Integrand size = 32, antiderivative size = 308 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(b e-a f)^2 x}{a b (b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\left (b^2 c d e^2+a^2 c d f^2+a b \left (d^2 e^2-4 c d e f+c^2 f^2\right )\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a b \sqrt {c} \sqrt {d} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {2 \sqrt {c} (b e-a f) (d e-c f) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:
(-a*f+b*e)^2*x/a/b/(-a*d+b*c)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)+(b^2*c*d*e^2 +a^2*c*d*f^2+a*b*(c^2*f^2-4*c*d*e*f+d^2*e^2))*(b*x^2+a)^(1/2)*EllipticE(d^ (1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))/a/b/c^(1/2)/d^(1/2)/( -a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-2*c^(1/2)*(-a* f+b*e)*(-c*f+d*e)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2) ),(1-b*c/a/d)^(1/2))/a/d^(1/2)/(-a*d+b*c)^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 = 11.08 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.09 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (b^2 c e^2 \left (c+d x^2\right )+a b \left (d^2 e^2 x^2-4 c d e f x^2+c^2 f \left (-2 e+f x^2\right )\right )+a^2 \left (d^2 e^2+2 c^2 f^2+c d f \left (-2 e+f x^2\right )\right )\right )+i c \left (b^2 c d e^2+a^2 c d f^2+a b \left (d^2 e^2-4 c d e f+c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (b d e^2+a f (-2 d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b c d (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]
Output:
(Sqrt[b/a]*(Sqrt[b/a]*d*x*(b^2*c*e^2*(c + d*x^2) + a*b*(d^2*e^2*x^2 - 4*c* d*e*f*x^2 + c^2*f*(-2*e + f*x^2)) + a^2*(d^2*e^2 + 2*c^2*f^2 + c*d*f*(-2*e + f*x^2))) + I*c*(b^2*c*d*e^2 + a^2*c*d*f^2 + a*b*(d^2*e^2 - 4*c*d*e*f + 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)*(b*d*e^2 + a*f*(-2*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)]))/(b*c*d*(b*c - a*d)^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Leaf count is larger than twice the leaf count of optimal. \(738\) vs. \(2(308)=616\).
Time = 0.84 (sec) , antiderivative size = 738, normalized size of antiderivative = 2.40, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 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 |
| \(\displaystyle -\frac {2 c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 b \sqrt {c} \sqrt {d} e^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {d} e^2 \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 \sqrt {c} e f \sqrt {a+b x^2} (a d+b c) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {4 \sqrt {c} \sqrt {d} e f \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} f^2 \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {b e^2 x}{a \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}-\frac {2 e f x}{\sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}+\frac {a f^2 x}{b \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)}\) |
Input:
Int[(e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]
Output:
(b*e^2*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) - (2*e*f*x)/((b* c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) + (a*f^2*x)/(b*(b*c - a*d)*Sqrt[ a + b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[d]*(b*c + a*d)*e^2*Sqrt[a + b*x^2]*Ell ipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*(b*c - a* d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (4*Sqrt[c]*S qrt[d]*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c )/(a*d)])/((b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d* x^2]) + (Sqrt[c]*(b*c + a*d)*f^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d] *x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x^ 2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (2*b*Sqrt[c]*Sqrt[d]*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c] *(b*c + a*d)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^ 2))]*Sqrt[c + d*x^2]) - (2*c^(3/2)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(S qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)^2*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]
Leaf count of result is larger than twice the leaf count of optimal. \(648\) vs. \(2(291)=582\).
Time = 8.49 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.11
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {2 b d \left (-\frac {\left (a^{2} c d \,f^{2}+a b \,c^{2} f^{2}-4 a b c d e f +a b \,d^{2} e^{2}+b^{2} d \,e^{2} c \right ) x^{3}}{2 b d a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (2 a^{2} c^{2} f^{2}-2 a^{2} c d e f +a^{2} d^{2} e^{2}-2 a b \,c^{2} e f +b^{2} c^{2} e^{2}\right ) x}{2 b d a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{d b}+\frac {a c}{d b}\right ) b d}}+\frac {\left (\frac {f^{2}}{b d}-\frac {a c \,f^{2}-b d \,e^{2}}{b d a c}-\frac {2 a^{2} c^{2} f^{2}-2 a^{2} c d e f +a^{2} d^{2} e^{2}-2 a b \,c^{2} e f +b^{2} c^{2} e^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\left (a^{2} c d \,f^{2}+a b \,c^{2} f^{2}-4 a b c d e f +a b \,d^{2} e^{2}+b^{2} d \,e^{2} c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(649\) |
| default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} f^{2} x^{3}+\sqrt {-\frac {b}{a}}\, a b \,c^{2} d \,f^{2} x^{3}-4 \sqrt {-\frac {b}{a}}\, a b c \,d^{2} e f \,x^{3}+\sqrt {-\frac {b}{a}}\, a b \,d^{3} e^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b^{2} c \,d^{2} e^{2} 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^{2} d \,f^{2}+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^{2} c \,d^{2} e f +\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^{3} f^{2}-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} d e f -\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^{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^{2} 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^{3} f^{2}+4 \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 e f -\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^{2} d \,e^{2}+2 \sqrt {-\frac {b}{a}}\, a^{2} c^{2} d \,f^{2} x -2 \sqrt {-\frac {b}{a}}\, a^{2} c \,d^{2} e f x +\sqrt {-\frac {b}{a}}\, a^{2} d^{3} e^{2} x -2 \sqrt {-\frac {b}{a}}\, a b \,c^{2} d e f x +\sqrt {-\frac {b}{a}}\, b^{2} c^{2} d \,e^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{d c \sqrt {-\frac {b}{a}}\, a \left (a d -b c \right )^{2} \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(924\) |
Input:
int((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(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)*(-2*b*d*(-1/2/ b/d*(a^2*c*d*f^2+a*b*c^2*f^2-4*a*b*c*d*e*f+a*b*d^2*e^2+b^2*c*d*e^2)/a/c/(a ^2*d^2-2*a*b*c*d+b^2*c^2)*x^3-1/2/b/d*(2*a^2*c^2*f^2-2*a^2*c*d*e*f+a^2*d^2 *e^2-2*a*b*c^2*e*f+b^2*c^2*e^2)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x)/((x^4+( a*d+b*c)/d/b*x^2+a*c/d/b)*b*d)^(1/2)+(f^2/b/d-1/b/d*(a*c*f^2-b*d*e^2)/a/c- (2*a^2*c^2*f^2-2*a^2*c*d*e*f+a^2*d^2*e^2-2*a*b*c^2*e*f+b^2*c^2*e^2)/a/c/(a ^2*d^2-2*a*b*c*d+b^2*c^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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b* c)/c/b)^(1/2))+(a^2*c*d*f^2+a*b*c^2*f^2-4*a*b*c*d*e*f+a*b*d^2*e^2+b^2*c*d* e^2)/a/(a^2*d^2-2*a*b*c*d+b^2*c^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)/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 ))))
Leaf count of result is larger than twice the leaf count of optimal. 942 vs. \(2 (291) = 582\).
Time = 0.12 (sec) , antiderivative size = 942, normalized size of antiderivative = 3.06 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="fricas ")
Output:
((4*a^2*b^2*c^2*d*e*f + (4*a*b^3*c*d^2*e*f - (b^4*c*d^2 + a*b^3*d^3)*e^2 - (a*b^3*c^2*d + a^2*b^2*c*d^2)*f^2)*x^4 - (a*b^3*c^2*d + a^2*b^2*c*d^2)*e^ 2 - (a^2*b^2*c^3 + a^3*b*c^2*d)*f^2 - ((b^4*c^2*d + 2*a*b^3*c*d^2 + a^2*b^ 2*d^3)*e^2 - 4*(a*b^3*c^2*d + a^2*b^2*c*d^2)*e*f + (a*b^3*c^3 + 2*a^2*b^2* c^2*d + a^3*b*c*d^2)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sq rt(-b/a)), a*d/(b*c)) + (((b^4*c*d^2 + (2*a^2*b^2 + a*b^3)*d^3)*e^2 - 2*(a ^3*b*d^3 + (a^2*b^2 + 2*a*b^3)*c*d^2)*e*f + (a*b^3*c^2*d + (2*a^3*b + a^2* b^2)*c*d^2)*f^2)*x^4 + (a*b^3*c^2*d + (2*a^3*b + a^2*b^2)*c*d^2)*e^2 - 2*( a^4*c*d^2 + (a^3*b + 2*a^2*b^2)*c^2*d)*e*f + (a^2*b^2*c^3 + (2*a^4 + a^3*b )*c^2*d)*f^2 + ((b^4*c^2*d + 2*(a^2*b^2 + a*b^3)*c*d^2 + (2*a^3*b + a^2*b^ 2)*d^3)*e^2 - 2*(a^4*d^3 + (a^2*b^2 + 2*a*b^3)*c^2*d + 2*(a^3*b + a^2*b^2) *c*d^2)*e*f + (a*b^3*c^3 + 2*(a^3*b + a^2*b^2)*c^2*d + (2*a^4 + a^3*b)*c*d ^2)*f^2)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b *c)) - ((4*a^2*b^2*c*d^2*e*f - (a*b^3*c*d^2 + a^2*b^2*d^3)*e^2 - (a^2*b^2* c^2*d + a^3*b*c*d^2)*f^2)*x^3 - (2*a^3*b*c^2*d*f^2 + (a*b^3*c^2*d + a^3*b* d^3)*e^2 - 2*(a^2*b^2*c^2*d + a^3*b*c*d^2)*e*f)*x)*sqrt(b*x^2 + a)*sqrt(d* x^2 + c))/(a^3*b^3*c^4*d - 2*a^4*b^2*c^3*d^2 + a^5*b*c^2*d^3 + (a^2*b^4*c^ 3*d^2 - 2*a^3*b^3*c^2*d^3 + a^4*b^2*c*d^4)*x^4 + (a^2*b^4*c^4*d - a^3*b^3* c^3*d^2 - a^4*b^2*c^2*d^3 + a^5*b*c*d^4)*x^2)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/(b*x**2+a)**(3/2)/(d*x**2+c)**(3/2),x)
Output:
Integral((e + f*x**2)**2/((a + b*x**2)**(3/2)*(c + d*x**2)**(3/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="maxima ")
Output:
integrate((f*x^2 + e)^2/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algorithm="giac")
Output:
integrate((f*x^2 + e)^2/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x)
Output:
int((e + f*x^2)^2/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
int((f*x^2+e)^2/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
Output:
( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*f**2*x + 2*int((sqrt(c + d*x**2)*sqr t(a + b*x**2)*x**2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b* c**2*x**2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d *x**6 + b**2*d**2*x**8),x)*a*b*c*d*e*f + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x **2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8),x)*a*b*d**2*e*f*x**2 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x **2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8),x)*b**2*c*d*e*f*x**2 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x **2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8),x)*b**2*d**2*e*f*x**4 + int((sqrt(c + d*x**2)*sqrt(a + b *x**2))/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2 *d**2*x**8),x)*a**2*c**2*f**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a **2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d* x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8 ),x)*a**2*c*d*f**2*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c* *2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d*x**...