Integrand size = 36, antiderivative size = 223 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\frac {53 x \left (2+3 x^2\right )}{\sqrt {2+5 x^2+3 x^4}}-\frac {3 x \left (49+53 x^2\right )}{\sqrt {2+5 x^2+3 x^4}}-\frac {53 \sqrt {2} \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {2+5 x^2+3 x^4}}+\frac {25 \left (1+x^2\right ) \sqrt {\frac {2+3 x^2}{1+x^2}} \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {2+5 x^2+3 x^4}}+\frac {124 \left (1+x^2\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {3} \sqrt {\frac {1+x^2}{2+3 x^2}} \sqrt {2+5 x^2+3 x^4}} \] Output:
53*x*(3*x^2+2)/(3*x^4+5*x^2+2)^(1/2)-3*x*(53*x^2+49)/(3*x^4+5*x^2+2)^(1/2) -53*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)+25/2*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)+124 /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 = 10.33 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.78 \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=-\frac {441 x+477 x^3+159 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} E\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )|\frac {2}{3}\right )-41 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )+31 i \sqrt {3} \sqrt {1+x^2} \sqrt {2+3 x^2} \operatorname {EllipticPi}\left (\frac {4}{3},i \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right ),\frac {2}{3}\right )}{3 \sqrt {2+5 x^2+3 x^4}} \] Input:
Integrate[(4 - 7*x^2 + x^4)/((1 + 2*x^2)*(2 + 5*x^2 + 3*x^4)^(3/2)),x]
Output:
-1/3*(441*x + 477*x^3 + (159*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*Elli pticE[I*ArcSinh[Sqrt[3/2]*x], 2/3] - (41*I)*Sqrt[3]*Sqrt[1 + x^2]*Sqrt[2 + 3*x^2]*EllipticF[I*ArcSinh[Sqrt[3/2]*x], 2/3] + (31*I)*Sqrt[3]*Sqrt[1 + x ^2]*Sqrt[2 + 3*x^2]*EllipticPi[4/3, I*ArcSinh[Sqrt[3/2]*x], 2/3])/Sqrt[2 + 5*x^2 + 3*x^4]
Time = 0.55 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.25, 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.
| \(\displaystyle \int \frac {x^4-7 x^2+4}{\left (2 x^2+1\right ) \left (3 x^4+5 x^2+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2258 |
| \(\displaystyle \int \left (\frac {12}{\left (x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}+\frac {31}{\left (2 x^2+1\right ) \sqrt {3 x^4+5 x^2+2}}-\frac {82}{\left (3 x^2+2\right ) \sqrt {3 x^4+5 x^2+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {93 \sqrt {\frac {3 x^2+2}{x^2+1}} \left (x^2+1\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {2} \sqrt {3 x^4+5 x^2+2}}+\frac {59 \sqrt {2} \left (3 x^2+2\right ) \operatorname {EllipticF}\left (\arctan (x),-\frac {1}{2}\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}-\frac {53 \sqrt {2} \left (3 x^2+2\right ) E\left (\arctan (x)\left |-\frac {1}{2}\right .\right )}{\sqrt {\frac {3 x^2+2}{x^2+1}} \sqrt {3 x^4+5 x^2+2}}+\frac {124 \left (x^2+1\right ) \operatorname {EllipticPi}\left (-\frac {1}{3},\arctan \left (\sqrt {\frac {3}{2}} x\right ),\frac {1}{3}\right )}{\sqrt {3} \sqrt {\frac {x^2+1}{3 x^2+2}} \sqrt {3 x^4+5 x^2+2}}-\frac {123 x \left (x^2+1\right )}{\sqrt {3 x^4+5 x^2+2}}+\frac {41 x \left (3 x^2+2\right )}{\sqrt {3 x^4+5 x^2+2}}\) |
Input:
Int[(4 - 7*x^2 + x^4)/((1 + 2*x^2)*(2 + 5*x^2 + 3*x^4)^(3/2)),x]
Output:
(-123*x*(1 + x^2))/Sqrt[2 + 5*x^2 + 3*x^4] + (41*x*(2 + 3*x^2))/Sqrt[2 + 5 *x^2 + 3*x^4] - (53*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]) + (59*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]) + (124*(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])
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 = 5.03 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.72
| method | result | size |
| elliptic | \(-\frac {6 \left (\frac {53}{2} x^{3}+\frac {49}{2} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {6 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {53 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(161\) |
| risch | \(-\frac {3 x \left (53 x^{2}+49\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {59 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}+\frac {53 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(173\) |
| default | \(\frac {-\frac {225}{8} x^{3}-\frac {195}{8} x}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {131 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}+\frac {83 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \left (\operatorname {EllipticF}\left (i x , \frac {\sqrt {6}}{2}\right )-\operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {3 \left (x^{3}+\frac {5}{6} x \right )}{\sqrt {3 x^{4}+5 x^{2}+2}}-\frac {93 \left (\frac {11}{4} x^{3}+\frac {31}{12} x \right )}{2 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {341 i \sqrt {x^{2}+1}\, \sqrt {6 x^{2}+4}\, \operatorname {EllipticE}\left (i x , \frac {\sqrt {6}}{2}\right )}{8 \sqrt {3 x^{4}+5 x^{2}+2}}-\frac {31 i \sqrt {x^{2}+1}\, \sqrt {1+\frac {3 x^{2}}{2}}\, \operatorname {EllipticPi}\left (i x , 2, \frac {i \sqrt {-3}\, \sqrt {2}}{2}\right )}{\sqrt {3 x^{4}+5 x^{2}+2}}\) | \(265\) |
Input:
int((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-6*(53/2*x^3+49/2*x)/(3*x^4+5*x^2+2)^(1/2)-6*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))-53*I*(x^2+1)^(1/2)*(6* x^2+4)^(1/2)/(3*x^4+5*x^2+2)^(1/2)*EllipticE(I*x,1/2*6^(1/2))-31*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))
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 1\right )}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(3/2),x, algorithm="fric as")
Output:
integral(sqrt(3*x^4 + 5*x^2 + 2)*(x^4 - 7*x^2 + 4)/(18*x^10 + 69*x^8 + 104 *x^6 + 77*x^4 + 28*x^2 + 4), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {x^{4} - 7 x^{2} + 4}{\left (\left (x^{2} + 1\right ) \left (3 x^{2} + 2\right )\right )^{\frac {3}{2}} \cdot \left (2 x^{2} + 1\right )}\, dx \] Input:
integrate((x**4-7*x**2+4)/(2*x**2+1)/(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 )), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 1\right )}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(3/2),x, algorithm="maxi ma")
Output:
integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(3/2)*(2*x^2 + 1)), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int { \frac {x^{4} - 7 \, x^{2} + 4}{{\left (3 \, x^{4} + 5 \, x^{2} + 2\right )}^{\frac {3}{2}} {\left (2 \, x^{2} + 1\right )}} \,d x } \] Input:
integrate((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(3/2),x, algorithm="giac ")
Output:
integrate((x^4 - 7*x^2 + 4)/((3*x^4 + 5*x^2 + 2)^(3/2)*(2*x^2 + 1)), x)
Timed out. \[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=\int \frac {x^4-7\,x^2+4}{\left (2\,x^2+1\right )\,{\left (3\,x^4+5\,x^2+2\right )}^{3/2}} \,d x \] Input:
int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)*(5*x^2 + 3*x^4 + 2)^(3/2)),x)
Output:
int((x^4 - 7*x^2 + 4)/((2*x^2 + 1)*(5*x^2 + 3*x^4 + 2)^(3/2)), x)
\[ \int \frac {4-7 x^2+x^4}{\left (1+2 x^2\right ) \left (2+5 x^2+3 x^4\right )^{3/2}} \, dx=4 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}}{18 x^{10}+69 x^{8}+104 x^{6}+77 x^{4}+28 x^{2}+4}d x \right )+\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{4}}{18 x^{10}+69 x^{8}+104 x^{6}+77 x^{4}+28 x^{2}+4}d x -7 \left (\int \frac {\sqrt {3 x^{4}+5 x^{2}+2}\, x^{2}}{18 x^{10}+69 x^{8}+104 x^{6}+77 x^{4}+28 x^{2}+4}d x \right ) \] Input:
int((x^4-7*x^2+4)/(2*x^2+1)/(3*x^4+5*x^2+2)^(3/2),x)
Output:
4*int(sqrt(3*x**4 + 5*x**2 + 2)/(18*x**10 + 69*x**8 + 104*x**6 + 77*x**4 + 28*x**2 + 4),x) + int((sqrt(3*x**4 + 5*x**2 + 2)*x**4)/(18*x**10 + 69*x** 8 + 104*x**6 + 77*x**4 + 28*x**2 + 4),x) - 7*int((sqrt(3*x**4 + 5*x**2 + 2 )*x**2)/(18*x**10 + 69*x**8 + 104*x**6 + 77*x**4 + 28*x**2 + 4),x)