Integrand size = 30, antiderivative size = 381 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\frac {\left (5 b c e+5 a d e+2 a c f-\frac {2 b c^2 f}{d}-\frac {2 a^2 d f}{b}\right ) x \sqrt {c+d x^2}}{15 d \sqrt {a+b x^2}}+\frac {(5 b d e-2 b c f+a d f) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 b d}+\frac {f x \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}{5 d}+\frac {\sqrt {a} \left (2 a^2 d^2 f-b^2 c (5 d e-2 c f)-a b d (5 d e+2 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 b^{3/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {a^{3/2} (10 b d e-b c f-a d f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{3/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(5*b*c*e+5*a*d*e+2*a*c*f-2*b*c^2*f/d-2*a^2*d*f/b)*x*(d*x^2+c)^(1/2)/d /(b*x^2+a)^(1/2)+1/15*(a*d*f-2*b*c*f+5*b*d*e)*x*(b*x^2+a)^(1/2)*(d*x^2+c)^ (1/2)/b/d+1/5*f*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)/d+1/15*a^(1/2)*(2*a^2*d^ 2*f-b^2*c*(-2*c*f+5*d*e)-a*b*d*(2*c*f+5*d*e))*(d*x^2+c)^(1/2)*EllipticE(b^ (1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/b^(3/2)/d^2/(b*x^2+a) ^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/15*a^(3/2)*(-a*d*f-b*c*f+10*b*d*e )*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1 /2))/b^(3/2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)
Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.71 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (a d f+b \left (5 d e+c f+3 d f x^2\right )\right )+i c \left (2 a^2 d^2 f+b^2 c (-5 d e+2 c f)-a b d (5 d e+2 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) (5 b d e-2 b c f+a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 b \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(a*d*f + b*(5*d*e + c*f + 3*d*f*x^2 )) + I*c*(2*a^2*d^2*f + b^2*c*(-5*d*e + 2*c*f) - a*b*d*(5*d*e + 2*c*f))*Sq rt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a *d)/(b*c)] - I*c*(-(b*c) + a*d)*(5*b*d*e - 2*b*c*f + a*d*f)*Sqrt[1 + (b*x^ 2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/ (15*b*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.48 (sec) , antiderivative size = 350, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {403, 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.
| \(\displaystyle \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\int \frac {\sqrt {b x^2+a} \left ((5 b d e+b c f-2 a d f) x^2+c (5 b e-a f)\right )}{\sqrt {d x^2+c}}dx}{5 b}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {\int \frac {a c (10 b d e-b c f-a d f)-\left (-c (5 d e-2 c f) b^2-a d (5 d e+2 c f) b+2 a^2 d^2 f\right ) x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-2 a d f+b c f+5 b d e)}{3 d}}{5 b}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 406 |
| \(\displaystyle \frac {\frac {a c (-a d f-b c f+10 b d e) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx-\left (2 a^2 d^2 f-a b d (2 c f+5 d e)+b^2 (-c) (5 d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-2 a d f+b c f+5 b d e)}{3 d}}{5 b}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 320 |
| \(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (-a d f-b c f+10 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (2 a^2 d^2 f-a b d (2 c f+5 d e)+b^2 (-c) (5 d e-2 c f)\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-2 a d f+b c f+5 b d e)}{3 d}}{5 b}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 388 |
| \(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (-a d f-b c f+10 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (2 a^2 d^2 f-a b d (2 c f+5 d e)+b^2 (-c) (5 d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-2 a d f+b c f+5 b d e)}{3 d}}{5 b}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
| \(\Big \downarrow \) 313 |
| \(\displaystyle \frac {\frac {\frac {c^{3/2} \sqrt {a+b x^2} (-a d f-b c f+10 b d e) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\left (2 a^2 d^2 f-a b d (2 c f+5 d e)+b^2 (-c) (5 d e-2 c f)\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{3 d}+\frac {x \sqrt {a+b x^2} \sqrt {c+d x^2} (-2 a d f+b c f+5 b d e)}{3 d}}{5 b}+\frac {f x \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{5 b}\) |
Input:
Int[Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2),x]
Output:
(f*x*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(5*b) + (((5*b*d*e + b*c*f - 2*a*d *f)*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d) + (-((2*a^2*d^2*f - b^2*c*(5* d*e - 2*c*f) - a*b*d*(5*d*e + 2*c*f))*((x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x ^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d *x^2]))) + (c^(3/2)*(10*b*d*e - b*c*f - a*d*f)*Sqrt[a + b*x^2]*EllipticF[A rcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2) )/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*d))/(5*b)
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 = 5.89 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.13
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f \,x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5}+\frac {\left (a d f +b c f +b d e -\frac {f \left (4 a d +4 b c \right )}{5}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a c e -\frac {\left (a d f +b c f +b d e -\frac {f \left (4 a d +4 b c \right )}{5}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {2 a c f}{5}+a d e +b c e -\frac {\left (a d f +b c f +b d e -\frac {f \left (4 a d +4 b c \right )}{5}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(431\) |
| risch | \(\frac {x \left (3 b d f \,x^{2}+a d f +b c f +5 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b d}-\frac {\left (-\frac {\left (2 f \,d^{2} a^{2}-2 f d c b a -5 a b \,d^{2} e +2 f \,c^{2} b^{2}-5 d \,b^{2} c e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a b \,c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {a^{2} c d f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {10 a c d e b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(536\) |
| default | \(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (3 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} f \,x^{7}+4 \sqrt {-\frac {b}{a}}\, a b \,d^{3} f \,x^{5}+4 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} f \,x^{5}+5 \sqrt {-\frac {b}{a}}\, b^{2} d^{3} e \,x^{5}+\sqrt {-\frac {b}{a}}\, a^{2} d^{3} f \,x^{3}+5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} f \,x^{3}+5 \sqrt {-\frac {b}{a}}\, a b \,d^{3} e \,x^{3}+\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d f \,x^{3}+5 \sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} e \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d f +5 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c \,d^{2} e +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3} f -5 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} d e -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c \,d^{2} f +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} d f +5 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c \,d^{2} e -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{3} f +5 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} d e +\sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} f x +\sqrt {-\frac {b}{a}}\, a b \,c^{2} d f x +5 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} e x \right )}{15 \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) b \,d^{2} \sqrt {-\frac {b}{a}}}\) | \(865\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x,method=_RETURNVERBOSE)
Output:
((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/5*f*x^3*(b* d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(a*d*f+b*c*f+b*d*e-1/5*f*(4*a*d+4*b*c ))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(a*c*e-1/3*(a*d*f+b*c*f+b*d*e -1/5*f*(4*a*d+4*b*c))/b/d*a*c)/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^ (1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a* d+b*c)/c/b)^(1/2))-(2/5*a*c*f+a*d*e+b*c*e-1/3*(a*d*f+b*c*f+b*d*e-1/5*f*(4* a*d+4*b*c))/b/d*(2*a*d+2*b*c))*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c )^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a)^(1/2),(- 1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)) ))
Time = 0.09 (sec) , antiderivative size = 313, normalized size of antiderivative = 0.82 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=-\frac {\sqrt {b d} {\left (5 \, {\left (b^{2} c^{2} d + a b c d^{2}\right )} e - 2 \, {\left (b^{2} c^{3} - a b c^{2} d + a^{2} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d} {\left (5 \, {\left (b^{2} c^{2} d + a b c d^{2} + 2 \, a b d^{3}\right )} e - {\left (2 \, b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} d^{3} + {\left (2 \, a^{2} + a b\right )} c d^{2}\right )} f\right )} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b^{2} d^{3} f x^{4} + {\left (5 \, b^{2} d^{3} e + {\left (b^{2} c d^{2} + a b d^{3}\right )} f\right )} x^{2} + 5 \, {\left (b^{2} c d^{2} + a b d^{3}\right )} e - 2 \, {\left (b^{2} c^{2} d - a b c d^{2} + a^{2} d^{3}\right )} f\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{15 \, b^{2} d^{3} x} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x, algorithm="fricas")
Output:
-1/15*(sqrt(b*d)*(5*(b^2*c^2*d + a*b*c*d^2)*e - 2*(b^2*c^3 - a*b*c^2*d + a ^2*c*d^2)*f)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - sq rt(b*d)*(5*(b^2*c^2*d + a*b*c*d^2 + 2*a*b*d^3)*e - (2*b^2*c^3 - 2*a*b*c^2* d + a^2*d^3 + (2*a^2 + a*b)*c*d^2)*f)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt( -c/d)/x), a*d/(b*c)) - (3*b^2*d^3*f*x^4 + (5*b^2*d^3*e + (b^2*c*d^2 + a*b* d^3)*f)*x^2 + 5*(b^2*c*d^2 + a*b*d^3)*e - 2*(b^2*c^2*d - a*b*c*d^2 + a^2*d ^3)*f)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*d^3*x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int \sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e),x)
Output:
Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2), x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int { \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e), x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int { \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e), x)
Timed out. \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\int \sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,\left (f\,x^2+e\right ) \,d x \] Input:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2),x)
Output:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2), x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right ) \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, a d f x +\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b c f x +5 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d e x +3 \sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, b d f \,x^{3}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} d^{2} f +2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d f +5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,d^{2} e -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c^{2} f +5 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b^{2} c d e -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a^{2} c d f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b \,c^{2} f +10 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a b c d e}{15 b d} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*d*f*x + sqrt(c + d*x**2)*sqrt(a + b*x **2)*b*c*f*x + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e*x + 3*sqrt(c + d* x**2)*sqrt(a + b*x**2)*b*d*f*x**3 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x** 2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*d**2*f + 2*int((sq rt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x** 4),x)*a*b*c*d*f + 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a* d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b*d**2*e - 2*int((sqrt(c + d*x**2)*sqrt (a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**2*c**2*f + 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**2*c*d*e - int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*c*d*f - int((sqrt(c + d*x**2)*sq rt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*b*c**2*f + 10* int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x **4),x)*a*b*c*d*e)/(15*b*d)