Integrand size = 30, antiderivative size = 262 \[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=-\frac {(6 b d-2 b f-3 a d f) x \sqrt {2+d x^2}}{3 d f \sqrt {3+f x^2}}+\frac {b x \sqrt {2+d x^2} \sqrt {3+f x^2}}{3 f}+\frac {\sqrt {2} (6 b d-2 b f-3 a d f) \sqrt {2+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{3 d f^{3/2} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}}-\frac {\sqrt {2} (b-a f) \sqrt {2+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{f^{3/2} \sqrt {\frac {2+d x^2}{3+f x^2}} \sqrt {3+f x^2}} \]
-1/3*(-3*a*d*f+6*b*d-2*b*f)*x*(d*x^2+2)^(1/2)/d/f/(f*x^2+3)^(1/2)+1/3*(-3* a*d*f+6*b*d-2*b*f)*(1/(3*f*x^2+9))^(1/2)*(3*f*x^2+9)^(1/2)*EllipticE(x*f^( 1/2)*3^(1/2)/(3*f*x^2+9)^(1/2),1/2*(4-6*d/f)^(1/2))*2^(1/2)*(d*x^2+2)^(1/2 )/d/f^(3/2)/((d*x^2+2)/(f*x^2+3))^(1/2)/(f*x^2+3)^(1/2)-(-a*f+b)*(1/(3*f*x ^2+9))^(1/2)*(3*f*x^2+9)^(1/2)*EllipticF(x*f^(1/2)*3^(1/2)/(3*f*x^2+9)^(1/ 2),1/2*(4-6*d/f)^(1/2))*2^(1/2)*(d*x^2+2)^(1/2)/f^(3/2)/((d*x^2+2)/(f*x^2+ 3))^(1/2)/(f*x^2+3)^(1/2)+1/3*b*x*(d*x^2+2)^(1/2)*(f*x^2+3)^(1/2)/f
Result contains complex when optimal does not.
Time = 1.17 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=\frac {b \sqrt {d} f x \sqrt {2+d x^2} \sqrt {3+f x^2}+i \sqrt {3} (6 b d-2 b f-3 a d f) E\left (i \text {arcsinh}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )+i \sqrt {3} (3 d-2 f) (-2 b+a f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right ),\frac {2 f}{3 d}\right )}{3 \sqrt {d} f^2} \]
(b*Sqrt[d]*f*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2] + I*Sqrt[3]*(6*b*d - 2*b*f - 3*a*d*f)*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + I*Sqrt [3]*(3*d - 2*f)*(-2*b + a*f)*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2* f)/(3*d)])/(3*Sqrt[d]*f^2)
Time = 0.35 (sec) , antiderivative size = 250, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {403, 25, 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.
| \(\displaystyle \int \frac {\sqrt {d x^2+2} \left (a+b x^2\right )}{\sqrt {f x^2+3}} \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int -\frac {(6 b d-3 a f d-2 b f) x^2+6 (b-a f)}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx}{3 f}+\frac {b x \sqrt {d x^2+2} \sqrt {f x^2+3}}{3 f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b x \sqrt {d x^2+2} \sqrt {f x^2+3}}{3 f}-\frac {\int \frac {(6 b d-3 a f d-2 b f) x^2+6 (b-a f)}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx}{3 f}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {b x \sqrt {d x^2+2} \sqrt {f x^2+3}}{3 f}-\frac {6 (b-a f) \int \frac {1}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx+(-3 a d f+6 b d-2 b f) \int \frac {x^2}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx}{3 f}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {b x \sqrt {d x^2+2} \sqrt {f x^2+3}}{3 f}-\frac {(-3 a d f+6 b d-2 b f) \int \frac {x^2}{\sqrt {d x^2+2} \sqrt {f x^2+3}}dx+\frac {3 \sqrt {2} \sqrt {d x^2+2} (b-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{\sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}}{3 f}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {b x \sqrt {d x^2+2} \sqrt {f x^2+3}}{3 f}-\frac {(-3 a d f+6 b d-2 b f) \left (\frac {x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}}-\frac {3 \int \frac {\sqrt {d x^2+2}}{\left (f x^2+3\right )^{3/2}}dx}{d}\right )+\frac {3 \sqrt {2} \sqrt {d x^2+2} (b-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{\sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}}{3 f}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {b x \sqrt {d x^2+2} \sqrt {f x^2+3}}{3 f}-\frac {\frac {3 \sqrt {2} \sqrt {d x^2+2} (b-a f) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right ),1-\frac {3 d}{2 f}\right )}{\sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}+(-3 a d f+6 b d-2 b f) \left (\frac {x \sqrt {d x^2+2}}{d \sqrt {f x^2+3}}-\frac {\sqrt {2} \sqrt {d x^2+2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {3}}\right )|1-\frac {3 d}{2 f}\right )}{d \sqrt {f} \sqrt {f x^2+3} \sqrt {\frac {d x^2+2}{f x^2+3}}}\right )}{3 f}\) |
(b*x*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2])/(3*f) - ((6*b*d - 2*b*f - 3*a*d*f)*( (x*Sqrt[2 + d*x^2])/(d*Sqrt[3 + f*x^2]) - (Sqrt[2]*Sqrt[2 + d*x^2]*Ellipti cE[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]*(b - a*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)
3.1.53.3.1 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 = 4.93 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.08
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (d \,x^{2}+2\right )}\, \left (\frac {b x \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}{3 f}+\frac {\left (2 a -\frac {2 b}{f}\right ) \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{2 \sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}-\frac {\left (a d +2 b -\frac {b \left (6 d +4 f \right )}{3 f}\right ) \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )-E\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}\, d}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}\) | \(282\) |
| risch | \(\frac {b x \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}}{3 f}+\frac {\left (-\frac {3 b \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}+\frac {3 a f \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, F\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}-\frac {\left (3 a d f -6 b d +2 b f \right ) \sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, \left (F\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )-E\left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right )\right )}{\sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}\, d}\right ) \sqrt {\left (f \,x^{2}+3\right ) \left (d \,x^{2}+2\right )}}{3 f \sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}\) | \(346\) |
| default | \(\frac {\sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}\, \left (b \,d^{2} f \,x^{5} \sqrt {-f}+3 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) a d f \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}+3 b \,d^{2} x^{3} \sqrt {-f}+2 b d f \,x^{3} \sqrt {-f}-6 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) b d \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}+2 \sqrt {2}\, E\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) b f \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}+3 \sqrt {2}\, F\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) b d \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}-2 \sqrt {2}\, F\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) b f \sqrt {d \,x^{2}+2}\, \sqrt {f \,x^{2}+3}+6 b d x \sqrt {-f}\right )}{3 \left (d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6\right ) f \sqrt {-f}\, d}\) | \(367\) |
((f*x^2+3)*(d*x^2+2))^(1/2)/(f*x^2+3)^(1/2)/(d*x^2+2)^(1/2)*(1/3*b/f*x*(d* f*x^4+3*d*x^2+2*f*x^2+6)^(1/2)+1/2*(2*a-2*b/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))-(a*d+2*b-1/3*b/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))-Elliptic E(1/3*x*(-3*f)^(1/2),1/2*(-4+2*(3*d+2*f)/f)^(1/2))))
Time = 0.09 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.69 \[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=\frac {3 \, \sqrt {3} {\left (6 \, b d - {\left (3 \, a d + 2 \, b\right )} f\right )} \sqrt {d f} x \sqrt {-\frac {1}{f}} E(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{f}}}{x}\right )\,|\,\frac {2 \, f}{3 \, d}) + \sqrt {3} {\left (2 \, a f^{3} - 2 \, b f^{2} - 18 \, b d + 3 \, {\left (3 \, a d + 2 \, b\right )} f\right )} \sqrt {d f} x \sqrt {-\frac {1}{f}} F(\arcsin \left (\frac {\sqrt {3} \sqrt {-\frac {1}{f}}}{x}\right )\,|\,\frac {2 \, f}{3 \, d}) + {\left (b d f^{2} x^{2} - 6 \, b d f + {\left (3 \, a d + 2 \, b\right )} f^{2}\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}{3 \, d f^{3} x} \]
1/3*(3*sqrt(3)*(6*b*d - (3*a*d + 2*b)*f)*sqrt(d*f)*x*sqrt(-1/f)*elliptic_e (arcsin(sqrt(3)*sqrt(-1/f)/x), 2/3*f/d) + sqrt(3)*(2*a*f^3 - 2*b*f^2 - 18* b*d + 3*(3*a*d + 2*b)*f)*sqrt(d*f)*x*sqrt(-1/f)*elliptic_f(arcsin(sqrt(3)* sqrt(-1/f)/x), 2/3*f/d) + (b*d*f^2*x^2 - 6*b*d*f + (3*a*d + 2*b)*f^2)*sqrt (d*x^2 + 2)*sqrt(f*x^2 + 3))/(d*f^3*x)
\[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=\int \frac {\left (a + b x^{2}\right ) \sqrt {d x^{2} + 2}}{\sqrt {f x^{2} + 3}}\, dx \]
\[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2}}{\sqrt {f x^{2} + 3}} \,d x } \]
\[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2}}{\sqrt {f x^{2} + 3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^2\right ) \sqrt {2+d x^2}}{\sqrt {3+f x^2}} \, dx=\int \frac {\left (b\,x^2+a\right )\,\sqrt {d\,x^2+2}}{\sqrt {f\,x^2+3}} \,d x \]