Integrand size = 32, antiderivative size = 456 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\left (24 a^2 d^2 f^2-a b d f (40 d e+27 c f)+b^2 \left (15 d^2 e^2+40 c d e f+3 c^2 f^2\right )\right ) x \sqrt {c+d x^2}}{15 b^3 d \sqrt {a+b x^2}}+\frac {2 f (5 b d e+3 b c f-3 a d f) x^3 \sqrt {c+d x^2}}{15 b^2 \sqrt {a+b x^2}}+\frac {d f^2 x^5 \sqrt {c+d x^2}}{5 b \sqrt {a+b x^2}}+\frac {\left (15 b^3 c d e^2-48 a^3 d^2 f^2+16 a^2 b d f (5 d e+3 c f)-a b^2 \left (30 d^2 e^2+70 c d e f+3 c^2 f^2\right )\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 \sqrt {a} b^{7/2} d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (24 a^2 d f^2+15 b^2 e (d e+2 c f)-a b f (40 d e+21 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 b^{7/2} \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
1/15*(24*a^2*d^2*f^2-a*b*d*f*(27*c*f+40*d*e)+b^2*(3*c^2*f^2+40*c*d*e*f+15* d^2*e^2))*x*(d*x^2+c)^(1/2)/b^3/d/(b*x^2+a)^(1/2)+2/15*f*(-3*a*d*f+3*b*c*f +5*b*d*e)*x^3*(d*x^2+c)^(1/2)/b^2/(b*x^2+a)^(1/2)+1/5*d*f^2*x^5*(d*x^2+c)^ (1/2)/b/(b*x^2+a)^(1/2)+1/15*(15*b^3*c*d*e^2-48*a^3*d^2*f^2+16*a^2*b*d*f*( 3*c*f+5*d*e)-a*b^2*(3*c^2*f^2+70*c*d*e*f+30*d^2*e^2))*(d*x^2+c)^(1/2)*Elli pticE(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^(7/ 2)/d/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/15*a^(1/2)*(24*a^2* d*f^2+15*b^2*e*(2*c*f+d*e)-a*b*f*(21*c*f+40*d*e))*(d*x^2+c)^(1/2)*InverseJ acobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(7/2)/(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 = 6.76 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (c+d x^2\right ) \left (15 b^3 c e^2-24 a^3 d f^2+a^2 b f \left (40 d e+21 c f-6 d f x^2\right )+a b^2 \left (6 c f \left (-5 e+f x^2\right )+d \left (-15 e^2+10 e f x^2+3 f^2 x^4\right )\right )\right )-i c \left (-15 b^3 c d e^2+48 a^3 d^2 f^2-16 a^2 b d f (5 d e+3 c f)+a b^2 \left (30 d^2 e^2+70 c d e f+3 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (15 b^2 d e^2+24 a^2 d f^2-a b f (40 d e+3 c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 b^4 d \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((c + d*x^2)^(3/2)*(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)*(15*b^3*c*e^2 - 24*a^3*d*f^2 + a^2*b *f*(40*d*e + 21*c*f - 6*d*f*x^2) + a*b^2*(6*c*f*(-5*e + f*x^2) + d*(-15*e^ 2 + 10*e*f*x^2 + 3*f^2*x^4))) - I*c*(-15*b^3*c*d*e^2 + 48*a^3*d^2*f^2 - 16 *a^2*b*d*f*(5*d*e + 3*c*f) + a*b^2*(30*d^2*e^2 + 70*c*d*e*f + 3*c^2*f^2))* 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*(-(b*c) + a*d)*(15*b^2*d*e^2 + 24*a^2*d*f^2 - a*b*f*(40 *d*e + 3*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)]))/(15*b^4*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(972\) vs. \(2(456)=912\).
Time = 1.34 (sec) , antiderivative size = 972, normalized size of antiderivative = 2.13, 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.
| \(\displaystyle \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2}}+\frac {2 e f x^2 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2}}+\frac {f^2 x^4 \left (c+d x^2\right )^{3/2}}{\left (a+b x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {f^2 \left (d x^2+c\right )^{3/2} x^3}{b \sqrt {b x^2+a}}+\frac {6 d f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 b^2}-\frac {2 e f \left (d x^2+c\right )^{3/2} x}{b \sqrt {b x^2+a}}+\frac {(7 b c-8 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{5 b^3}+\frac {8 d e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 b^2}+\frac {(b c-a d) e^2 \sqrt {d x^2+c} x}{a b \sqrt {b x^2+a}}-\frac {d (b c-2 a d) e^2 \sqrt {b x^2+a} x}{a b^2 \sqrt {d x^2+c}}+\frac {\left (b^2 c^2-16 a b d c+16 a^2 d^2\right ) f^2 \sqrt {b x^2+a} x}{5 b^4 \sqrt {d x^2+c}}+\frac {2 d (7 b c-8 a d) e f \sqrt {b x^2+a} x}{3 b^3 \sqrt {d x^2+c}}+\frac {\sqrt {c} \sqrt {d} (b c-2 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 )}{a b^2 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {\sqrt {c} \left (b^2 c^2-16 a b d c+16 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 )}{5 b^4 \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 \sqrt {c} \sqrt {d} (7 b c-8 a d) 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 )}{3 b^3 \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}+\frac {c^{3/2} \sqrt {d} 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 )}{a b \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}-\frac {c^{3/2} (7 b c-8 a d) 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 )}{5 b^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} (3 b c-4 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 )}{3 a b^2 \sqrt {d} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x]
Output:
-((d*(b*c - 2*a*d)*e^2*x*Sqrt[a + b*x^2])/(a*b^2*Sqrt[c + d*x^2])) + (2*d* (7*b*c - 8*a*d)*e*f*x*Sqrt[a + b*x^2])/(3*b^3*Sqrt[c + d*x^2]) + ((b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*f^2*x*Sqrt[a + b*x^2])/(5*b^4*Sqrt[c + d*x^2]) + ((b*c - a*d)*e^2*x*Sqrt[c + d*x^2])/(a*b*Sqrt[a + b*x^2]) + (8*d*e*f*x* Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*b^2) + ((7*b*c - 8*a*d)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*b^3) + (6*d*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d *x^2])/(5*b^2) - (2*e*f*x*(c + d*x^2)^(3/2))/(b*Sqrt[a + b*x^2]) - (f^2*x^ 3*(c + d*x^2)^(3/2))/(b*Sqrt[a + b*x^2]) + (Sqrt[c]*Sqrt[d]*(b*c - 2*a*d)* e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d) ])/(a*b^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*Sqrt [c]*Sqrt[d]*(7*b*c - 8*a*d)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]* x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2)) ]*Sqrt[c + d*x^2]) - (Sqrt[c]*(b^2*c^2 - 16*a*b*c*d + 16*a^2*d^2)*f^2*Sqrt [a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*b^ 4*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (c^(3/2 )*Sqrt[d]*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - ( b*c)/(a*d)])/(a*b*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*c^(3/2)*(3*b*c - 4*a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]* x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a*b^2*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(7*b*c - 8*a*d)*f^2*Sqrt[a + b*...
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]
Leaf count of result is larger than twice the leaf count of optimal. \(850\) vs. \(2(421)=842\).
Time = 19.02 (sec) , antiderivative size = 851, normalized size of antiderivative = 1.87
| method | result | size |
| risch | \(-\frac {f x \left (-3 b d f \,x^{2}+9 a d f -6 b c f -10 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{3}}+\frac {\left (-\frac {\left (33 a^{2} d^{2} f^{2}-33 a b c d \,f^{2}-50 a b \,d^{2} e f +3 b^{2} c^{2} f^{2}+40 b^{2} c d e f +15 b^{2} 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 {15 \left (a^{4} d^{2} f^{2}-2 a^{3} b c d \,f^{2}-2 a^{3} b \,d^{2} e f +a^{2} b^{2} c^{2} f^{2}+4 a^{2} b^{2} c d e f +a^{2} b^{2} d^{2} e^{2}-2 a \,b^{3} c^{2} e f -2 a \,b^{3} c d \,e^{2}+b^{4} c^{2} 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 (15 a^{3} d^{2} f^{2}-39 a^{2} b c d \,f^{2}-30 a^{2} b \,d^{2} e f +21 a \,b^{2} c^{2} f^{2}+70 a \,b^{2} c d e f +15 a \,b^{2} d^{2} e^{2}-30 b^{3} c^{2} e f -30 b^{3} c 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 )}}{15 b^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(851\) |
| 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^{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 ) x}{a \,b^{4} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {d \,f^{2} x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 b^{2}}+\frac {\left (-\frac {d f \left (a d f -2 b c f -2 b d e \right )}{b^{2}}-\frac {d \,f^{2} \left (4 a d +4 b c \right )}{5 b^{2}}\right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (-\frac {a^{3} d^{2} f^{2}-2 a^{2} b c d \,f^{2}-2 a^{2} b \,d^{2} e f +a \,b^{2} c^{2} f^{2}+4 a \,b^{2} c d e f +a \,b^{2} d^{2} e^{2}-2 b^{3} c^{2} e f -2 b^{3} c d \,e^{2}}{b^{4}}+\frac {\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 (a d -b c \right )}{b^{4} a}+\frac {c \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 )}{b^{3} a}-\frac {\left (-\frac {d f \left (a d f -2 b c f -2 b d e \right )}{b^{2}}-\frac {d \,f^{2} \left (4 a d +4 b c \right )}{5 b^{2}}\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 {a^{2} d^{2} f^{2}-2 a b c d \,f^{2}-2 a b \,d^{2} e f +b^{2} c^{2} f^{2}+4 b^{2} c d e f +b^{2} d^{2} e^{2}}{b^{3}}+\frac {\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 ) d}{b^{3} a}-\frac {3 d \,f^{2} a c}{5 b^{2}}-\frac {\left (-\frac {d f \left (a d f -2 b c f -2 b d e \right )}{b^{2}}-\frac {d \,f^{2} \left (4 a d +4 b c \right )}{5 b^{2}}\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}}\) | \(923\) |
| default | \(\text {Expression too large to display}\) | \(1288\) |
Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/15*f/b^3*x*(-3*b*d*f*x^2+9*a*d*f-6*b*c*f-10*b*d*e)*(b*x^2+a)^(1/2)*(d*x ^2+c)^(1/2)+1/15/b^3*(-(33*a^2*d^2*f^2-33*a*b*c*d*f^2-50*a*b*d^2*e*f+3*b^2 *c^2*f^2+40*b^2*c*d*e*f+15*b^2*d^2*e^2)*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)))+15*(a^4*d^2*f^2-2*a^3*b*c*d*f^2-2*a^3*b*d^2*e*f+a^2*b^2*c^2*f^2 +4*a^2*b^2*c*d*e*f+a^2*b^2*d^2*e^2-2*a*b^3*c^2*e*f-2*a*b^3*c*d*e^2+b^4*c^2 *e^2)/b*(-(b*d*x^2+b*c)/a/(a*d-b*c)*x/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(1/a +b*c/(a*d-b*c)/a)/(-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))-b/(a*d-b*c)/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))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))))-(15*a^3*d^ 2*f^2-39*a^2*b*c*d*f^2-30*a^2*b*d^2*e*f+21*a*b^2*c^2*f^2+70*a*b^2*c*d*e*f+ 15*a*b^2*d^2*e^2-30*b^3*c^2*e*f-30*b^3*c*d*e^2)/b/(-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)))*((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+ a)^(1/2)/(d*x^2+c)^(1/2)
Time = 0.11 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.78 \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="fricas ")
Output:
1/15*(((15*(b^4*c^2*d - 2*a*b^3*c*d^2)*e^2 - 10*(7*a*b^3*c^2*d - 8*a^2*b^2 *c*d^2)*e*f - 3*(a*b^3*c^3 - 16*a^2*b^2*c^2*d + 16*a^3*b*c*d^2)*f^2)*x^3 + (15*(a*b^3*c^2*d - 2*a^2*b^2*c*d^2)*e^2 - 10*(7*a^2*b^2*c^2*d - 8*a^3*b*c *d^2)*e*f - 3*(a^2*b^2*c^3 - 16*a^3*b*c^2*d + 16*a^4*c*d^2)*f^2)*x)*sqrt(b *d)*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((15*(b^4*c^2 *d - 2*a*b^3*c*d^2 - a*b^3*d^3)*e^2 - 10*(7*a*b^3*c^2*d - 4*a^2*b^2*d^3 - (8*a^2*b^2 - 3*a*b^3)*c*d^2)*e*f - 3*(a*b^3*c^3 - 16*a^2*b^2*c^2*d + 8*a^3 *b*d^3 + (16*a^3*b - 7*a^2*b^2)*c*d^2)*f^2)*x^3 + (15*(a*b^3*c^2*d - 2*a^2 *b^2*c*d^2 - a^2*b^2*d^3)*e^2 - 10*(7*a^2*b^2*c^2*d - 4*a^3*b*d^3 - (8*a^3 *b - 3*a^2*b^2)*c*d^2)*e*f - 3*(a^2*b^2*c^3 - 16*a^3*b*c^2*d + 8*a^4*d^3 + (16*a^4 - 7*a^3*b)*c*d^2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin( sqrt(-c/d)/x), a*d/(b*c)) + (3*a*b^3*d^3*f^2*x^6 + 2*(5*a*b^3*d^3*e*f + 3* (a*b^3*c*d^2 - a^2*b^2*d^3)*f^2)*x^4 - 15*(a*b^3*c*d^2 - 2*a^2*b^2*d^3)*e^ 2 + 10*(7*a^2*b^2*c*d^2 - 8*a^3*b*d^3)*e*f + 3*(a^2*b^2*c^2*d - 16*a^3*b*c *d^2 + 16*a^4*d^3)*f^2 + (15*a*b^3*d^3*e^2 + 40*(a*b^3*c*d^2 - a^2*b^2*d^3 )*e*f + 3*(a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 8*a^3*b*d^3)*f^2)*x^2)*sqrt(b*x ^2 + a)*sqrt(d*x^2 + c))/(a*b^5*d^2*x^3 + a^2*b^4*d^2*x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {\left (c + d x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((d*x**2+c)**(3/2)*(f*x**2+e)**2/(b*x**2+a)**(3/2),x)
Output:
Integral((c + d*x**2)**(3/2)*(e + f*x**2)**2/(a + b*x**2)**(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="maxima ")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^2/(b*x^2 + a)^(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x, algorithm="giac")
Output:
integrate((d*x^2 + c)^(3/2)*(f*x^2 + e)^2/(b*x^2 + a)^(3/2), x)
Timed out. \[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\int \frac {{\left (d\,x^2+c\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}} \,d x \] Input:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2),x)
Output:
int(((c + d*x^2)^(3/2)*(e + f*x^2)^2)/(a + b*x^2)^(3/2), x)
\[ \int \frac {\left (c+d x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((d*x^2+c)^(3/2)*(f*x^2+e)^2/(b*x^2+a)^(3/2),x)
Output:
(9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2*x - 6*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a**2*d**2*f**2*x**3 - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2) *a*b*c**2*f**2*x - 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*e*f*x + 6* sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*f**2*x**3 + 10*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a*b*d**2*e*f*x**3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a *b*d**2*f**2*x**5 + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*e*f*x + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e**2*x + 24*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**4*d**3*f**2 - 36*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*b*c*d**2*f**2 - 40*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*b*d**3*e*f + 24*int((sq rt(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*b*d**3*f**2*x**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**2*c**2*d*f **2 + 55*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**2* c*d**2*e*f - 36*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a**2*c + ...