Integrand size = 32, antiderivative size = 609 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\frac {\left (8 a^3 d^3 f^2-a^2 b d^2 f (28 d e+5 c f)+a b^2 d \left (35 d^2 e^2+28 c d e f-5 c^2 f^2\right )+b^3 c \left (35 d^2 e^2-28 c d e f+8 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{105 b^2 d^3 \sqrt {a+b x^2}}-\frac {\left (\frac {4 a^2 d f^2}{b}-2 a f (7 d e+c f)-b \left (35 d e^2+14 c e f-\frac {4 c^2 f^2}{d}\right )\right ) x \sqrt {a+b x^2} \sqrt {c+d x^2}}{105 b d}+\frac {f (14 b d e+b c f+a d f) x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}}{35 b d}+\frac {1}{7} f^2 x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}-\frac {\sqrt {a} \left (8 a^3 d^3 f^2-a^2 b d^2 f (28 d e+5 c f)+a b^2 d \left (35 d^2 e^2+28 c d e f-5 c^2 f^2\right )+b^3 c \left (35 d^2 e^2-28 c d e f+8 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 b^{5/2} d^3 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {2 a^{3/2} \left (2 a^2 d^2 f^2-a b d f (7 d e+c f)+b^2 \left (35 d^2 e^2-7 c d e f+2 c^2 f^2\right )\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{5/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/105*(8*a^3*d^3*f^2-a^2*b*d^2*f*(5*c*f+28*d*e)+a*b^2*d*(-5*c^2*f^2+28*c*d *e*f+35*d^2*e^2)+b^3*c*(8*c^2*f^2-28*c*d*e*f+35*d^2*e^2))*x*(d*x^2+c)^(1/2 )/b^2/d^3/(b*x^2+a)^(1/2)-1/105*(4*a^2*d*f^2/b-2*a*f*(c*f+7*d*e)-b*(35*d*e ^2+14*c*e*f-4*c^2*f^2/d))*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1/35*f*(a* d*f+b*c*f+14*b*d*e)*x^3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d+1/7*f^2*x^5*(b *x^2+a)^(1/2)*(d*x^2+c)^(1/2)-1/105*a^(1/2)*(8*a^3*d^3*f^2-a^2*b*d^2*f*(5* c*f+28*d*e)+a*b^2*d*(-5*c^2*f^2+28*c*d*e*f+35*d^2*e^2)+b^3*c*(8*c^2*f^2-28 *c*d*e*f+35*d^2*e^2))*(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^(5/2)/d^3/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b *x^2+a))^(1/2)+2/105*a^(3/2)*(2*a^2*d^2*f^2-a*b*d*f*(c*f+7*d*e)+b^2*(2*c^2 *f^2-7*c*d*e*f+35*d^2*e^2))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2) *x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/d^2/(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 = 3.41 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.69 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 d^2 f^2-a b d f \left (14 d e+2 c f+3 d f x^2\right )-b^2 \left (-4 c^2 f^2+c d f \left (14 e+3 f x^2\right )+d^2 \left (35 e^2+42 e f x^2+15 f^2 x^4\right )\right )\right )-i c \left (8 a^3 d^3 f^2-a^2 b d^2 f (28 d e+5 c f)+a b^2 d \left (35 d^2 e^2+28 c d e f-5 c^2 f^2\right )+b^3 c \left (35 d^2 e^2-28 c d e f+8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (4 a^2 d^2 f^2+a b d f (-14 d e+c f)+b^2 \left (-35 d^2 e^2+28 c d e f-8 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2,x]
Output:
(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a^2*d^2*f^2 - a*b*d*f*(14*d*e + 2*c*f + 3*d*f*x^2) - b^2*(-4*c^2*f^2 + c*d*f*(14*e + 3*f*x^2) + d^2*(35* e^2 + 42*e*f*x^2 + 15*f^2*x^4)))) - I*c*(8*a^3*d^3*f^2 - a^2*b*d^2*f*(28*d *e + 5*c*f) + a*b^2*d*(35*d^2*e^2 + 28*c*d*e*f - 5*c^2*f^2) + b^3*c*(35*d^ 2*e^2 - 28*c*d*e*f + 8*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E llipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*c*(-(b*c) + a*d)*(4*a^2* d^2*f^2 + a*b*d*f*(-14*d*e + c*f) + b^2*(-35*d^2*e^2 + 28*c*d*e*f - 8*c^2* f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a ]*x], (a*d)/(b*c)])/(105*a^2*(b/a)^(5/2)*d^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^ 2])
Time = 1.39 (sec) , antiderivative size = 1043, normalized size of antiderivative = 1.71, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 2009}
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.
| \(\displaystyle \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (e^2 \sqrt {a+b x^2} \sqrt {c+d x^2}+2 e f x^2 \sqrt {a+b x^2} \sqrt {c+d x^2}+f^2 x^4 \sqrt {a+b x^2} \sqrt {c+d x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^5+\frac {(b c+a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{35 b d}+\frac {2}{5} e f \sqrt {b x^2+a} \sqrt {d x^2+c} x^3+\frac {1}{3} e^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x-\frac {2 \left (2 b^2 c^2-a b d c+2 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{105 b^2 d^2}+\frac {2 (b c+a d) e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{15 b d}+\frac {(b c+a d) e^2 \sqrt {b x^2+a} x}{3 b \sqrt {d x^2+c}}+\frac {(b c+a d) \left (8 b^2 c^2-13 a b d c+8 a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{105 b^3 d^2 \sqrt {d x^2+c}}-\frac {4 \left (b^2 c^2-a b d c+a^2 d^2\right ) e f \sqrt {b x^2+a} x}{15 b^2 d \sqrt {d x^2+c}}-\frac {\sqrt {c} (b c+a d) e^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} (b c+a d) \left (8 b^2 c^2-13 a b d c+8 a^2 d^2\right ) f^2 \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{105 b^3 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {4 \sqrt {c} \left (b^2 c^2-a b d c+a^2 d^2\right ) e f \sqrt {b x^2+a} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b^2 d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 c^{3/2} e^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {2 c^{3/2} \left (2 b^2 c^2-a b d c+2 a^2 d^2\right ) f^2 \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{105 b^2 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {2 c^{3/2} (b c+a d) e f \sqrt {b x^2+a} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 b d^{3/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(e + f*x^2)^2,x]
Output:
((b*c + a*d)*e^2*x*Sqrt[a + b*x^2])/(3*b*Sqrt[c + d*x^2]) - (4*(b^2*c^2 - a*b*c*d + a^2*d^2)*e*f*x*Sqrt[a + b*x^2])/(15*b^2*d*Sqrt[c + d*x^2]) + ((b *c + a*d)*(8*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*f^2*x*Sqrt[a + b*x^2])/(105 *b^3*d^2*Sqrt[c + d*x^2]) + (e^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/3 + (2 *(b*c + a*d)*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(15*b*d) - (2*(2*b^2*c ^2 - a*b*c*d + 2*a^2*d^2)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(105*b^2* d^2) + (2*e*f*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/5 + ((b*c + a*d)*f^2*x^ 3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(35*b*d) + (f^2*x^5*Sqrt[a + b*x^2]*Sqr t[c + d*x^2])/7 - (Sqrt[c]*(b*c + a*d)*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTa n[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*Sqrt[d]*Sqrt[(c*(a + b*x^2) )/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (4*Sqrt[c]*(b^2*c^2 - a*b*c*d + a^2* d^2)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/ (a*d)])/(15*b^2*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x ^2]) - (Sqrt[c]*(b*c + a*d)*(8*b^2*c^2 - 13*a*b*c*d + 8*a^2*d^2)*f^2*Sqrt[ a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(105*b ^3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*c^( 3/2)*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/ (a*d)])/(3*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*c^(3/2)*(b*c + a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/ Sqrt[c]], 1 - (b*c)/(a*d)])/(15*b*d^(3/2)*Sqrt[(c*(a + b*x^2))/(a*(c + ...
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ )^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) ^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x]
Time = 8.46 (sec) , antiderivative size = 704, normalized size of antiderivative = 1.16
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f^{2} x^{5} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{7}+\frac {\left (a d \,f^{2}+b c \,f^{2}+2 d b e f -\frac {f^{2} \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b d}+\frac {\left (\frac {2 a c \,f^{2}}{7}+2 a d e f +2 b c e f +b d \,e^{2}-\frac {\left (a d \,f^{2}+b c \,f^{2}+2 d b e f -\frac {f^{2} \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (a c \,e^{2}-\frac {\left (\frac {2 a c \,f^{2}}{7}+2 a d e f +2 b c e f +b d \,e^{2}-\frac {\left (a d \,f^{2}+b c \,f^{2}+2 d b e f -\frac {f^{2} \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 a c e f +a d \,e^{2}+b c \,e^{2}-\frac {3 \left (a d \,f^{2}+b c \,f^{2}+2 d b e f -\frac {f^{2} \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (\frac {2 a c \,f^{2}}{7}+2 a d e f +2 b c e f +b d \,e^{2}-\frac {\left (a d \,f^{2}+b c \,f^{2}+2 d b e f -\frac {f^{2} \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(704\) |
| risch | \(-\frac {x \left (-15 f^{2} x^{4} b^{2} d^{2}-3 a b \,d^{2} f^{2} x^{2}-3 b^{2} c d \,f^{2} x^{2}-42 b^{2} d^{2} e f \,x^{2}+4 a^{2} d^{2} f^{2}-2 a b c d \,f^{2}-14 a b \,d^{2} e f +4 b^{2} c^{2} f^{2}-14 b^{2} c d e f -35 b^{2} d^{2} e^{2}\right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{105 b^{2} d^{2}}+\frac {\left (-\frac {\left (8 a^{3} d^{3} f^{2}-5 a^{2} b c \,d^{2} f^{2}-28 a^{2} b \,d^{3} e f -5 a \,b^{2} c^{2} d \,f^{2}+28 a \,b^{2} c \,d^{2} e f +35 a \,b^{2} d^{3} e^{2}+8 b^{3} c^{3} f^{2}-28 b^{3} c^{2} d e f +35 b^{3} c \,d^{2} e^{2}\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 {4 a \,b^{2} c^{3} f^{2} \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 {4 a^{3} c \,d^{2} f^{2} \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 {70 a \,b^{2} c \,d^{2} e^{2} \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 {2 a^{2} b \,c^{2} d \,f^{2} \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 {14 a \,b^{2} c^{2} d e 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 {14 a^{2} b c \,d^{2} e 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}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{105 b^{2} d^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(1001\) |
| default | \(\text {Expression too large to display}\) | \(1722\) |
Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,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)*(1/7*f^2*x^5*( b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/7*f^2* (6*a*d+6*b*c))/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(2/7*a*c*f^ 2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/5*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/7*f^2*(6*a* d+6*b*c))/b/d*(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^2-1/3*(2/7*a*c*f^2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/5*(a*d*f^2+b*c*f^2+2* d*b*e*f-1/7*f^2*(6*a*d+6*b*c))/b/d*(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)*Elli pticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(2*a*c*e*f+a*d*e^2+b*c*e^2- 3/5*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/7*f^2*(6*a*d+6*b*c))/b/d*a*c-1/3*(2/7*a*c *f^2+2*a*d*e*f+2*b*c*e*f+b*d*e^2-1/5*(a*d*f^2+b*c*f^2+2*d*b*e*f-1/7*f^2*(6 *a*d+6*b*c))/b/d*(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*(Ellipti cF(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 = 593, normalized size of antiderivative = 0.97 \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=-\frac {{\left (35 \, {\left (b^{3} c^{2} d^{2} + a b^{2} c d^{3}\right )} e^{2} - 28 \, {\left (b^{3} c^{3} d - a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} e f + {\left (8 \, b^{3} c^{4} - 5 \, a b^{2} c^{3} d - 5 \, a^{2} b c^{2} d^{2} + 8 \, a^{3} c d^{3}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (35 \, {\left (b^{3} c^{2} d^{2} + a b^{2} c d^{3} + 2 \, a b^{2} d^{4}\right )} e^{2} - 14 \, {\left (2 \, b^{3} c^{3} d - 2 \, a b^{2} c^{2} d^{2} + a^{2} b d^{4} + {\left (2 \, a^{2} b + a b^{2}\right )} c d^{3}\right )} e f + {\left (8 \, b^{3} c^{4} - 5 \, a b^{2} c^{3} d + 4 \, a^{3} d^{4} - {\left (5 \, a^{2} b - 4 \, a b^{2}\right )} c^{2} d^{2} + 2 \, {\left (4 \, a^{3} - a^{2} b\right )} c d^{3}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, b^{3} d^{4} f^{2} x^{6} + 3 \, {\left (14 \, b^{3} d^{4} e f + {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} f^{2}\right )} x^{4} + 35 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} e^{2} - 28 \, {\left (b^{3} c^{2} d^{2} - a b^{2} c d^{3} + a^{2} b d^{4}\right )} e f + {\left (8 \, b^{3} c^{3} d - 5 \, a b^{2} c^{2} d^{2} - 5 \, a^{2} b c d^{3} + 8 \, a^{3} d^{4}\right )} f^{2} + {\left (35 \, b^{3} d^{4} e^{2} + 14 \, {\left (b^{3} c d^{3} + a b^{2} d^{4}\right )} e f - 2 \, {\left (2 \, b^{3} c^{2} d^{2} - a b^{2} c d^{3} + 2 \, a^{2} b d^{4}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{105 \, b^{3} d^{4} x} \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="fricas ")
Output:
-1/105*((35*(b^3*c^2*d^2 + a*b^2*c*d^3)*e^2 - 28*(b^3*c^3*d - a*b^2*c^2*d^ 2 + a^2*b*c*d^3)*e*f + (8*b^3*c^4 - 5*a*b^2*c^3*d - 5*a^2*b*c^2*d^2 + 8*a^ 3*c*d^3)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/ (b*c)) - (35*(b^3*c^2*d^2 + a*b^2*c*d^3 + 2*a*b^2*d^4)*e^2 - 14*(2*b^3*c^3 *d - 2*a*b^2*c^2*d^2 + a^2*b*d^4 + (2*a^2*b + a*b^2)*c*d^3)*e*f + (8*b^3*c ^4 - 5*a*b^2*c^3*d + 4*a^3*d^4 - (5*a^2*b - 4*a*b^2)*c^2*d^2 + 2*(4*a^3 - a^2*b)*c*d^3)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*b^3*d^4*f^2*x^6 + 3*(14*b^3*d^4*e*f + (b^3*c*d^3 + a*b^2 *d^4)*f^2)*x^4 + 35*(b^3*c*d^3 + a*b^2*d^4)*e^2 - 28*(b^3*c^2*d^2 - a*b^2* c*d^3 + a^2*b*d^4)*e*f + (8*b^3*c^3*d - 5*a*b^2*c^2*d^2 - 5*a^2*b*c*d^3 + 8*a^3*d^4)*f^2 + (35*b^3*d^4*e^2 + 14*(b^3*c*d^3 + a*b^2*d^4)*e*f - 2*(2*b ^3*c^2*d^2 - a*b^2*c*d^3 + 2*a^2*b*d^4)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x ^2 + c))/(b^3*d^4*x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int \sqrt {a + b x^{2}} \sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}\, dx \] Input:
integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)*(f*x**2+e)**2,x)
Output:
Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)*(e + f*x**2)**2, x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int { \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="maxima ")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2, x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int { \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*(f*x^2 + e)^2, x)
Timed out. \[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx=\int \sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2 \,d x \] Input:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2,x)
Output:
int((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)*(e + f*x^2)^2, x)
\[ \int \sqrt {a+b x^2} \sqrt {c+d x^2} \left (e+f x^2\right )^2 \, dx =\text {Too large to display} \] Input:
int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^2,x)
Output:
( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*d**2*f**2*x + 2*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x + 14*sqrt(c + d*x**2)*sqrt(a + b*x**2) *a*b*d**2*e*f*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*f**2*x**3 - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f**2*x + 14*sqrt(c + d*x**2 )*sqrt(a + b*x**2)*b**2*c*d*e*f*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b* *2*c*d*f**2*x**3 + 35*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e**2*x + 42*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*d**2*e*f*x**3 + 15*sqrt(c + d*x **2)*sqrt(a + b*x**2)*b**2*d**2*f**2*x**5 + 8*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**3*d**3*f**2 - 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**2*b*c*d**2*f**2 - 28*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*b*d**3*e*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**2*c**2*d*f**2 + 28*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**2*c*d**2*e*f + 35*i nt((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**2*d**3*e**2 + 8*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**3*c**3*f**2 - 28*int((sqr t(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**3*c**2*d*e*f + 35*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/...