Integrand size = 24, antiderivative size = 70 \[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\frac {x \sqrt {-2+b x^2} \sqrt {2+b x^2}}{\sqrt {3}}-\frac {4 \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {-2+b x^2}}\right ),\frac {1}{2}\right )}{\sqrt {3} \sqrt {b}} \] Output:
1/3*3^(1/2)*x*(b*x^2-2)^(1/2)*(b*x^2+2)^(1/2)-4/3*3^(1/2)*EllipticF(2^(1/2 )*b^(1/2)*x/(b*x^2-2)^(1/2),1/2*2^(1/2))/b^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87 \[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\frac {x \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {b^2 x^4}{4}\right )}{\sqrt {1-\frac {b^2 x^4}{4}}} \] Input:
Integrate[Sqrt[2 + b*x^2]*Sqrt[-6 + 3*b*x^2],x]
Output:
(x*Sqrt[2 + b*x^2]*Sqrt[-6 + 3*b*x^2]*Hypergeometric2F1[-1/2, 1/4, 5/4, (b ^2*x^4)/4])/Sqrt[1 - (b^2*x^4)/4]
Time = 0.19 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {285, 27, 288, 762}
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.
\(\displaystyle \int \sqrt {b x^2+2} \sqrt {3 b x^2-6} \, dx\) |
\(\Big \downarrow \) 285 |
\(\displaystyle \frac {x \sqrt {b x^2-2} \sqrt {b x^2+2}}{\sqrt {3}}-8 \int \frac {1}{\sqrt {3} \sqrt {b x^2-2} \sqrt {b x^2+2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x \sqrt {b x^2-2} \sqrt {b x^2+2}}{\sqrt {3}}-\frac {8 \int \frac {1}{\sqrt {b x^2-2} \sqrt {b x^2+2}}dx}{\sqrt {3}}\) |
\(\Big \downarrow \) 288 |
\(\displaystyle \frac {x \sqrt {b x^2-2} \sqrt {b x^2+2}}{\sqrt {3}}-\frac {8 \sqrt {2-b x^2} \int \frac {1}{\sqrt {4-b^2 x^4}}dx}{\sqrt {3} \sqrt {b x^2-2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {x \sqrt {b x^2-2} \sqrt {b x^2+2}}{\sqrt {3}}-\frac {4 \sqrt {\frac {2}{3}} \sqrt {2-b x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {b} \sqrt {b x^2-2}}\) |
Input:
Int[Sqrt[2 + b*x^2]*Sqrt[-6 + 3*b*x^2],x]
Output:
(x*Sqrt[-2 + b*x^2]*Sqrt[2 + b*x^2])/Sqrt[3] - (4*Sqrt[2/3]*Sqrt[2 - b*x^2 ]*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[2]], -1])/(Sqrt[b]*Sqrt[-2 + b*x^2])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[x*(a + b*x^2)^p*((c + d*x^2)^p/(4*p + 1)), x] + Simp[4*a*c*(p/(4*p + 1)) Int[(a + b*x^2)^(p - 1)*(c + d*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d} , x] && EqQ[b*c + a*d, 0] && GtQ[p, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(c + d*x^2)^FracPart[p]/((-1)^IntPart[p]*(-c - d*x^2)^FracPart[p]) Int[ ((-a)*c - b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && GtQ[a, 0] && LtQ[c, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Result contains complex when optimal does not.
Time = 0.78 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.44
method | result | size |
default | \(-\frac {\sqrt {b \,x^{2}+2}\, \sqrt {3}\, \sqrt {2 b \,x^{2}-4}\, \sqrt {2}\, \left (-b^{2} x^{5} \sqrt {-b}+4 \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {-b}}{2}, i\right ) \sqrt {-2 b \,x^{2}+4}\, \sqrt {b \,x^{2}+2}+4 \sqrt {-b}\, x \right )}{6 \left (b^{2} x^{4}-4\right ) \sqrt {-b}}\) | \(101\) |
elliptic | \(\frac {\sqrt {3 b^{2} x^{4}-12}\, \sqrt {3 b \,x^{2}-6}\, \left (\frac {x \sqrt {3 b^{2} x^{4}-12}}{3}-\frac {4 \sqrt {2 b \,x^{2}+4}\, \sqrt {-2 b \,x^{2}+4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2 b}}{2}, i\right )}{\sqrt {-2 b}\, \sqrt {3 b^{2} x^{4}-12}}\right )}{3 \sqrt {b \,x^{2}+2}\, \left (b \,x^{2}-2\right )}\) | \(109\) |
risch | \(\frac {x \sqrt {b \,x^{2}+2}\, \left (b \,x^{2}-2\right ) \sqrt {\left (b \,x^{2}+2\right ) \left (3 b \,x^{2}-6\right )}\, \sqrt {3}}{3 \sqrt {\left (b \,x^{2}+2\right ) \left (b \,x^{2}-2\right )}\, \sqrt {3 b \,x^{2}-6}}-\frac {4 \sqrt {2 b \,x^{2}+4}\, \sqrt {-2 b \,x^{2}+4}\, \operatorname {EllipticF}\left (\frac {x \sqrt {-2 b}}{2}, i\right ) \sqrt {\left (b \,x^{2}+2\right ) \left (3 b \,x^{2}-6\right )}\, \sqrt {3}}{3 \sqrt {-2 b}\, \sqrt {b^{2} x^{4}-4}\, \sqrt {b \,x^{2}+2}\, \sqrt {3 b \,x^{2}-6}}\) | \(158\) |
Input:
int((b*x^2+2)^(1/2)*(3*b*x^2-6)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/6*(b*x^2+2)^(1/2)*3^(1/2)*(2*b*x^2-4)^(1/2)*2^(1/2)*(-b^2*x^5*(-b)^(1/2 )+4*EllipticF(1/2*x*2^(1/2)*(-b)^(1/2),I)*(-2*b*x^2+4)^(1/2)*(b*x^2+2)^(1/ 2)+4*(-b)^(1/2)*x)/(b^2*x^4-4)/(-b)^(1/2)
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.81 \[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\frac {\sqrt {3 \, b x^{2} - 6} \sqrt {b x^{2} + 2} b x + \frac {4 \, \sqrt {3} \sqrt {2} \sqrt {b^{2}} F(\arcsin \left (\frac {\sqrt {2}}{\sqrt {b} x}\right )\,|\,-1)}{\sqrt {b}}}{3 \, b} \] Input:
integrate((b*x^2+2)^(1/2)*(3*b*x^2-6)^(1/2),x, algorithm="fricas")
Output:
1/3*(sqrt(3*b*x^2 - 6)*sqrt(b*x^2 + 2)*b*x + 4*sqrt(3)*sqrt(2)*sqrt(b^2)*e lliptic_f(arcsin(sqrt(2)/(sqrt(b)*x)), -1)/sqrt(b))/b
\[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\sqrt {3} \int \sqrt {b x^{2} - 2} \sqrt {b x^{2} + 2}\, dx \] Input:
integrate((b*x**2+2)**(1/2)*(3*b*x**2-6)**(1/2),x)
Output:
sqrt(3)*Integral(sqrt(b*x**2 - 2)*sqrt(b*x**2 + 2), x)
\[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\int { \sqrt {3 \, b x^{2} - 6} \sqrt {b x^{2} + 2} \,d x } \] Input:
integrate((b*x^2+2)^(1/2)*(3*b*x^2-6)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(3*b*x^2 - 6)*sqrt(b*x^2 + 2), x)
\[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\int { \sqrt {3 \, b x^{2} - 6} \sqrt {b x^{2} + 2} \,d x } \] Input:
integrate((b*x^2+2)^(1/2)*(3*b*x^2-6)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(3*b*x^2 - 6)*sqrt(b*x^2 + 2), x)
Timed out. \[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\int \sqrt {b\,x^2+2}\,\sqrt {3\,b\,x^2-6} \,d x \] Input:
int((b*x^2 + 2)^(1/2)*(3*b*x^2 - 6)^(1/2),x)
Output:
int((b*x^2 + 2)^(1/2)*(3*b*x^2 - 6)^(1/2), x)
\[ \int \sqrt {2+b x^2} \sqrt {-6+3 b x^2} \, dx=\frac {\sqrt {3}\, \left (\sqrt {b \,x^{2}+2}\, \sqrt {b \,x^{2}-2}\, x -8 \left (\int \frac {\sqrt {b \,x^{2}+2}\, \sqrt {b \,x^{2}-2}}{b^{2} x^{4}-4}d x \right )\right )}{3} \] Input:
int((b*x^2+2)^(1/2)*(3*b*x^2-6)^(1/2),x)
Output:
(sqrt(3)*(sqrt(b*x**2 + 2)*sqrt(b*x**2 - 2)*x - 8*int((sqrt(b*x**2 + 2)*sq rt(b*x**2 - 2))/(b**2*x**4 - 4),x)))/3