Integrand size = 32, antiderivative size = 459 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\left (3 a^2 d^2 f^2+a b d f (40 d e-27 c f)+b^2 \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )\right ) x \sqrt {a+b x^2}}{15 b d^3 \sqrt {c+d x^2}}+\frac {2 f (5 b d e-3 b c f+3 a d f) x^3 \sqrt {a+b x^2}}{15 d^2 \sqrt {c+d x^2}}+\frac {b f^2 x^5 \sqrt {a+b x^2}}{5 d \sqrt {c+d x^2}}-\frac {\left (3 a^2 c d^2 f^2+2 b^2 c \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )-a b d \left (15 d^2 e^2-70 c d e f+48 c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 b \sqrt {c} d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} \left (3 a d f (10 d e-7 c f)+b \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 d^{7/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
1/15*(3*a^2*d^2*f^2+a*b*d*f*(-27*c*f+40*d*e)+b^2*(24*c^2*f^2-40*c*d*e*f+15 *d^2*e^2))*x*(b*x^2+a)^(1/2)/b/d^3/(d*x^2+c)^(1/2)+2/15*f*(3*a*d*f-3*b*c*f +5*b*d*e)*x^3*(b*x^2+a)^(1/2)/d^2/(d*x^2+c)^(1/2)+1/5*b*f^2*x^5*(b*x^2+a)^ (1/2)/d/(d*x^2+c)^(1/2)-1/15*(3*a^2*c*d^2*f^2+2*b^2*c*(24*c^2*f^2-40*c*d*e *f+15*d^2*e^2)-a*b*d*(48*c^2*f^2-70*c*d*e*f+15*d^2*e^2))*(b*x^2+a)^(1/2)*E llipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/b/c^(1/2)/ d^(7/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)+1/15*c^(1/2)*(3*a* d*f*(-7*c*f+10*d*e)+b*(24*c^2*f^2-40*c*d*e*f+15*d^2*e^2))*(b*x^2+a)^(1/2)* InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/d^(7/2)/(c*(b *x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
Result contains complex when optimal does not.
Time = 6.74 (sec) , antiderivative size = 384, normalized size of antiderivative = 0.84 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 a d \left (5 d^2 e^2+7 c^2 f^2+2 c d f \left (-5 e+f x^2\right )\right )+b c \left (-24 c^2 f^2+2 c d f \left (20 e-3 f x^2\right )+d^2 \left (-15 e^2+10 e f x^2+3 f^2 x^4\right )\right )\right )-i c \left (3 a^2 c d^2 f^2+a b d \left (-15 d^2 e^2+70 c d e f-48 c^2 f^2\right )+2 b^2 c \left (15 d^2 e^2-40 c d e f+24 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i c (-b c+a d) \left (3 a d f (-5 d e+4 c f)+b \left (-15 d^2 e^2+40 c d e f-24 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 \sqrt {\frac {b}{a}} c d^4 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
Output:
(Sqrt[b/a]*d*x*(a + b*x^2)*(3*a*d*(5*d^2*e^2 + 7*c^2*f^2 + 2*c*d*f*(-5*e + f*x^2)) + b*c*(-24*c^2*f^2 + 2*c*d*f*(20*e - 3*f*x^2) + d^2*(-15*e^2 + 10 *e*f*x^2 + 3*f^2*x^4))) - I*c*(3*a^2*c*d^2*f^2 + a*b*d*(-15*d^2*e^2 + 70*c *d*e*f - 48*c^2*f^2) + 2*b^2*c*(15*d^2*e^2 - 40*c*d*e*f + 24*c^2*f^2))*Sqr t[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a* d)/(b*c)] + (2*I)*c*(-(b*c) + a*d)*(3*a*d*f*(-5*d*e + 4*c*f) + b*(-15*d^2* e^2 + 40*c*d*e*f - 24*c^2*f^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*El lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*Sqrt[b/a]*c*d^4*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(951\) vs. \(2(459)=918\).
Time = 1.31 (sec) , antiderivative size = 951, normalized size of antiderivative = 2.07, 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.
| \(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}+\frac {2 e f x^2 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}+\frac {f^2 x^4 \left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^{3/2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {6 b f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x^3}{5 d^2}-\frac {f^2 \left (b x^2+a\right )^{3/2} x^3}{d \sqrt {d x^2+c}}-\frac {(8 b c-7 a d) f^2 \sqrt {b x^2+a} \sqrt {d x^2+c} x}{5 d^3}+\frac {8 b e f \sqrt {b x^2+a} \sqrt {d x^2+c} x}{3 d^2}-\frac {2 e f \left (b x^2+a\right )^{3/2} x}{d \sqrt {d x^2+c}}-\frac {(b c-a d) e^2 \sqrt {b x^2+a} x}{c d \sqrt {d x^2+c}}+\frac {(2 b c-a d) e^2 \sqrt {b x^2+a} x}{c d \sqrt {d x^2+c}}-\frac {\left (-\frac {d a^2}{b}+16 c a-\frac {16 b c^2}{d}\right ) f^2 \sqrt {b x^2+a} x}{5 d^2 \sqrt {d x^2+c}}-\frac {2 (8 b c-7 a d) e f \sqrt {b x^2+a} x}{3 d^2 \sqrt {d x^2+c}}-\frac {(2 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 )}{\sqrt {c} 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 {\sqrt {c} \left (16 b^2 c^2-16 a b d c+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 d^{7/2} \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} (8 b c-7 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 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 {b \sqrt {c} 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 )}{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 {c^{3/2} (8 b c-7 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 d^{7/2} \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} (4 b c-3 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 d^{5/2} \sqrt {\frac {c \left (b x^2+a\right )}{a \left (d x^2+c\right )}} \sqrt {d x^2+c}}\) |
Input:
Int[((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x]
Output:
-(((b*c - a*d)*e^2*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2])) + ((2*b*c - a *d)*e^2*x*Sqrt[a + b*x^2])/(c*d*Sqrt[c + d*x^2]) - (2*(8*b*c - 7*a*d)*e*f* x*Sqrt[a + b*x^2])/(3*d^2*Sqrt[c + d*x^2]) - ((16*a*c - (16*b*c^2)/d - (a^ 2*d)/b)*f^2*x*Sqrt[a + b*x^2])/(5*d^2*Sqrt[c + d*x^2]) - (2*e*f*x*(a + b*x ^2)^(3/2))/(d*Sqrt[c + d*x^2]) - (f^2*x^3*(a + b*x^2)^(3/2))/(d*Sqrt[c + d *x^2]) + (8*b*e*f*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(3*d^2) - ((8*b*c - 7 *a*d)*f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d^3) + (6*b*f^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(5*d^2) - ((2*b*c - a*d)*e^2*Sqrt[a + b*x^2]*El lipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*d^(3/2)*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(8*b*c - 7*a*d)*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b* c)/(a*d)])/(3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 ]) - (Sqrt[c]*(16*b^2*c^2 - 16*a*b*c*d + a^2*d^2)*f^2*Sqrt[a + b*x^2]*Elli pticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(5*b*d^(7/2)*Sqrt[(c* (a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*Sqrt[c]*e^2*Sqrt[a + b *x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(d^(3/2)*Sq rt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*Sqrt[c]*(4*b*c - 3*a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b* c)/(a*d)])/(3*d^(5/2)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 ]) + (c^(3/2)*(8*b*c - 7*a*d)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqr...
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 = 15.30 (sec) , antiderivative size = 846, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(\frac {f x \left (3 b d f \,x^{2}+6 a d f -9 b c f +10 b d e \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 d^{3}}+\frac {\left (-\frac {\left (3 a^{2} d^{2} f^{2}-33 a b c d \,f^{2}+40 a b \,d^{2} e f +33 b^{2} c^{2} f^{2}-50 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^{2} c^{2} f^{2} d^{2}-2 a^{2} c e f \,d^{3}+a^{2} e^{2} d^{4}-2 a b \,c^{3} d \,f^{2}+4 a b \,c^{2} d^{2} e f -2 a b c \,d^{3} e^{2}+b^{2} c^{4} f^{2}-2 b^{2} c^{3} d e f +b^{2} c^{2} d^{2} e^{2}\right ) \left (\frac {\left (b d \,x^{2}+a d \right ) x}{c \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {1}{c}-\frac {a d}{\left (a d -b c \right ) c}\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 \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 ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{d}-\frac {\left (21 a^{2} c \,d^{2} f^{2}-30 a^{2} d^{3} e f -39 a b \,c^{2} d \,f^{2}+70 a b c \,d^{2} e f -30 a b \,d^{3} e^{2}+15 b^{2} c^{3} f^{2}-30 b^{2} c^{2} d e f +15 b^{2} c \,d^{2} 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 )}{d \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 d^{3} \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(846\) |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (b d \,x^{2}+a d \right ) \left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) x}{c \,d^{4} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {b \,f^{2} x^{3} \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 d^{2}}+\frac {\left (\frac {f b \left (2 a d f -b c f +2 b d e \right )}{d^{2}}-\frac {b \,f^{2} \left (4 a d +4 b c \right )}{5 d^{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^{2} c \,d^{2} f^{2}-2 a^{2} d^{3} e f -2 a b \,c^{2} d \,f^{2}+4 a b c \,d^{2} e f -2 a b \,d^{3} e^{2}+b^{2} c^{3} f^{2}-2 b^{2} c^{2} d e f +b^{2} c \,d^{2} e^{2}}{d^{4}}+\frac {\left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) \left (a d -b c \right )}{d^{4} c}-\frac {a \left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right )}{d^{3} c}-\frac {\left (\frac {f b \left (2 a d f -b c f +2 b d e \right )}{d^{2}}-\frac {b \,f^{2} \left (4 a d +4 b c \right )}{5 d^{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}+4 a b \,d^{2} e f +b^{2} c^{2} f^{2}-2 b^{2} c d e f +b^{2} d^{2} e^{2}}{d^{3}}-\frac {\left (a \,c^{2} d \,f^{2}-2 a c e f \,d^{2}+a \,d^{3} e^{2}-b \,c^{3} f^{2}+2 b \,c^{2} d e f -b c \,d^{2} e^{2}\right ) b}{d^{3} c}-\frac {3 a b c \,f^{2}}{5 d^{2}}-\frac {\left (\frac {f b \left (2 a d f -b c f +2 b d e \right )}{d^{2}}-\frac {b \,f^{2} \left (4 a d +4 b c \right )}{5 d^{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}}\) | \(924\) |
| default | \(\text {Expression too large to display}\) | \(1334\) |
Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/15*f*x*(3*b*d*f*x^2+6*a*d*f-9*b*c*f+10*b*d*e)*(b*x^2+a)^(1/2)*(d*x^2+c)^ (1/2)/d^3+1/15/d^3*(-(3*a^2*d^2*f^2-33*a*b*c*d*f^2+40*a*b*d^2*e*f+33*b^2*c ^2*f^2-50*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^2*c^2*d^2*f^2-2*a^2*c*d^3*e*f+a^2*d^4*e^2-2*a*b*c^3*d*f^2+4 *a*b*c^2*d^2*e*f-2*a*b*c*d^3*e^2+b^2*c^4*f^2-2*b^2*c^3*d*e*f+b^2*c^2*d^2*e ^2)/d*((b*d*x^2+a*d)/c/(a*d-b*c)*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/c-1/ (a*d-b*c)/c*a*d)/(-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)/(-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))))-(21*a^2*c*d^2*f ^2-30*a^2*d^3*e*f-39*a*b*c^2*d*f^2+70*a*b*c*d^2*e*f-30*a*b*d^3*e^2+15*b^2* c^3*f^2-30*b^2*c^2*d*e*f+15*b^2*c*d^2*e^2)/d/(-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 = 771, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="fricas ")
Output:
-1/15*(((15*(2*b^2*c^2*d^3 - a*b*c*d^4)*e^2 - 10*(8*b^2*c^3*d^2 - 7*a*b*c^ 2*d^3)*e*f + 3*(16*b^2*c^4*d - 16*a*b*c^3*d^2 + a^2*c^2*d^3)*f^2)*x^3 + (1 5*(2*b^2*c^3*d^2 - a*b*c^2*d^3)*e^2 - 10*(8*b^2*c^4*d - 7*a*b*c^3*d^2)*e*f + 3*(16*b^2*c^5 - 16*a*b*c^4*d + a^2*c^3*d^2)*f^2)*x)*sqrt(b*d)*sqrt(-c/d )*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - ((15*(2*b^2*c^2*d^3 - a*b* c*d^4 + a*b*d^5)*e^2 - 10*(8*b^2*c^3*d^2 - 7*a*b*c^2*d^3 + 4*a*b*c*d^4 - 3 *a^2*d^5)*e*f + 3*(16*b^2*c^4*d - 16*a*b*c^3*d^2 - 7*a^2*c*d^4 + (a^2 + 8* a*b)*c^2*d^3)*f^2)*x^3 + (15*(2*b^2*c^3*d^2 - a*b*c^2*d^3 + a*b*c*d^4)*e^2 - 10*(8*b^2*c^4*d - 7*a*b*c^3*d^2 + 4*a*b*c^2*d^3 - 3*a^2*c*d^4)*e*f + 3* (16*b^2*c^5 - 16*a*b*c^4*d - 7*a^2*c^2*d^3 + (a^2 + 8*a*b)*c^3*d^2)*f^2)*x )*sqrt(b*d)*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b^ 2*c*d^4*f^2*x^6 + 2*(5*b^2*c*d^4*e*f - 3*(b^2*c^2*d^3 - a*b*c*d^4)*f^2)*x^ 4 + 15*(2*b^2*c^2*d^3 - a*b*c*d^4)*e^2 - 10*(8*b^2*c^3*d^2 - 7*a*b*c^2*d^3 )*e*f + 3*(16*b^2*c^4*d - 16*a*b*c^3*d^2 + a^2*c^2*d^3)*f^2 + (15*b^2*c*d^ 4*e^2 - 40*(b^2*c^2*d^3 - a*b*c*d^4)*e*f + 3*(8*b^2*c^3*d^2 - 9*a*b*c^2*d^ 3 + a^2*c*d^4)*f^2)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b*c*d^6*x^3 + b *c^2*d^5*x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (e + f x^{2}\right )^{2}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((b*x**2+a)**(3/2)*(f*x**2+e)**2/(d*x**2+c)**(3/2),x)
Output:
Integral((a + b*x**2)**(3/2)*(e + f*x**2)**2/(c + d*x**2)**(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="maxima ")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/(d*x^2 + c)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (f x^{2} + e\right )}^{2}}{{\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((b*x^2 + a)^(3/2)*(f*x^2 + e)^2/(d*x^2 + c)^(3/2), x)
Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,{\left (f\,x^2+e\right )}^2}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2),x)
Output:
int(((a + b*x^2)^(3/2)*(e + f*x^2)^2)/(c + d*x^2)^(3/2), x)
\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (e+f x^2\right )^2}{\left (c+d x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((b*x^2+a)^(3/2)*(f*x^2+e)^2/(d*x^2+c)^(3/2),x)
Output:
( - 9*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f**2*x + 15*sqrt(c + d*x* *2)*sqrt(a + b*x**2)*a**2*d**2*e*f*x + 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 + 15*sqrt(c + d*x**2)* sqrt(a + b*x**2)*a*b*d**2*e**2*x - 6*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b** 2*c**2*f**2*x**3 + 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e*f*x**3 + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c*d*f**2*x**5 + 12*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b* c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*c**2*d**2*f**2 - 15*int( (sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x **4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*c*d**3*e*f + 12* int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d* *2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*c*d**3*f**2* x**2 - 15*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x **2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a**2*b*d* *4*e*f*x**2 - 36*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2* a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6),x)*a* b**2*c**3*d*f**2 + 55*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d*x**4 + b*d**2*x**6), x)*a*b**2*c**2*d**2*e*f - 36*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**...