Integrand size = 32, antiderivative size = 23 \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {9}{5},\arcsin \left (\sqrt {\frac {5}{3}} x\right ),-\frac {3}{5}\right )}{\sqrt {10}} \] Output:
1/10*EllipticPi(1/3*15^(1/2)*x,-9/5,1/5*I*15^(1/2))*10^(1/2)
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {9}{5},\arcsin \left (\sqrt {\frac {5}{3}} x\right ),-\frac {3}{5}\right )}{\sqrt {10}} \] Input:
Integrate[1/(Sqrt[3 - 5*x^2]*Sqrt[2 + 2*x^2]*(1 + 3*x^2)),x]
Output:
EllipticPi[-9/5, ArcSin[Sqrt[5/3]*x], -3/5]/Sqrt[10]
Time = 0.15 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, 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.
\(\displaystyle \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2 x^2+2} \left (3 x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 412 |
\(\displaystyle \frac {\operatorname {EllipticPi}\left (-\frac {9}{5},\arcsin \left (\sqrt {\frac {5}{3}} x\right ),-\frac {3}{5}\right )}{\sqrt {10}}\) |
Input:
Int[1/(Sqrt[3 - 5*x^2]*Sqrt[2 + 2*x^2]*(1 + 3*x^2)),x]
Output:
EllipticPi[-9/5, ArcSin[Sqrt[5/3]*x], -3/5]/Sqrt[10]
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 = 4.52 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {\operatorname {EllipticPi}\left (\frac {\sqrt {5}\, \sqrt {3}\, x}{3}, -\frac {9}{5}, \frac {i \sqrt {5}\, \sqrt {3}}{5}\right ) \sqrt {5}\, \sqrt {2}}{10}\) | \(29\) |
elliptic | \(\frac {\sqrt {-\left (5 x^{2}-3\right ) \left (x^{2}+1\right )}\, \sqrt {3}\, \sqrt {5}\, \sqrt {1-\frac {5 x^{2}}{3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {5}\, \sqrt {3}\, x}{3}, -\frac {9}{5}, \frac {i \sqrt {5}\, \sqrt {3}}{5}\right )}{5 \sqrt {-5 x^{2}+3}\, \sqrt {-10 x^{4}-4 x^{2}+6}}\) | \(77\) |
Input:
int(1/(-5*x^2+3)^(1/2)/(2*x^2+2)^(1/2)/(3*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/10*EllipticPi(1/3*5^(1/2)*3^(1/2)*x,-9/5,1/5*I*5^(1/2)*3^(1/2))*5^(1/2)* 2^(1/2)
\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )} \sqrt {2 \, x^{2} + 2} \sqrt {-5 \, x^{2} + 3}} \,d x } \] Input:
integrate(1/(-5*x^2+3)^(1/2)/(2*x^2+2)^(1/2)/(3*x^2+1),x, algorithm="frica s")
Output:
integral(-1/2*sqrt(2*x^2 + 2)*sqrt(-5*x^2 + 3)/(15*x^6 + 11*x^4 - 7*x^2 - 3), x)
\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\frac {\sqrt {2} \int \frac {1}{3 x^{2} \sqrt {3 - 5 x^{2}} \sqrt {x^{2} + 1} + \sqrt {3 - 5 x^{2}} \sqrt {x^{2} + 1}}\, dx}{2} \] Input:
integrate(1/(-5*x**2+3)**(1/2)/(2*x**2+2)**(1/2)/(3*x**2+1),x)
Output:
sqrt(2)*Integral(1/(3*x**2*sqrt(3 - 5*x**2)*sqrt(x**2 + 1) + sqrt(3 - 5*x* *2)*sqrt(x**2 + 1)), x)/2
\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )} \sqrt {2 \, x^{2} + 2} \sqrt {-5 \, x^{2} + 3}} \,d x } \] Input:
integrate(1/(-5*x^2+3)^(1/2)/(2*x^2+2)^(1/2)/(3*x^2+1),x, algorithm="maxim a")
Output:
integrate(1/((3*x^2 + 1)*sqrt(2*x^2 + 2)*sqrt(-5*x^2 + 3)), x)
\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\int { \frac {1}{{\left (3 \, x^{2} + 1\right )} \sqrt {2 \, x^{2} + 2} \sqrt {-5 \, x^{2} + 3}} \,d x } \] Input:
integrate(1/(-5*x^2+3)^(1/2)/(2*x^2+2)^(1/2)/(3*x^2+1),x, algorithm="giac" )
Output:
integrate(1/((3*x^2 + 1)*sqrt(2*x^2 + 2)*sqrt(-5*x^2 + 3)), x)
Timed out. \[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=\int \frac {1}{\sqrt {2\,x^2+2}\,\left (3\,x^2+1\right )\,\sqrt {3-5\,x^2}} \,d x \] Input:
int(1/((2*x^2 + 2)^(1/2)*(3*x^2 + 1)*(3 - 5*x^2)^(1/2)),x)
Output:
int(1/((2*x^2 + 2)^(1/2)*(3*x^2 + 1)*(3 - 5*x^2)^(1/2)), x)
\[ \int \frac {1}{\sqrt {3-5 x^2} \sqrt {2+2 x^2} \left (1+3 x^2\right )} \, dx=-\frac {\sqrt {2}\, \left (\int \frac {\sqrt {-5 x^{2}+3}\, \sqrt {x^{2}+1}}{15 x^{6}+11 x^{4}-7 x^{2}-3}d x \right )}{2} \] Input:
int(1/(-5*x^2+3)^(1/2)/(2*x^2+2)^(1/2)/(3*x^2+1),x)
Output:
( - sqrt(2)*int((sqrt( - 5*x**2 + 3)*sqrt(x**2 + 1))/(15*x**6 + 11*x**4 - 7*x**2 - 3),x))/2