Integrand size = 43, antiderivative size = 159 \[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=-\frac {41}{243} x \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}+\frac {1}{9} x^3 \sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}+\frac {2042 \sqrt {2} \sqrt {1-9 x^2} E\left (\arcsin (3 x)\left |-\frac {1}{18}\right .\right )}{729 \sqrt {-1+3 x} \sqrt {1+3 x}}-\frac {2083 \sqrt {2} \sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{729 \sqrt {-1+3 x} \sqrt {1+3 x}} \] Output:
-41/243*x*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)+1/9*x^3*(-1+3*x)^(1/2 )*(1+3*x)^(1/2)*(x^2+2)^(1/2)+2042/729*2^(1/2)*(-9*x^2+1)^(1/2)*EllipticE( 3*x,1/6*I*2^(1/2))/(-1+3*x)^(1/2)/(1+3*x)^(1/2)-2083/729*2^(1/2)*(-9*x^2+1 )^(1/2)*EllipticF(3*x,1/6*I*2^(1/2))/(-1+3*x)^(1/2)/(1+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 33.58 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.86 \[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {\sqrt {\frac {-1+3 x}{1+3 x}} \left (9 x (1+3 x) \left (2+x^2\right ) \left (-41+27 x^2\right )+6126 \left (2+x^2\right ) \left (1+\frac {2}{-1+3 x}\right )-\frac {2042 i (1+3 x) \sqrt {\frac {2+x^2}{(1-3 x)^2}} E\left (\arcsin \left (\frac {\sqrt {-\frac {i}{3}+\sqrt {2}+\frac {19 i}{3-9 x}}}{2^{3/4}}\right )|\frac {12 \sqrt {2}}{17 i+6 \sqrt {2}}\right )}{\sqrt {\frac {i (1+3 x)}{\left (17 i+6 \sqrt {2}\right ) (-1+3 x)}}}+36190 i (-1+3 x) \sqrt {\frac {i (1+3 x)}{\left (17 i+6 \sqrt {2}\right ) (-1+3 x)}} \sqrt {\frac {2+x^2}{(1-3 x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {-\frac {i}{3}+\sqrt {2}+\frac {19 i}{3-9 x}}}{2^{3/4}}\right ),\frac {12 \sqrt {2}}{17 i+6 \sqrt {2}}\right )\right )}{2187 \sqrt {2+x^2}} \] Input:
Integrate[(x^2*(2 + 3*x^2 + 5*x^4))/(Sqrt[-1 + 3*x]*Sqrt[1 + 3*x]*Sqrt[2 + x^2]),x]
Output:
(Sqrt[(-1 + 3*x)/(1 + 3*x)]*(9*x*(1 + 3*x)*(2 + x^2)*(-41 + 27*x^2) + 6126 *(2 + x^2)*(1 + 2/(-1 + 3*x)) - ((2042*I)*(1 + 3*x)*Sqrt[(2 + x^2)/(1 - 3* x)^2]*EllipticE[ArcSin[Sqrt[-1/3*I + Sqrt[2] + (19*I)/(3 - 9*x)]/2^(3/4)], (12*Sqrt[2])/(17*I + 6*Sqrt[2])])/Sqrt[(I*(1 + 3*x))/((17*I + 6*Sqrt[2])* (-1 + 3*x))] + (36190*I)*(-1 + 3*x)*Sqrt[(I*(1 + 3*x))/((17*I + 6*Sqrt[2]) *(-1 + 3*x))]*Sqrt[(2 + x^2)/(1 - 3*x)^2]*EllipticF[ArcSin[Sqrt[-1/3*I + S qrt[2] + (19*I)/(3 - 9*x)]/2^(3/4)], (12*Sqrt[2])/(17*I + 6*Sqrt[2])]))/(2 187*Sqrt[2 + x^2])
Time = 1.24 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2038, 7293, 2009}
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.
| \(\displaystyle \int \frac {x^2 \left (5 x^4+3 x^2+2\right )}{\sqrt {3 x-1} \sqrt {3 x+1} \sqrt {x^2+2}} \, dx\) |
| \(\Big \downarrow \) 2038 |
| \(\displaystyle \frac {\sqrt {9 x^2-1} \int \frac {x^2 \left (5 x^4+3 x^2+2\right )}{\sqrt {x^2+2} \sqrt {9 x^2-1}}dx}{\sqrt {3 x-1} \sqrt {3 x+1}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\sqrt {9 x^2-1} \int \left (\frac {5 x^6}{\sqrt {x^2+2} \sqrt {9 x^2-1}}+\frac {3 x^4}{\sqrt {x^2+2} \sqrt {9 x^2-1}}+\frac {2 x^2}{\sqrt {x^2+2} \sqrt {9 x^2-1}}\right )dx}{\sqrt {3 x-1} \sqrt {3 x+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {9 x^2-1} \left (-\frac {2083 \sqrt {2} \sqrt {1-9 x^2} \operatorname {EllipticF}\left (\arcsin (3 x),-\frac {1}{18}\right )}{729 \sqrt {9 x^2-1}}+\frac {2042 \sqrt {2} \sqrt {1-9 x^2} E\left (\arcsin (3 x)\left |-\frac {1}{18}\right .\right )}{729 \sqrt {9 x^2-1}}-\frac {41}{243} \sqrt {x^2+2} \sqrt {9 x^2-1} x+\frac {1}{9} \sqrt {x^2+2} \sqrt {9 x^2-1} x^3\right )}{\sqrt {3 x-1} \sqrt {3 x+1}}\) |
Input:
Int[(x^2*(2 + 3*x^2 + 5*x^4))/(Sqrt[-1 + 3*x]*Sqrt[1 + 3*x]*Sqrt[2 + x^2]) ,x]
Output:
(Sqrt[-1 + 9*x^2]*((-41*x*Sqrt[2 + x^2]*Sqrt[-1 + 9*x^2])/243 + (x^3*Sqrt[ 2 + x^2]*Sqrt[-1 + 9*x^2])/9 + (2042*Sqrt[2]*Sqrt[1 - 9*x^2]*EllipticE[Arc Sin[3*x], -1/18])/(729*Sqrt[-1 + 9*x^2]) - (2083*Sqrt[2]*Sqrt[1 - 9*x^2]*E llipticF[ArcSin[3*x], -1/18])/(729*Sqrt[-1 + 9*x^2])))/(Sqrt[-1 + 3*x]*Sqr t[1 + 3*x])
Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p _)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_), x_Symbol] :> Simp[(a1 + b1*x^(n/2)) ^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*a2 + b1*b2*x^n)^FracPart[p] ) Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x] && EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && !(Eq Q[n, 2] && IGtQ[q, 0])
Time = 6.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.77
| method | result | size |
| default | \(-\frac {\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}\, \left (-2187 x^{7}+1304 i \sqrt {x^{2}+2}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-2042 i \sqrt {x^{2}+2}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-810 x^{5}+6759 x^{3}-738 x \right )}{2187 \left (9 x^{4}+17 x^{2}-2\right )}\) | \(123\) |
| risch | \(\frac {x \left (27 x^{2}-41\right ) \sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}{243}+\frac {\left (\frac {41 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )}{243 \sqrt {9 x^{4}+17 x^{2}-2}}-\frac {1021 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )\right )}{2187 \sqrt {9 x^{4}+17 x^{2}-2}}\right ) \sqrt {\left (-1+3 x \right ) \left (1+3 x \right ) \left (x^{2}+2\right )}}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\) | \(195\) |
| elliptic | \(\frac {\sqrt {\left (x^{2}+2\right ) \left (9 x^{2}-1\right )}\, \left (-\frac {41 x \sqrt {9 x^{4}+17 x^{2}-2}}{243}+\frac {41 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )}{243 \sqrt {9 x^{4}+17 x^{2}-2}}-\frac {1021 i \sqrt {2}\, \sqrt {2 x^{2}+4}\, \sqrt {-9 x^{2}+1}\, \left (\operatorname {EllipticF}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )-\operatorname {EllipticE}\left (\frac {i x \sqrt {2}}{2}, 3 i \sqrt {2}\right )\right )}{2187 \sqrt {9 x^{4}+17 x^{2}-2}}+\frac {x^{3} \sqrt {9 x^{4}+17 x^{2}-2}}{9}\right )}{\sqrt {-1+3 x}\, \sqrt {1+3 x}\, \sqrt {x^{2}+2}}\) | \(196\) |
Input:
int(x^2*(5*x^4+3*x^2+2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x,metho d=_RETURNVERBOSE)
Output:
-1/2187*(-1+3*x)^(1/2)*(1+3*x)^(1/2)*(x^2+2)^(1/2)*(-2187*x^7+1304*I*(x^2+ 2)^(1/2)*(-9*x^2+1)^(1/2)*EllipticF(1/2*I*x*2^(1/2),3*I*2^(1/2))-2042*I*(x ^2+2)^(1/2)*(-9*x^2+1)^(1/2)*EllipticE(1/2*I*x*2^(1/2),3*I*2^(1/2))-810*x^ 5+6759*x^3-738*x)/(9*x^4+17*x^2-2)
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.40 \[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {9 \, {\left (243 \, x^{4} - 369 \, x^{2} + 2042\right )} \sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1} + 2042 \, x E(\arcsin \left (\frac {1}{3 \, x}\right )\,|\,-18) + 4600 \, x F(\arcsin \left (\frac {1}{3 \, x}\right )\,|\,-18)}{19683 \, x} \] Input:
integrate(x^2*(5*x^4+3*x^2+2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x , algorithm="fricas")
Output:
1/19683*(9*(243*x^4 - 369*x^2 + 2042)*sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1) + 2042*x*elliptic_e(arcsin(1/3/x), -18) + 4600*x*elliptic_f(arcsin(1 /3/x), -18))/x
\[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int \frac {x^{2} \cdot \left (5 x^{4} + 3 x^{2} + 2\right )}{\sqrt {3 x - 1} \sqrt {3 x + 1} \sqrt {x^{2} + 2}}\, dx \] Input:
integrate(x**2*(5*x**4+3*x**2+2)/(-1+3*x)**(1/2)/(1+3*x)**(1/2)/(x**2+2)** (1/2),x)
Output:
Integral(x**2*(5*x**4 + 3*x**2 + 2)/(sqrt(3*x - 1)*sqrt(3*x + 1)*sqrt(x**2 + 2)), x)
\[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {{\left (5 \, x^{4} + 3 \, x^{2} + 2\right )} x^{2}}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1}} \,d x } \] Input:
integrate(x^2*(5*x^4+3*x^2+2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x , algorithm="maxima")
Output:
integrate((5*x^4 + 3*x^2 + 2)*x^2/(sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1)), x)
\[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int { \frac {{\left (5 \, x^{4} + 3 \, x^{2} + 2\right )} x^{2}}{\sqrt {x^{2} + 2} \sqrt {3 \, x + 1} \sqrt {3 \, x - 1}} \,d x } \] Input:
integrate(x^2*(5*x^4+3*x^2+2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x , algorithm="giac")
Output:
integrate((5*x^4 + 3*x^2 + 2)*x^2/(sqrt(x^2 + 2)*sqrt(3*x + 1)*sqrt(3*x - 1)), x)
Timed out. \[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\int \frac {x^2\,\left (5\,x^4+3\,x^2+2\right )}{\sqrt {3\,x-1}\,\sqrt {3\,x+1}\,\sqrt {x^2+2}} \,d x \] Input:
int((x^2*(3*x^2 + 5*x^4 + 2))/((3*x - 1)^(1/2)*(3*x + 1)^(1/2)*(x^2 + 2)^( 1/2)),x)
Output:
int((x^2*(3*x^2 + 5*x^4 + 2))/((3*x - 1)^(1/2)*(3*x + 1)^(1/2)*(x^2 + 2)^( 1/2)), x)
\[ \int \frac {x^2 \left (2+3 x^2+5 x^4\right )}{\sqrt {-1+3 x} \sqrt {1+3 x} \sqrt {2+x^2}} \, dx=\frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x^{3}}{9}-\frac {41 \sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x}{243}+\frac {2042 \left (\int \frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}\, x^{2}}{9 x^{4}+17 x^{2}-2}d x \right )}{243}-\frac {82 \left (\int \frac {\sqrt {3 x +1}\, \sqrt {3 x -1}\, \sqrt {x^{2}+2}}{9 x^{4}+17 x^{2}-2}d x \right )}{243} \] Input:
int(x^2*(5*x^4+3*x^2+2)/(-1+3*x)^(1/2)/(1+3*x)^(1/2)/(x^2+2)^(1/2),x)
Output:
(27*sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2)*x**3 - 41*sqrt(3*x + 1)*sqr t(3*x - 1)*sqrt(x**2 + 2)*x + 2042*int((sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x **2 + 2)*x**2)/(9*x**4 + 17*x**2 - 2),x) - 82*int((sqrt(3*x + 1)*sqrt(3*x - 1)*sqrt(x**2 + 2))/(9*x**4 + 17*x**2 - 2),x))/243