Integrand size = 32, antiderivative size = 302 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {2 f (3 b d e-b c f-a d f) x \sqrt {c+d x^2}}{3 b d^2 \sqrt {a+b x^2}}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}-\frac {2 \sqrt {a} f (3 b d e-b c f-a d f) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 b^{3/2} d^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}+\frac {\sqrt {a} \left (3 b d e^2-a c f^2\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 b^{3/2} c d \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:
2/3*f*(-a*d*f-b*c*f+3*b*d*e)*x*(d*x^2+c)^(1/2)/b/d^2/(b*x^2+a)^(1/2)+1/3*f ^2*x*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/b/d-2/3*a^(1/2)*f*(-a*d*f-b*c*f+3*b*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))/b^(3/2)/d^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)+1/3 *a^(1/2)*(-a*c*f^2+3*b*d*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^(3/2)/c/d/(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 = 6.34 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.78 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} d f^2 x \left (a+b x^2\right ) \left (c+d x^2\right )+2 i c f (-3 b d e+b c f+a d f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (a c d f^2+b \left (3 d^2 e^2-6 c d e f+2 c^2 f^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 b \sqrt {\frac {b}{a}} d^2 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:
Integrate[(e + f*x^2)^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(Sqrt[b/a]*d*f^2*x*(a + b*x^2)*(c + d*x^2) + (2*I)*c*f*(-3*b*d*e + b*c*f + a*d*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*(a*c*d*f^2 + b*(3*d^2*e^2 - 6*c*d*e*f + 2*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)])/(3*b*Sqrt[b/a]*d^2*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
Time = 0.59 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.60, 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.
| \(\displaystyle \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx\) |
| \(\Big \downarrow \) 433 |
| \(\displaystyle \int \left (\frac {e^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {2 e f x^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {f^2 x^4}{\sqrt {a+b x^2} \sqrt {c+d x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \sqrt {c} f^2 \sqrt {a+b x^2} (a d+b c) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 b^2 d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {c^{3/2} f^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 b d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {\sqrt {c} e^2 \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 \sqrt {c} e f \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {2 f^2 x \sqrt {a+b x^2} (a d+b c)}{3 b^2 d \sqrt {c+d x^2}}+\frac {2 e f x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}+\frac {f^2 x \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 b d}\) |
Input:
Int[(e + f*x^2)^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]),x]
Output:
(2*e*f*x*Sqrt[a + b*x^2])/(b*Sqrt[c + d*x^2]) - (2*(b*c + a*d)*f^2*x*Sqrt[ a + b*x^2])/(3*b^2*d*Sqrt[c + d*x^2]) + (f^2*x*Sqrt[a + b*x^2]*Sqrt[c + d* x^2])/(3*b*d) - (2*Sqrt[c]*e*f*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x )/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^ 2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*(b*c + a*d)*f^2*Sqrt[a + b*x^2]*Ellipti cE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b^2*d^(3/2)*Sqrt[(c*( a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (Sqrt[c]*e^2*Sqrt[a + b*x^ 2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqr t[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*f^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*d^( 3/2)*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]
Time = 8.90 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.08
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {f^{2} x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b d}+\frac {\left (e^{2}-\frac {a c \,f^{2}}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (2 e f -\frac {f^{2} \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(325\) |
| risch | \(\frac {f^{2} x \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 b d}-\frac {\left (-\frac {2 f \left (a d f +b c f -3 b d e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}+\frac {a c \,f^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {3 b d \,e^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b d \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\) | \(397\) |
| default | \(\frac {\left (\sqrt {-\frac {b}{a}}\, b \,d^{2} f^{2} x^{5}+\sqrt {-\frac {b}{a}}\, a \,d^{2} f^{2} x^{3}+\sqrt {-\frac {b}{a}}\, b c d \,f^{2} x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d \,f^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f^{2}-6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e f +3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,d^{2} e^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a c d \,f^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b \,c^{2} f^{2}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c d e f +\sqrt {-\frac {b}{a}}\, a c d \,f^{2} x \right ) \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{3 \sqrt {-\frac {b}{a}}\, d^{2} b \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\) | \(524\) |
Input:
int((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),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)*(1/3*f^2/b/d*x *(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(e^2-1/3*a/b*c/d*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)*Elli pticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(2*e*f-1/3*f^2/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*(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 = 224, normalized size of antiderivative = 0.74 \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (3 \, b c^{2} d e f - {\left (b c^{3} + a c^{2} d\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (3 \, b d^{3} e^{2} + 6 \, b c^{2} d e f - {\left (2 \, b c^{3} + 2 \, a c^{2} d + a c d^{2}\right )} f^{2}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (b c d^{2} f^{2} x^{2} + 6 \, b c d^{2} e f - 2 \, {\left (b c^{2} d + a c d^{2}\right )} f^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, b^{2} c d^{3} x} \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="fricas ")
Output:
-1/3*(2*(3*b*c^2*d*e*f - (b*c^3 + a*c^2*d)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*ell iptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (3*b*d^3*e^2 + 6*b*c^2*d*e*f - (2*b*c^3 + 2*a*c^2*d + a*c*d^2)*f^2)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f(arc sin(sqrt(-c/d)/x), a*d/(b*c)) - (b*c*d^2*f^2*x^2 + 6*b*c*d^2*e*f - 2*(b*c^ 2*d + a*c*d^2)*f^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(b^2*c*d^3*x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {\left (e + f x^{2}\right )^{2}}{\sqrt {a + b x^{2}} \sqrt {c + d x^{2}}}\, dx \] Input:
integrate((f*x**2+e)**2/(b*x**2+a)**(1/2)/(d*x**2+c)**(1/2),x)
Output:
Integral((e + f*x**2)**2/(sqrt(a + b*x**2)*sqrt(c + d*x**2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="maxima ")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )}^{2}}{\sqrt {b x^{2} + a} \sqrt {d x^{2} + c}} \,d x } \] Input:
integrate((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
Output:
integrate((f*x^2 + e)^2/(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)), x)
Timed out. \[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\int \frac {{\left (f\,x^2+e\right )}^2}{\sqrt {b\,x^2+a}\,\sqrt {d\,x^2+c}} \,d x \] Input:
int((e + f*x^2)^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)),x)
Output:
int((e + f*x^2)^2/((a + b*x^2)^(1/2)*(c + d*x^2)^(1/2)), x)
\[ \int \frac {\left (e+f x^2\right )^2}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, f^{2} x -2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a d \,f^{2}-2 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b c \,f^{2}+6 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b d e f -\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) a c \,f^{2}+3 \left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b d \,x^{4}+a d \,x^{2}+b c \,x^{2}+a c}d x \right ) b d \,e^{2}}{3 b d} \] Input:
int((f*x^2+e)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2),x)
Output:
(sqrt(c + d*x**2)*sqrt(a + b*x**2)*f**2*x - 2*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*d*f**2 - 2*in t((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*c*f**2 + 6*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*d*e*f - int((sqrt(c + d*x**2)*sqrt( a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a*c*f**2 + 3*int((s qrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x )*b*d*e**2)/(3*b*d)