Integrand size = 32, antiderivative size = 194 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=-\frac {\sqrt {d} \sqrt {3+f x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {2}}\right ),1-\frac {2 f}{3 d}\right )}{\sqrt {3} (2 b-a d) \sqrt {2+d x^2} \sqrt {\frac {3+f x^2}{2+d x^2}}}+\frac {2 b \sqrt {3+f x^2} \operatorname {EllipticPi}\left (1-\frac {2 b}{a d},\arctan \left (\frac {\sqrt {d} x}{\sqrt {2}}\right ),1-\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {d} (2 b-a d) \sqrt {2+d x^2} \sqrt {\frac {3+f x^2}{2+d x^2}}} \] Output:
-1/3*d^(1/2)*(f*x^2+3)^(1/2)*InverseJacobiAM(arctan(1/2*d^(1/2)*x*2^(1/2)) ,1/3*(9-6*f/d)^(1/2))*3^(1/2)/(-a*d+2*b)/(d*x^2+2)^(1/2)/((f*x^2+3)/(d*x^2 +2))^(1/2)+2/3*b*(f*x^2+3)^(1/2)*EllipticPi(d^(1/2)*x*2^(1/2)/(2*d*x^2+4)^ (1/2),1-2*b/a/d,1/3*(9-6*f/d)^(1/2))*3^(1/2)/a/d^(1/2)/(-a*d+2*b)/(d*x^2+2 )^(1/2)/((f*x^2+3)/(d*x^2+2))^(1/2)
Time = 2.24 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\frac {\operatorname {EllipticPi}\left (\frac {2 b}{a d},\arcsin \left (\frac {\sqrt {-d} x}{\sqrt {2}}\right ),\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \] Input:
Integrate[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]
Output:
EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3 ]*a*Sqrt[-d])
Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.25, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {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.
\(\displaystyle \int \frac {1}{\sqrt {d x^2+2} \sqrt {f x^2+3} \left (a+b x^2\right )} \, dx\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {\operatorname {EllipticPi}\left (\frac {2 b}{a d},\arcsin \left (\frac {\sqrt {-d} x}{\sqrt {2}}\right ),\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}}\) |
Input:
Int[1/((a + b*x^2)*Sqrt[2 + d*x^2]*Sqrt[3 + f*x^2]),x]
Output:
EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3 ]*a*Sqrt[-d])
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])
Time = 4.69 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.27
method | result | size |
default | \(\frac {\operatorname {EllipticPi}\left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {3 b}{f a}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right ) \sqrt {2}}{2 a \sqrt {-f}}\) | \(53\) |
elliptic | \(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (x^{2} d +2\right )}\, \sqrt {1+\frac {f \,x^{2}}{3}}\, \sqrt {1+\frac {x^{2} d}{2}}\, \operatorname {EllipticPi}\left (\sqrt {-\frac {f}{3}}\, x , \frac {3 b}{f a}, \frac {\sqrt {-\frac {d}{2}}}{\sqrt {-\frac {f}{3}}}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {x^{2} d +2}\, a \sqrt {-\frac {f}{3}}\, \sqrt {d f \,x^{4}+3 x^{2} d +2 f \,x^{2}+6}}\) | \(115\) |
Input:
int(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*EllipticPi(1/3*x*3^(1/2)*(-f)^(1/2),3/f*b/a,1/2*2^(1/2)*(-d)^(1/2)*3^( 1/2)/(-f)^(1/2))*2^(1/2)/a/(-f)^(1/2)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\text {Timed out} \] Input:
integrate(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x, algorithm="fricas ")
Output:
Timed out
\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right ) \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}}\, dx \] Input:
integrate(1/(b*x**2+a)/(d*x**2+2)**(1/2)/(f*x**2+3)**(1/2),x)
Output:
Integral(1/((a + b*x**2)*sqrt(d*x**2 + 2)*sqrt(f*x**2 + 3)), x)
\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}} \,d x } \] Input:
integrate(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x, algorithm="maxima ")
Output:
integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)), x)
\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )} \sqrt {d x^{2} + 2} \sqrt {f x^{2} + 3}} \,d x } \] Input:
integrate(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x, algorithm="giac")
Output:
integrate(1/((b*x^2 + a)*sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)), x)
Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3}} \,d x \] Input:
int(1/((a + b*x^2)*(d*x^2 + 2)^(1/2)*(f*x^2 + 3)^(1/2)),x)
Output:
int(1/((a + b*x^2)*(d*x^2 + 2)^(1/2)*(f*x^2 + 3)^(1/2)), x)
\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx=\int \frac {\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}{b d f \,x^{6}+a d f \,x^{4}+3 b d \,x^{4}+2 b f \,x^{4}+3 a d \,x^{2}+2 a f \,x^{2}+6 b \,x^{2}+6 a}d x \] Input:
int(1/(b*x^2+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x)
Output:
int((sqrt(f*x**2 + 3)*sqrt(d*x**2 + 2))/(a*d*f*x**4 + 3*a*d*x**2 + 2*a*f*x **2 + 6*a + b*d*f*x**6 + 3*b*d*x**4 + 2*b*f*x**4 + 6*b*x**2),x)