*d*x**2 + 2*f*x**2 + 6),x)*b*f)/(15*d*f)