Integrand size = 28, antiderivative size = 204 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=\frac {\sqrt {-b+\sqrt {b^2+4 a c}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2+4 a c}\right ) e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {-b+\sqrt {b^2+4 a c}}}\right ),\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} d \sqrt {-a+b x^2+c x^4}} \]
1/2*EllipticPi(x*2^(1/2)*c^(1/2)/(-b+(4*a*c+b^2)^(1/2))^(1/2),1/2*e*(b-(4* a*c+b^2)^(1/2))/c/d,((b-(4*a*c+b^2)^(1/2))/(b+(4*a*c+b^2)^(1/2)))^(1/2))*( 1+2*c*x^2/(b-(4*a*c+b^2)^(1/2)))^(1/2)*(-b+(4*a*c+b^2)^(1/2))^(1/2)*(1+2*c *x^2/(b+(4*a*c+b^2)^(1/2)))^(1/2)/d*2^(1/2)/c^(1/2)/(c*x^4+b*x^2-a)^(1/2)
Result contains complex when optimal does not.
Time = 10.21 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=-\frac {i \sqrt {\frac {b+\sqrt {b^2+4 a c}+2 c x^2}{b+\sqrt {b^2+4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2+4 a c}}} d \sqrt {-a+b x^2+c x^4}} \]
((-I)*Sqrt[(b + Sqrt[b^2 + 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*Sqrt [1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*EllipticPi[((b + Sqrt[b^2 + 4*a*c] )*e)/(2*c*d), I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*x], (b + S qrt[b^2 + 4*a*c])/(b - Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 + 4*a*c])]*d*Sqrt[-a + b*x^2 + c*x^4])
Time = 0.37 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1544, 412}
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.
\(\displaystyle \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx\) |
\(\Big \downarrow \) 1544 |
\(\displaystyle \frac {\sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \int \frac {1}{\sqrt {\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}+1} \sqrt {\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}+1} \left (e x^2+d\right )}dx}{\sqrt {-a+b x^2+c x^4}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {\sqrt {\sqrt {4 a c+b^2}-b} \sqrt {\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}+1} \sqrt {\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}+1} \operatorname {EllipticPi}\left (\frac {\left (b-\sqrt {b^2+4 a c}\right ) e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2+4 a c}-b}}\right ),\frac {b-\sqrt {b^2+4 a c}}{b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} d \sqrt {-a+b x^2+c x^4}}\) |
(Sqrt[-b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]* Sqrt[1 + (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[((b - Sqrt[b^2 + 4* a*c])*e)/(2*c*d), ArcSin[(Sqrt[2]*Sqrt[c]*x)/Sqrt[-b + Sqrt[b^2 + 4*a*c]]] , (b - Sqrt[b^2 + 4*a*c])/(b + Sqrt[b^2 + 4*a*c])])/(Sqrt[2]*Sqrt[c]*d*Sqr t[-a + b*x^2 + c*x^4])
3.4.91.3.1 Defintions of rubi rules used
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt[1 + 2*c*(x^2/(b - q))]*( Sqrt[1 + 2*c*(x^2/(b + q))]/Sqrt[a + b*x^2 + c*x^4]) Int[1/((d + e*x^2)*S qrt[1 + 2*c*(x^2/(b - q))]*Sqrt[1 + 2*c*(x^2/(b + q))]), x], x]] /; FreeQ[{ a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[c/a]
Time = 1.05 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97
method | result | size |
default | \(\frac {\sqrt {1-\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {1-\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \Pi \left (\sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}\, x , \frac {2 a e}{\left (-b +\sqrt {4 a c +b^{2}}\right ) d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2 \sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}}\right )}{d \sqrt {\frac {b}{2 a}-\frac {\sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}\) | \(198\) |
elliptic | \(\frac {\sqrt {1-\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {1-\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \Pi \left (\sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}\, x , \frac {2 a e}{\left (-b +\sqrt {4 a c +b^{2}}\right ) d}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {4 a c +b^{2}}}{a}}}{2 \sqrt {-\frac {-b +\sqrt {4 a c +b^{2}}}{2 a}}}\right )}{d \sqrt {\frac {b}{2 a}-\frac {\sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {c \,x^{4}+b \,x^{2}-a}}\) | \(198\) |
1/d/(1/2*b/a-1/2/a*(4*a*c+b^2)^(1/2))^(1/2)*(1-1/2*b*x^2/a+1/2*x^2/a*(4*a* c+b^2)^(1/2))^(1/2)*(1-1/2*b*x^2/a-1/2*x^2/a*(4*a*c+b^2)^(1/2))^(1/2)/(c*x ^4+b*x^2-a)^(1/2)*EllipticPi((-1/2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2)*x,2/(-b +(4*a*c+b^2)^(1/2))*a*e/d,1/2*2^(1/2)*((b+(4*a*c+b^2)^(1/2))/a)^(1/2)/(-1/ 2*(-b+(4*a*c+b^2)^(1/2))/a)^(1/2))
Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{\left (d + e x^{2}\right ) \sqrt {- a + b x^{2} + c x^{4}}}\, dx \]
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - a} {\left (e x^{2} + d\right )}} \,d x } \]
\[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=\int { \frac {1}{\sqrt {c x^{4} + b x^{2} - a} {\left (e x^{2} + d\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {-a+b x^2+c x^4}} \, dx=\int \frac {1}{\left (e\,x^2+d\right )\,\sqrt {c\,x^4+b\,x^2-a}} \,d x \]