Integrand size = 36, antiderivative size = 265 \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=-\frac {145 x \left (2+3 x^2\right )}{64 \sqrt {2+5 x^2+3 x^4}}+\frac {31 x \sqrt {2+5 x^2+3 x^4}}{16 \left (1+2 x^2\right )^2}+\frac {153 x \sqrt {2+5 x^2+3 x^4}}{32 \left (1+2 x^2\right )}+\frac {145 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{32 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {79 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{32 \sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {95 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{32 \sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
-145/64*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)+31/16*x*(3*x^4+5*x^2+2)^(1/2)/(2 *x^2+1)^2+153*x*(3*x^4+5*x^2+2)^(1/2)/(64*x^2+32)+145/64*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)-79/64*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacob iAM(arctan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)-95/96*(x^2+1)*EllipticP i(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 = 10.53 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.80 \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\frac {\frac {744 x \left (2+5 x^2+3 x^4\right )}{\left (1+2 x^2\right )^2}+\frac {1836 x \left (2+5 x^2+3 x^4\right )}{1+2 x^2}+870 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-269 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )+95 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{384 \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[((4 - 7*x^2 + x^4)*Sqrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2)^3,x]
Output:
((744*x*(2 + 5*x^2 + 3*x^4))/(1 + 2*x^2)^2 + (1836*x*(2 + 5*x^2 + 3*x^4))/ (1 + 2*x^2) + (870*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticE[I*Ar cSinh[Sqrt[3/2]*x], 2/3] - (269*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*E llipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (95*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[ 2 + 3*x^2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(384*Sqrt[2 + 5*x ^2 + 3*x^4])
Time = 0.85 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.24, 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.
| \(\displaystyle \int \frac {\left (x^4-7 x^2+4\right ) \sqrt {3 x^4+5 x^2+2}}{\left (2 x^2+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (\frac {3 x^2}{8 \sqrt {3 x^4+5 x^2+2}}+\frac {15}{8 \left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {27}{4 \left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}+\frac {31}{16 \left (2 x^2+1\right )^3 \sqrt {3 x^4+5 x^2+2}}-\frac {41}{16 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {79 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{32 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {145 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{32 \sqrt {2} \sqrt {3 x^4+5 x^2+2}}-\frac {11 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {961 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{32 \sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {153 \sqrt {3 x^4+5 x^2+2} x}{32 \left (2 x^2+1\right )}+\frac {31 \sqrt {3 x^4+5 x^2+2} x}{16 \left (2 x^2+1\right )^2}-\frac {145 \left (3 x^2+2\right ) x}{64 \sqrt {3 x^4+5 x^2+2}}\) |
Input:
Int[((4 - 7*x^2 + x^4)*Sqrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2)^3,x]
Output:
(-145*x*(2 + 3*x^2))/(64*Sqrt[2 + 5*x^2 + 3*x^4]) + (31*x*Sqrt[2 + 5*x^2 + 3*x^4])/(16*(1 + 2*x^2)^2) + (153*x*Sqrt[2 + 5*x^2 + 3*x^4])/(32*(1 + 2*x ^2)) + (145*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticE[ArcTan[x], -1/ 2])/(32*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) - (79*(1 + x^2)*Sqrt[(2 + 3*x^2)/ (1 + x^2)]*EllipticF[ArcTan[x], -1/2])/(32*Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4] ) + (961*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(32*Sqrt[3] *Sqrt[(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (11*Sqrt[3]*(1 + x ^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(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 = 10.58 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x \left (306 x^{2}+215\right )}{32 \left (2 x^{2}+1\right )^{2}}+\frac {601 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{256 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {145 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{64 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {95 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{128 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(182\) |
| default | \(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{16 \left (2 x^{2}+1\right )^{2}}+\frac {153 x \sqrt {3 x^{4}+5 x^{2}+2}}{32 \left (2 x^{2}+1\right )}+\frac {21 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{256 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {145 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{64 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {95 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{128 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(188\) |
| elliptic | \(\frac {31 x \sqrt {3 x^{4}+5 x^{2}+2}}{16 \left (2 x^{2}+1\right )^{2}}+\frac {153 x \sqrt {3 x^{4}+5 x^{2}+2}}{32 \left (2 x^{2}+1\right )}+\frac {21 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{256 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {145 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{64 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {95 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{128 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(188\) |
Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^3,x,method=_RETURNVERBOS E)
Output:
1/32*(3*x^4+5*x^2+2)^(1/2)*x*(306*x^2+215)/(2*x^2+1)^2+601/256*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))-145/ 64*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))-EllipticE(I*x,1/2*6^(1/2)))+95/128*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))
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^3,x, algorithm="fr icas")
Output:
integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(8*x^6 + 12*x^4 + 6*x^2 + 1), x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\int \frac {\sqrt {\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )} \left (x^{4} - 7 x^{2} + 4\right )}{\left (2 x^{2} + 1\right )^{3}}\, dx \] Input:
integrate((x**4-7*x**2+4)*(3*x**4+5*x**2+2)**(1/2)/(2*x**2+1)**3,x)
Output:
Integral(sqrt((x**2 + 1)*(3*x**2 + 2))*(x**4 - 7*x**2 + 4)/(2*x**2 + 1)**3 , x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^3,x, algorithm="ma xima")
Output:
integrate(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^3, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\int { \frac {\sqrt {3 \, x^{4} + 5 \, x^{2} + 2} {\left (x^{4} - 7 \, x^{2} + 4\right )}}{{\left (2 \, x^{2} + 1\right )}^{3}} \,d x } \] Input:
integrate((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^3,x, algorithm="gi ac")
Output:
integrate(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(2*x^2 + 1)^3, x)
Timed out. \[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\int \frac {\left (x^4-7\,x^2+4\right )\,\sqrt {3\,x^4+5\,x^2+2}}{{\left (2\,x^2+1\right )}^3} \,d x \] Input:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(1/2))/(2*x^2 + 1)^3,x)
Output:
int(((x^4 - 7*x^2 + 4)*(5*x^2 + 3*x^4 + 2)^(1/2))/(2*x^2 + 1)^3, x)
\[ \int \frac {\left (4-7 x^2+x^4\right ) \sqrt {2+5 x^2+3 x^4}}{\left (1+2 x^2\right )^3} \, dx=\frac {-149 \sqrt {3 x^{4}+5 x^{2}+2}\, x^{3}-190 \sqrt {3 x^{4}+5 x^{2}+2}\, x +720 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{4}+720 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{2}+180 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right )+2912 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{4}+2912 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+728 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )-2028 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{4}-2028 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right ) x^{2}-507 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{24 x^{10}+76 x^{8}+94 x^{6}+57 x^{4}+17 x^{2}+2}d x \right )+2184 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{4}+2184 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}+546 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{6}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )-2184 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{4}-2184 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right ) x^{2}-546 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{12 x^{8}+32 x^{6}+31 x^{4}+13 x^{2}+2}d x \right )}{264 x^{4}+264 x^{2}+66} \] Input:
int((x^4-7*x^2+4)*(3*x^4+5*x^2+2)^(1/2)/(2*x^2+1)^3,x)
Output:
( - 149*sqrt(3*x**4 + 5*x**2 + 2)*x**3 - 190*sqrt(3*x**4 + 5*x**2 + 2)*x + 720*int(sqrt(3*x**4 + 5*x**2 + 2)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x)*x**4 + 720*int(sqrt(3*x**4 + 5*x**2 + 2)/(24*x**10 + 76 *x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x)*x**2 + 180*int(sqrt(3*x**4 + 5 *x**2 + 2)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x) + 291 2*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x**4 + 13*x**2 + 2 ),x)*x**4 + 2912*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32*x**6 + 31*x** 4 + 13*x**2 + 2),x)*x**2 + 728*int(sqrt(3*x**4 + 5*x**2 + 2)/(12*x**8 + 32 *x**6 + 31*x**4 + 13*x**2 + 2),x) - 2028*int((sqrt(3*x**4 + 5*x**2 + 2)*x* *6)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x)*x**4 - 2028* int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(24*x**10 + 76*x**8 + 94*x**6 + 57*x* *4 + 17*x**2 + 2),x)*x**2 - 507*int((sqrt(3*x**4 + 5*x**2 + 2)*x**6)/(24*x **10 + 76*x**8 + 94*x**6 + 57*x**4 + 17*x**2 + 2),x) + 2184*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**4 + 2184*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 + 546*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) - 2184*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**4 - 2184*int((s qrt(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 - 546*int((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(12*x**8 + 32*x**6 ...