Integrand size = 36, antiderivative size = 256 \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=-\frac {547 x \left (2+3 x^2\right )}{144 \sqrt {2+5 x^2+3 x^4}}+\frac {1}{12} x \sqrt {2+5 x^2+3 x^4}+\frac {31 x \sqrt {2+5 x^2+3 x^4}}{8 \left (1+2 x^2\right )}+\frac {547 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{72 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {367 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{24 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {41 \sqrt {3} \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{8 \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
-547/144*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+1/12*x*(3*x^4+5*x^2+2)^(1/2)+31 *x*(3*x^4+5*x^2+2)^(1/2)/(16*x^2+8)+547/144*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^ 2+1))^(1/2)*EllipticE(x/(x^2+1)^(1/2),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2) -367/48*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(arctan(x ),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)+41/8*(x^2+1)*EllipticPi(x*6^(1/2)/( 6*x^2+4)^(1/2),-1/3,1/3*3^(1/2))*3^(1/2)/((x^2+1)/(3*x^2+2))^(1/2)/(3*x^4+ 5*x^2+2)^(1/2)
Result contains complex when optimal does not.
Time = 10.46 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.84 \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\frac {12 x \left (190+483 x^2+305 x^4+12 x^6\right )+1094 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-733 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )-369 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{288 \left (1+2 x^2\right ) \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[((4 - 7*x^2 + x^4)*Sqrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2)^2,x]
Output:
(12*x*(190 + 483*x^2 + 305*x^4 + 12*x^6) + (1094*I)*Sqrt[3]*Sqrt[1 + x^2]* (1 + 2*x^2)*Sqrt[2 + 3*x^2]*EllipticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (733* I)*Sqrt[3]*Sqrt[1 + x^2]*(1 + 2*x^2)*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[S qrt[3/2]*x], 2/3] - (369*I)*Sqrt[3]*Sqrt[1 + x^2]*(1 + 2*x^2)*Sqrt[2 + 3*x ^2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(288*(1 + 2*x^2)*Sqrt[2 + 5*x^2 + 3*x^4])
Time = 0.62 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2258, 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.
| \(\displaystyle \int \frac {\left (x^4-7 x^2+4\right ) \sqrt {3 x^4+5 x^2+2}}{\left (2 x^2+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (\frac {3 x^4}{4 \sqrt {3 x^4+5 x^2+2}}-\frac {19 x^2}{4 \sqrt {3 x^4+5 x^2+2}}+\frac {27}{4 \left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {31}{16 \left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}-\frac {11}{16 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {367 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{24 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {547 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{72 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {41 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{8 \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {31 \sqrt {3 x^4+5 x^2+2} x}{8 \left (2 x^2+1\right )}+\frac {1}{12} \sqrt {3 x^4+5 x^2+2} x-\frac {547 \left (3 x^2+2\right ) x}{144 \sqrt {3 x^4+5 x^2+2}}\) |
Input:
Int[((4 - 7*x^2 + x^4)*Sqrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2)^2,x]
Output:
(-547*x*(2 + 3*x^2))/(144*Sqrt[2 + 5*x^2 + 3*x^4]) + (x*Sqrt[2 + 5*x^2 + 3 *x^4])/12 + (31*x*Sqrt[2 + 5*x^2 + 3*x^4])/(8*(1 + 2*x^2)) + (547*(1 + x^2 )*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/2])/(72*Sqrt[2]*Sqrt [2 + 5*x^2 + 3*x^4]) - (367*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*Elliptic F[ArcTan[x], -1/2])/(24*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) + (41*Sqrt[3]*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(8*Sqrt[(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4])
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ (p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e *x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 7.24 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.70
| method | result | size |
| elliptic | \(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{8 \left (2 x^{2}+1\right )}+\frac {x \sqrt {3 x^{4}+5 x^{2}+2}}{12}-\frac {1105 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{576 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {547 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{144 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {123 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{32 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(179\) |
| risch | \(\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x \left (4 x^{2}+95\right )}{48 x^{2}+24}+\frac {361 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{192 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {547 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{144 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {123 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{32 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(182\) |
| default | \(\frac {x \sqrt {3 x^{4}+5 x^{2}+2}}{12}-\frac {395 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{192 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {5 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{36 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{8 \left (2 x^{2}+1\right )}+\frac {63 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{16 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {123 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{32 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(235\) |
Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^2,x,method=_RETURNVERBOS E)
Output:
31/8*x*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)+1/12*x*(3*x^4+5*x^2+2)^(1/2)-1105/5 76*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticF(I*x,1/2 *6^(1/2))+547/144*I*(x^2+1)^(1/2)*(6*x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*El lipticE(I*x,1/2*6^(1/2))-123/32*I*(x^2+1)^(1/2)*(1+3/2*x^2)^(1/2)/(3*x^4+5 *x^2+2)^(1/2)*EllipticPi(I*x,2,1/2*I*(-3)^(1/2)*2^(1/2))
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^2,x, algorithm="fr icas")
Output:
integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(4*x^4 + 4*x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\int \frac {\sqrt {\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )} \left (x^{4} - 7 x^{2} + 4\right )}{\left (2 x^{2} + 1\right )^{2}}\, dx \] Input:
integrate((x**4-7*x**2+4)*(3*x**4+5*x**2+2)**(1/2)/(2*x**2+1)**2,x)
Output:
Integral(sqrt((x**2 + 1)*(3*x**2 + 2))*(x**4 - 7*x**2 + 4)/(2*x**2 + 1)**2 , x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^2,x, algorithm="ma xima")
Output:
integrate(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^2, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^2,x, algorithm="gi ac")
Output:
integrate(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^2, x)
Timed out. \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\int \frac {\left (x^4-7\,x^2+4\right )\,\sqrt {3\,x^4+5\,x^2+2}}{{\left (2\,x^2+1\right )}^2} \,d x \] Input:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(1/2))/(2*x^2 + 1)^2,x)
Output:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(1/2))/(2*x^2 + 1)^2, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^2} \, dx=\frac {3 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}-50 \sqrt {3 x^{4}+5 x^{2}+2}\, x +488 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+244 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )-186 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}-93 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )+780 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+390 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )}{36 x^{2}+18} \] Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^2,x)
Output:
(3*sqrt(3*x**4 + 5*x**2 + 2)*x**3 - 50*sqrt(3*x**4 + 5*x**2 + 2)*x + 488*i nt(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x )*x**2 + 244*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x) - 186*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(12*x**8 + 32*x **6 + 31*x**4 + 13*x**2 + 2),x)*x**2 - 93*int((sqrt(3*x**4 + 5*x**2 + 2)*x **6)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x) + 780*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2),x)*x**2 + 390*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 + 32*x**6 + 31*x**4 + 1 3*x**2 + 2),x))/(18*(2*x**2 + 1))