Integrand size = 27, antiderivative size = 197 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2-c x^4}} \, dx=\frac {\sqrt {b+\sqrt {b^2+4 a c}} \sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticPi}\left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} d \sqrt {a+b x^2-c x^4}} \]
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 = 205, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2-c x^4}} \, dx=-\frac {i \sqrt {1+\frac {2 c x^2}{-b+\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \operatorname {EllipticPi}\left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} x\right ),-\frac {b+\sqrt {b^2+4 a c}}{-b+\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {-\frac {c}{b+\sqrt {b^2+4 a c}}} d \sqrt {a+b x^2-c x^4}} \]
((-I)*Sqrt[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[-1/2*((b + Sqrt[b^2 + 4*a*c])*e)/(c*d), I* ArcSinh[Sqrt[2]*Sqrt[-(c/(b + Sqrt[b^2 + 4*a*c]))]*x], -((b + Sqrt[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 = 0.32 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, 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 |
\(\displaystyle \frac {\sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \int \frac {1}{\sqrt {1-\frac {2 c x^2}{b-\sqrt {b^2+4 a c}}} \sqrt {1-\frac {2 c x^2}{b+\sqrt {b^2+4 a c}}} \left (e x^2+d\right )}dx}{\sqrt {a+b x^2-c x^4}}\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {\sqrt {\sqrt {4 a c+b^2}+b} \sqrt {1-\frac {2 c x^2}{b-\sqrt {4 a c+b^2}}} \sqrt {1-\frac {2 c x^2}{\sqrt {4 a c+b^2}+b}} \operatorname {EllipticPi}\left (-\frac {\left (b+\sqrt {b^2+4 a c}\right ) e}{2 c d},\arcsin \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2+4 a c}}}\right ),\frac {b+\sqrt {b^2+4 a c}}{b-\sqrt {b^2+4 a c}}\right )}{\sqrt {2} \sqrt {c} d \sqrt {a+b x^2-c x^4}}\) |
(Sqrt[b + Sqrt[b^2 + 4*a*c]]*Sqrt[1 - (2*c*x^2)/(b - Sqrt[b^2 + 4*a*c])]*S qrt[1 - (2*c*x^2)/(b + Sqrt[b^2 + 4*a*c])]*EllipticPi[-1/2*((b + Sqrt[b^2 + 4*a*c])*e)/(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*Sq rt[a + b*x^2 - c*x^4])
3.4.88.3.1 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 = 1.04 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \Pi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}\right )}{d \sqrt {-\frac {b}{a}+\frac {\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}\) | \(201\) |
elliptic | \(\frac {\sqrt {2}\, \sqrt {1+\frac {b \,x^{2}}{2 a}-\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {1+\frac {b \,x^{2}}{2 a}+\frac {x^{2} \sqrt {4 a c +b^{2}}}{2 a}}\, \Pi \left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}{2}, -\frac {2 a e}{\left (-b +\sqrt {4 a c +b^{2}}\right ) d}, \frac {\sqrt {-\frac {b +\sqrt {4 a c +b^{2}}}{2 a}}\, \sqrt {2}}{\sqrt {\frac {-b +\sqrt {4 a c +b^{2}}}{a}}}\right )}{d \sqrt {-\frac {b}{a}+\frac {\sqrt {4 a c +b^{2}}}{a}}\, \sqrt {-c \,x^{4}+b \,x^{2}+a}}\) | \(201\) |
1/d*2^(1/2)/(-b/a+1/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*x*2^(1/2)*((-b+(4*a*c+b^2)^(1/2))/a)^ (1/2),-2/(-b+(4*a*c+b^2)^(1/2))*a*e/d,(-1/2*(b+(4*a*c+b^2)^(1/2))/a)^(1/2) *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. \[ \int \frac {1}{\left (d+e x^2\right ) \sqrt {a+b x^2-c x^4}} \, dx=\int \frac {1}{\left (e\,x^2+d\right )\,\sqrt {-c\,x^4+b\,x^2+a}} \,d x \]