Integrand size = 32, antiderivative size = 304 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {f (6 b d e+b c f-4 a d f) x \sqrt {c+d x^2}}{3 b^2 d \sqrt {a+b x^2}}+\frac {f^2 x^3 \sqrt {c+d x^2}}{3 b \sqrt {a+b x^2}}+\frac {\left (3 b^2 d e^2+8 a^2 d f^2-a b f (12 d e+c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 \sqrt {a} b^{5/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {2 \sqrt {a} f (3 b e-2 a f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{5/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/3*f*(-4*a*d*f+b*c*f+6*b*d*e)*x*(d*x^2+c)^(1/2)/b^2/d/(b*x^2+a)^(1/2)+1/3 *f^2*x^3*(d*x^2+c)^(1/2)/b/(b*x^2+a)^(1/2)+1/3*(3*b^2*d*e^2+8*a^2*d*f^2-a* b*f*(c*f+12*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))/a^(1/2)/b^(5/2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/ (b*x^2+a))^(1/2)+2/3*a^(1/2)*f*(-2*a*f+3*b*e)*(d*x^2+c)^(1/2)*InverseJacob iAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(5/2)/(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 = 4.12 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (c+d x^2\right ) \left (3 b^2 e^2+4 a^2 f^2+a b f \left (-6 e+f x^2\right )\right )+i c \left (3 b^2 d e^2+8 a^2 d f^2-a b f (12 d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c \left (3 b^2 d e^2+4 a^2 d f^2-a b f (6 d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 b^3 d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(Sqrt[c + d*x^2]*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*d*x*(c + d*x^2)*(3*b^2*e^2 + 4*a^2*f^2 + a*b*f*(-6*e + f*x^2)) + I*c*(3*b^2*d*e^2 + 8*a^2*d*f^2 - a*b*f*(12*d*e + c*f))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/ (b*c)] - I*c*(3*b^2*d*e^2 + 4*a^2*d*f^2 - a*b*f*(6*d*e + c*f))*Sqrt[1 + (b *x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c) ]))/(3*b^3*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(633\) vs. \(2(304)=608\).
Time = 0.83 (sec) , antiderivative size = 633, normalized size of antiderivative = 2.08, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}}+\frac {2 e f x^2 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}}+\frac {f^2 x^4 \sqrt {c+d x^2}}{\left (a+b x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {c} f^2 \sqrt {a+b x^2} (b c-8 a d) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^3 \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 \sqrt {c} \sqrt {d} e f \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b^2 \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 c^{3/2} e f \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {e^2 \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {f^2 x \sqrt {a+b x^2} (b c-8 a d)}{3 b^3 \sqrt {c+d x^2}}+\frac {4 d e f x \sqrt {a+b x^2}}{b^2 \sqrt {c+d x^2}}+\frac {4 f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b^2}-\frac {2 e f x \sqrt {c+d x^2}}{b \sqrt {a+b x^2}}-\frac {f^2 x^3 \sqrt {c+d x^2}}{b \sqrt {a+b x^2}}\) |
Input:
Int[(Sqrt[c + d*x^2]*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x]
Output:
(4*d*e*f*x*Sqrt[a + b*x^2])/(b^2*Sqrt[c + d*x^2]) + ((b*c - 8*a*d)*f^2*x*S qrt[a + b*x^2])/(3*b^3*Sqrt[c + d*x^2]) - (2*e*f*x*Sqrt[c + d*x^2])/(b*Sqr t[a + b*x^2]) - (f^2*x^3*Sqrt[c + d*x^2])/(b*Sqrt[a + b*x^2]) + (4*f^2*x*S qrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b^2) + (e^2*Sqrt[c + d*x^2]*EllipticE[A rcTan[(Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*Sqrt[b]*Sqrt[a + b* x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (4*Sqrt[c]*Sqrt[d]*e*f*Sqrt[ a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b^2*S qrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(b*c - 8* a*d)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/ (a*d)])/(3*b^3*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^ 2]) + (2*c^(3/2)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]] , 1 - (b*c)/(a*d)])/(a*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqr t[c + d*x^2]) - (4*c^(3/2)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x )/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
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 = 9.23 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.75
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+b c \right ) \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) x}{a \,b^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2}}+\frac {\left (\frac {a^{2} d \,f^{2}-a b c \,f^{2}-2 a b d e f +2 b^{2} c e f +b^{2} d \,e^{2}}{b^{3}}-\frac {\left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \left (a d -b c \right )}{b^{3} a}-\frac {c \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{b^{2} a}-\frac {a c \,f^{2}}{3 b^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (-\frac {f \left (a d f -b c f -2 b d e \right )}{b^{2}}-\frac {\left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) d}{b^{2} a}-\frac {f^{2} \left (2 a d +2 b c \right )}{3 b^{2}}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(532\) |
| default | \(\frac {\sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}\, \left (\sqrt {-\frac {b}{a}}\, a b \,d^{2} f^{2} x^{5}+4 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} f^{2} x^{3}+\sqrt {-\frac {b}{a}}\, a b c d \,f^{2} x^{3}-6 \sqrt {-\frac {b}{a}}\, a b \,d^{2} e f \,x^{3}+3 \sqrt {-\frac {b}{a}}\, b^{2} d^{2} e^{2} x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}-6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e f +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}+12 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e f -3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}+4 \sqrt {-\frac {b}{a}}\, a^{2} c d \,f^{2} x -6 \sqrt {-\frac {b}{a}}\, a b c d e f x +3 \sqrt {-\frac {b}{a}}\, b^{2} c d \,e^{2} x \right )}{3 b^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {-\frac {b}{a}}\, d a}\) | \(680\) |
| risch | \(\frac {f^{2} x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b^{2}}-\frac {\left (-\frac {f \left (5 a d f -b c f -6 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {3 \left (a^{3} d \,f^{2}-a^{2} c \,f^{2} b -2 a^{2} b d e f +2 a c e f \,b^{2}+a \,b^{2} d \,e^{2}-b^{3} c \,e^{2}\right ) \left (-\frac {\left (b d \,x^{2}+b c \right ) x}{a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {1}{a}+\frac {b c}{\left (a d -b c \right ) a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{b}-\frac {\left (3 a^{2} d \,f^{2}-4 a b c \,f^{2}-6 a b d e f +6 b^{2} c e f +3 b^{2} d \,e^{2}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{b \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(691\) |
Input:
int((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),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)*((b*d*x^2+b*c) *(a^2*f^2-2*a*b*e*f+b^2*e^2)/a/b^3*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+1/3*f ^2/b^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+((a^2*d*f^2-a*b*c*f^2-2*a*b*d *e*f+2*b^2*c*e*f+b^2*d*e^2)/b^3-(a^2*f^2-2*a*b*e*f+b^2*e^2)/b^3*(a*d-b*c)/ a-1/b^2*c*(a^2*f^2-2*a*b*e*f+b^2*e^2)/a-1/3*a/b^2*c*f^2)/(-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)*Ellipt icF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(-f/b^2*(a*d*f-b*c*f-2*b*d*e) -(a^2*f^2-2*a*b*e*f+b^2*e^2)/b^2*d/a-1/3*f^2/b^2*(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 = 0.10 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.45 \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {{\left ({\left (3 \, b^{3} c d e^{2} - 12 \, a b^{2} c d e f - {\left (a b^{2} c^{2} - 8 \, a^{2} b c d\right )} f^{2}\right )} x^{3} + {\left (3 \, a b^{2} c d e^{2} - 12 \, a^{2} b c d e f - {\left (a^{2} b c^{2} - 8 \, a^{3} c d\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, b^{3} c d e^{2} - 6 \, {\left (2 \, a b^{2} c d + a b^{2} d^{2}\right )} e f - {\left (a b^{2} c^{2} - 8 \, a^{2} b c d - 4 \, a^{2} b d^{2}\right )} f^{2}\right )} x^{3} + {\left (3 \, a b^{2} c d e^{2} - 6 \, {\left (2 \, a^{2} b c d + a^{2} b d^{2}\right )} e f - {\left (a^{2} b c^{2} - 8 \, a^{3} c d - 4 \, a^{3} d^{2}\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) + {\left (a b^{2} d^{2} f^{2} x^{4} - 3 \, a b^{2} d^{2} e^{2} + 12 \, a^{2} b d^{2} e f + {\left (a^{2} b c d - 8 \, a^{3} d^{2}\right )} f^{2} + {\left (6 \, a b^{2} d^{2} e f + {\left (a b^{2} c d - 4 \, a^{2} b d^{2}\right )} f^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a b^{4} d^{2} x^{3} + a^{2} b^{3} d^{2} x\right )}} \] Input:
integrate((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="fricas ")
Output:
1/3*(((3*b^3*c*d*e^2 - 12*a*b^2*c*d*e*f - (a*b^2*c^2 - 8*a^2*b*c*d)*f^2)*x ^3 + (3*a*b^2*c*d*e^2 - 12*a^2*b*c*d*e*f - (a^2*b*c^2 - 8*a^3*c*d)*f^2)*x) *sqrt(b*d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((3*b^ 3*c*d*e^2 - 6*(2*a*b^2*c*d + a*b^2*d^2)*e*f - (a*b^2*c^2 - 8*a^2*b*c*d - 4 *a^2*b*d^2)*f^2)*x^3 + (3*a*b^2*c*d*e^2 - 6*(2*a^2*b*c*d + a^2*b*d^2)*e*f - (a^2*b*c^2 - 8*a^3*c*d - 4*a^3*d^2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*ellipti c_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) + (a*b^2*d^2*f^2*x^4 - 3*a*b^2*d^2*e^ 2 + 12*a^2*b*d^2*e*f + (a^2*b*c*d - 8*a^3*d^2)*f^2 + (6*a*b^2*d^2*e*f + (a *b^2*c*d - 4*a^2*b*d^2)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a*b^4* d^2*x^3 + a^2*b^3*d^2*x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {c + d x^{2}} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(1/2)*(f*x**2+e)**2/(b*x**2+a)**(3/2),x)
Output:
Integral(sqrt(c + d*x**2)*(e + f*x**2)**2/(a + b*x**2)**(3/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="maxima ")
Output:
integrate(sqrt(d*x^2 + c)*(f*x^2 + e)^2/(b*x^2 + a)^(3/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {d x^{2} + c} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(d*x^2 + c)*(f*x^2 + e)^2/(b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {d\,x^2+c}\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(((c + d*x^2)^(1/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x)
Output:
int(((c + d*x^2)^(1/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2), x)
\[ \int \frac {\sqrt {c+d x^2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((d*x^2+c)^(1/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x)
Output:
( - 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*c*f**2*x + 2*sqrt(c + d*x**2)*sq rt(a + b*x**2)*a*d*f**2*x**3 + 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*c*e*f *x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b*d*e**2*x - 8*int((sqrt(c + d*x* *2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d* x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**3*d**2*f**2 + 5*int((sqrt(c + d*x* *2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d* x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c*d*f**2 + 12*int((sqrt(c + d* x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b* d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**2*e*f - 8*int((sqrt(c + d *x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b *d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**2*f**2*x**2 - 6*int((sqr t(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d*e*f + 5*int((sqr t(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d*f**2*x**2 - 3*in t((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a*b*c *x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*d**2*e**2 + 12 *int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x**2 + 2*a* b*c*x**2 + 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*d**2*e*f*x* *2 - 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + a**2*d*x*...