Integrand size = 44, antiderivative size = 371 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {C x \sqrt {e+f x^2}}{b f \sqrt {c+d x^2}}-\frac {\sqrt {c} C \sqrt {e+f x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {c f}{d e}\right )}{b \sqrt {d} f \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {\sqrt {c} (b B c-a c C-A b d) \sqrt {e+f x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{b \sqrt {d} (b c-a d) e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}}+\frac {c^{3/2} \left (A b^2-a (b B-a C)\right ) \sqrt {e+f x^2} \operatorname {EllipticPi}\left (1-\frac {b c}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {c f}{d e}\right )}{a b \sqrt {d} (b c-a d) e \sqrt {c+d x^2} \sqrt {\frac {c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}} \] Output:
C*x*(f*x^2+e)^(1/2)/b/f/(d*x^2+c)^(1/2)-c^(1/2)*C*(f*x^2+e)^(1/2)*Elliptic E(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-c*f/d/e)^(1/2))/b/d^(1/2)/f/(d*x^ 2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)+c^(1/2)*(-A*b*d+B*b*c-C*a*c)*(f *x^2+e)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-c*f/d/e)^(1/2)) /b/d^(1/2)/(-a*d+b*c)/e/(d*x^2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)+c^ (3/2)*(A*b^2-a*(B*b-C*a))*(f*x^2+e)^(1/2)*EllipticPi(d^(1/2)*x/c^(1/2)/(1+ d*x^2/c)^(1/2),1-b*c/a/d,(1-c*f/d/e)^(1/2))/a/b/d^(1/2)/(-a*d+b*c)/e/(d*x^ 2+c)^(1/2)/(c*(f*x^2+e)/e/(d*x^2+c))^(1/2)
Result contains complex when optimal does not.
Time = 7.31 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.54 \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {i \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \left (a b C e E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right )|\frac {c f}{d e}\right )-a (b C e-b B f+a C f) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )+\left (A b^2+a (-b B+a C)\right ) f \operatorname {EllipticPi}\left (\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {\frac {d}{c}} x\right ),\frac {c f}{d e}\right )\right )}{a b^2 \sqrt {\frac {d}{c}} f \sqrt {c+d x^2} \sqrt {e+f x^2}} \] Input:
Integrate[(A + B*x^2 + C*x^4)/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2] ),x]
Output:
((-I)*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*(a*b*C*e*EllipticE[I*ArcSinh [Sqrt[d/c]*x], (c*f)/(d*e)] - a*(b*C*e - b*B*f + a*C*f)*EllipticF[I*ArcSin h[Sqrt[d/c]*x], (c*f)/(d*e)] + (A*b^2 + a*(-(b*B) + a*C))*f*EllipticPi[(b* c)/(a*d), I*ArcSinh[Sqrt[d/c]*x], (c*f)/(d*e)]))/(a*b^2*Sqrt[d/c]*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 1.10 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.92, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {7276, 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.
| \(\displaystyle \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a^2 C-a b B+A b^2}{b^2 \left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {b B-a C}{b^2 \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {C x^2}{b \sqrt {c+d x^2} \sqrt {e+f x^2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \left (A b^2-a (b B-a C)\right ) \operatorname {EllipticPi}\left (\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a b^2 \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {e} \sqrt {c+d x^2} (b B-a C) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right ),1-\frac {d e}{c f}\right )}{b^2 c \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}-\frac {C \sqrt {e} \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {f} x}{\sqrt {e}}\right )|1-\frac {d e}{c f}\right )}{b d \sqrt {f} \sqrt {e+f x^2} \sqrt {\frac {e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac {C x \sqrt {c+d x^2}}{b d \sqrt {e+f x^2}}\) |
Input:
Int[(A + B*x^2 + C*x^4)/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]),x]
Output:
(C*x*Sqrt[c + d*x^2])/(b*d*Sqrt[e + f*x^2]) - (C*Sqrt[e]*Sqrt[c + d*x^2]*E llipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b*d*Sqrt[f]*Sqrt[ (e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + ((b*B - a*C)*Sqrt[e]*S qrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(b ^2*c*Sqrt[f]*Sqrt[(e*(c + d*x^2))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (Sqr t[-c]*(A*b^2 - a*(b*B - a*C))*Sqrt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*Elli pticPi[(b*c)/(a*d), ArcSin[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*b^2*Sqr t[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.92 (sec) , antiderivative size = 328, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\left (A \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) b^{2} f +B \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a b f -B \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a b f -C \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a^{2} f -C \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a b e +C \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {\frac {c f}{d e}}\right ) a b e +C \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) a^{2} f \right ) \sqrt {\frac {f \,x^{2}+e}{e}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}{f a \sqrt {-\frac {d}{c}}\, b^{2} \left (d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e \right )}\) | \(328\) |
| elliptic | \(\frac {\sqrt {\left (x^{2} d +c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {B \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right ) C a}{b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {C e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}+\frac {C e \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1+\frac {c f +d e}{e d}}\right )}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}\, f}+\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) A}{a \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}-\frac {\sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) B}{b \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}+\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \operatorname {EllipticPi}\left (x \sqrt {-\frac {d}{c}}, \frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) C}{b^{2} \sqrt {-\frac {d}{c}}\, \sqrt {d f \,x^{4}+c f \,x^{2}+d e \,x^{2}+c e}}\right )}{\sqrt {x^{2} d +c}\, \sqrt {f \,x^{2}+e}}\) | \(705\) |
Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x,method=_RE TURNVERBOSE)
Output:
(A*EllipticPi(x*(-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*b^2*f+ B*EllipticF(x*(-1/c*d)^(1/2),(c*f/d/e)^(1/2))*a*b*f-B*EllipticPi(x*(-1/c*d )^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*a*b*f-C*EllipticF(x*(-1/c*d)^ (1/2),(c*f/d/e)^(1/2))*a^2*f-C*EllipticF(x*(-1/c*d)^(1/2),(c*f/d/e)^(1/2)) *a*b*e+C*EllipticE(x*(-1/c*d)^(1/2),(c*f/d/e)^(1/2))*a*b*e+C*EllipticPi(x* (-1/c*d)^(1/2),b*c/a/d,(-f/e)^(1/2)/(-1/c*d)^(1/2))*a^2*f)*((f*x^2+e)/e)^( 1/2)*((d*x^2+c)/c)^(1/2)*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/f/a/(-1/c*d)^(1/2 )/b^2/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, alg orithm="fricas")
Output:
Timed out
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {A + B x^{2} + C x^{4}}{\left (a + b x^{2}\right ) \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \] Input:
integrate((C*x**4+B*x**2+A)/(b*x**2+a)/(d*x**2+c)**(1/2)/(f*x**2+e)**(1/2) ,x)
Output:
Integral((A + B*x**2 + C*x**4)/((a + b*x**2)*sqrt(c + d*x**2)*sqrt(e + f*x **2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, alg orithm="maxima")
Output:
integrate((C*x^4 + B*x^2 + A)/((b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e) ), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {C x^{4} + B x^{2} + A}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + c} \sqrt {f x^{2} + e}} \,d x } \] Input:
integrate((C*x^4+B*x^2+A)/(b*x^2+a)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x, alg orithm="giac")
Output:
integrate((C*x^4 + B*x^2 + A)/((b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(f*x^2 + e) ), x)
Timed out. \[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{\left (b\,x^2+a\right )\,\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}} \,d x \] Input:
int((A + B*x^2 + C*x^4)/((a + b*x^2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
Output:
int((A + B*x^2 + C*x^4)/((a + b*x^2)*(c + d*x^2)^(1/2)*(e + f*x^2)^(1/2)), x)
\[ \int \frac {A+B x^2+C x^4}{\left (a+b x^2\right ) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, x^{4}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) c +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}\, x^{2}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) b +\left (\int \frac {\sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{b d f \,x^{6}+a d f \,x^{4}+b c f \,x^{4}+b d e \,x^{4}+a c f \,x^{2}+a d e \,x^{2}+b c e \,x^{2}+a c e}d x \right ) a \] Input:
int((C*x^4+B*x^2+A)/(b*x^2+a)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2),x)
Output:
int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*x**4)/(a*c*e + a*c*f*x**2 + a*d*e*x **2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)*c + int((sqrt(e + f*x**2)*sqrt(c + d*x**2)*x**2)/(a*c*e + a*c*f*x**2 + a*d* e*x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x )*b + int((sqrt(e + f*x**2)*sqrt(c + d*x**2))/(a*c*e + a*c*f*x**2 + a*d*e* x**2 + a*d*f*x**4 + b*c*e*x**2 + b*c*f*x**4 + b*d*e*x**4 + b*d*f*x**6),x)* a