Integrand size = 36, antiderivative size = 255 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {31 x}{2 \left (1+2 x^2\right ) \sqrt {2+5 x^2+3 x^4}}-\frac {166 x \left (2+3 x^2\right )}{\sqrt {2+5 x^2+3 x^4}}+\frac {x \left (1003+996 x^2\right )}{2 \sqrt {2+5 x^2+3 x^4}}+\frac {166 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2+5 x^2+3 x^4}}+\frac {231 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}-\frac {746 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
31/2*x/(2*x^2+1)/(3*x^4+5*x^2+2)^(1/2)-166*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/ 2)+1/2*x*(996*x^2+1003)/(3*x^4+5*x^2+2)^(1/2)+166*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)+231/2*2^(1/2)*(x^2+1)*((3*x^2+2)/(x^2+1))^(1/2)*InverseJacobiAM(arc tan(x),1/2*I*2^(1/2))/(3*x^4+5*x^2+2)^(1/2)-746/3*(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 = 10.54 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.00 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {3102 x+9006 x^3+5976 x^5+996 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-339 i \sqrt {3} \sqrt {1+x^2} \left (1+2 x^2\right ) \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )+373 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )+746 i \sqrt {3} x^2 \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{6 \left (1+2 x^2\right ) \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[(4 - 7*x^2 + x^4)/((1 + 2*x^2)^2*(2 + 5*x^2 + 3*x^4)^(3/2)),x]
Output:
(3102*x + 9006*x^3 + 5976*x^5 + (996*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] - (339*I)*Sqrt[3]*S qrt[1 + x^2]*(1 + 2*x^2)*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (373*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticPi[4/3, I*Ar cSinh[Sqrt[3/2]*x], 2/3] + (746*I)*Sqrt[3]*x^2*Sqrt[1 + x^2]*Sqrt[2 + 3*x^ 2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/(6*(1 + 2*x^2)*Sqrt[2 + 5 *x^2 + 3*x^4])
Time = 0.83 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.87, 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.
| \(\displaystyle \int \frac {x^4-7 x^2+4}{\left (2 x^2+1\right )^2 \left (3 x^4+5 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (-\frac {12}{\left (x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {140}{\left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {246}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {31}{\left (2 x^2+1\right )^2 \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {210 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {3 x^4+5 x^2+2}}+\frac {93 \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}-\frac {141 \sqrt {2} \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {31 \sqrt {2} \left (x^2+1\right ) \sqrt {\frac {3 x^2+2}{x^2+1}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {3 x^4+5 x^2+2}}+\frac {135 \sqrt {2} \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}-\frac {62 \sqrt {3} \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}-\frac {560 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}+\frac {62 \sqrt {3 x^4+5 x^2+2} x}{2 x^2+1}+\frac {369 \left (x^2+1\right ) x}{\sqrt {3 x^4+5 x^2+2}}-\frac {154 \left (3 x^2+2\right ) x}{\sqrt {3 x^4+5 x^2+2}}\) |
Input:
Int[(4 - 7*x^2 + x^4)/((1 + 2*x^2)^2*(2 + 5*x^2 + 3*x^4)^(3/2)),x]
Output:
(369*x*(1 + x^2))/Sqrt[2 + 5*x^2 + 3*x^4] - (154*x*(2 + 3*x^2))/Sqrt[2 + 5 *x^2 + 3*x^4] + (62*x*Sqrt[2 + 5*x^2 + 3*x^4])/(1 + 2*x^2) + (135*Sqrt[2]* (2 + 3*x^2)*EllipticE[ArcTan[x], -1/2])/(Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[ 2 + 5*x^2 + 3*x^4]) + (31*Sqrt[2]*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*El lipticE[ArcTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4] - (141*Sqrt[2]*(2 + 3*x^ 2)*EllipticF[ArcTan[x], -1/2])/(Sqrt[(2 + 3*x^2)/(1 + x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) + (93*(1 + x^2)*Sqrt[(2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x] , -1/2])/(Sqrt[2]*Sqrt[2 + 5*x^2 + 3*x^4]) + (210*Sqrt[2]*(1 + x^2)*Sqrt[( 2 + 3*x^2)/(1 + x^2)]*EllipticF[ArcTan[x], -1/2])/Sqrt[2 + 5*x^2 + 3*x^4] - (560*(1 + x^2)*EllipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(Sqrt[3]*Sqrt [(1 + x^2)/(2 + 3*x^2)]*Sqrt[2 + 5*x^2 + 3*x^4]) - (62*Sqrt[3]*(1 + x^2)*E llipticPi[-1/3, ArcTan[Sqrt[3/2]*x], 1/3])/(Sqrt[(1 + x^2)/(2 + 3*x^2)]*Sq rt[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 = 7.06 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(\frac {x \left (996 x^{4}+1501 x^{2}+517\right )}{\left (2 x^{2}+1\right ) \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {657 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{4 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {166 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {373 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(186\) |
| elliptic | \(\frac {62 x \sqrt {3 x^{4}+5 x^{2}+2}}{2 x^{2}+1}-\frac {6 \left (-\frac {135}{2} x^{3}-\frac {131}{2} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {7 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{4 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {166 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {373 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}\) | \(187\) |
| default | \(-\frac {3 \left (-\frac {5}{4} x^{3}-\frac {13}{12} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {9 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {5 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {62 x \sqrt {3 x^{4}+5 x^{2}+2}}{2 x^{2}+1}-\frac {93 \left (-\frac {29}{4} x^{3}-\frac {85}{12} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {1323 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {373 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {66 x^{3}+62 x}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(293\) |
Input:
int((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(3/2),x,method=_RETURNVERBOS E)
Output:
x*(996*x^4+1501*x^2+517)/(2*x^2+1)/(3*x^4+5*x^2+2)^(1/2)+657/4*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))-166* 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)))+373/2*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))
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(3/2),x, algorithm="fr icas")
Output:
integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(36*x^12 + 156*x^10 + 2 77*x^8 + 258*x^6 + 133*x^4 + 36*x^2 + 4), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {x^{4} - 7 x^{2} + 4}{\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}} \left (2 x^{2} + 1\right )^{2}}\, dx \] Input:
integrate((x**4-7*x**2+4)/(2*x**2+1)**2/(3*x**4+5*x**2+2)**(3/2),x)
Output:
Integral((x**4 - 7*x**2 + 4)/(((x**2 + 1)*(3*x**2 + 2))**(3/2)*(2*x**2 + 1 )**2), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(3/2),x, algorithm="ma xima")
Output:
integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(3/2)*(2*x^2 + 1)^2), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 1\right )}^{2}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(3/2),x, algorithm="gi ac")
Output:
integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(3/2)*(2*x^2 + 1)^2), x)
Timed out. \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {x^4-7\,x^2+4}{{\left (2\,x^2+1\right )}^2\,{\left (3\,x^4+5\,x^2+2\right )}^{3/2}} \,d x \] Input:
int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)^2*(5*x^2 + 3*x^4 + 2)^(3/2)),x)
Output:
int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)^2*(5*x^2 + 3*x^4 + 2)^(3/2)), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right )^2 \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=4 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{36 x^{12}+156 x^{10}+277 x^{8}+258 x^{6}+133 x^{4}+36 x^{2}+4}d x \right )+\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{36 x^{12}+156 x^{10}+277 x^{8}+258 x^{6}+133 x^{4}+36 x^{2}+4}d x -7 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{36 x^{12}+156 x^{10}+277 x^{8}+258 x^{6}+133 x^{4}+36 x^{2}+4}d x \right ) \] Input:
int((x^4-7*x^2+4)/(2*x^2+1)^2/(3*x^4+5*x^2+2)^(3/2),x)
Output:
4*int(sqrt(3*x**4 + 5*x**2 + 2)/(36*x**12 + 156*x**10 + 277*x**8 + 258*x** 6 + 133*x**4 + 36*x**2 + 4),x) + int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(36* x**12 + 156*x**10 + 277*x**8 + 258*x**6 + 133*x**4 + 36*x**2 + 4),x) - 7*i nt((sqrt(3*x**4 + 5*x**2 + 2)*x**2)/(36*x**12 + 156*x**10 + 277*x**8 + 258 *x**6 + 133*x**4 + 36*x**2 + 4),x)